diff --git "a/JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt" "b/JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt" @@ -0,0 +1,2414 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf,len=2413 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='03137v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='NT] 9 Jan 2023 Gaps on the intersection numbers of sections on a rational elliptic surface Renato Dias Costa Abstract Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If not, we say that k a gap number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We try to answer when gap numbers exist, how they are distributed and how to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 The Mordell-Weil Lattice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 The case ∆ = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 15 5 Main Results 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 No gap numbers in rank r ≥ 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Gaps with probability 1 in rank r ≤ 2 .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 Surfaces with a 1-gap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20 6 Appendix 23 1 1 Introduction Description of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface over an algebraically closed field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' a smooth, rational projective surface with a fibration π : X → P1 whose general fiber is a smooth curve of genus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume also that π is relatively minimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' no fiber contains an exceptional curve in its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use E/K to denote the generic fiber of π, which is an elliptic curve over the function field K := k(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the Mordell-Weil theorem, the set E(K) of K-points is a finitely generated Abelian group, whose rank we denote by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The points on E(K) are in bijective correspondence with the sections of π, as well as with the exceptional curves on X, so we use these terms interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This paper addresses the following question: given sections P1, P2 ∈ E(K), what values can the intersection number P1 · P2 possibly attain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Original motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The problem originates from a previous investigation of conic bundles on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' morphisms ϕ : X → P1 whose general fiber is a smooth curve of genus zero [Cos].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More specifically, one of the ways to produce a conic bundle is by finding a pair of sections P1, P2 ∈ E(K) with P1 · P2 = 1, so that the linear system |P1 + P2| induces a conic bundle ϕ|P1+P2| : X → P1 having P1 + P2 as a reducible fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We may ask under which conditions such a pair exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' An immediate necessary condition is that r ≥ 1, for if r = 0 any two distinct sections must be disjoint [SS19, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, given that r ≥ 1, does X admit such a pair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The first observation is that r ≥ 1 implies an infinite number of sections, so we should expect infinitely many values for P1·P2 as P1, P2 run through E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then the question is ultimately: what values may P1·P2 assume?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Mordell-Weil lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The computation of intersection numbers on a surface is a difficult prob- lem in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' However, as we are concerned with sections on an elliptic surface, the information we need is considerably more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The reason for this lies in the Mordell-Weil lattice, a concept first established in [Elk90], [Shi89], [Shi90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' It involves the definition of a Q-valued pair- ing ⟨·, ·⟩ on E(K), called the height pairing [SS19, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5], inducing a positive-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩), named the Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' A key aspect of its construction is the connection with the Néron-Severi lattice, so that the height pairing and the intersection pairing of sections are strongly intertwined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the case of rational elliptic surfaces, the possibilities for the Mordell-Weil lattice have already been classified in [OS91], which gives us a good starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Representation of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The use of Mordell-Weil lattices in our investigation leads to a classical problem in number theory, which is the representation of integers by positive-definite quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, the free part of E(K) is generated by r terms, so the height h(P) := ⟨P, P⟩ induces a positive-definite quadratic form on r variables with coefficients in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If O ∈ E(K) is the neutral section and R is the set of reducible fibers of π, then by the height formula (2) h(P) = 2 + 2(P · O) − � v∈R contrv(P), where the sum over v is a rational number which can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By clearing denominators, we see that the possible values of P · O depend on a certain range of integers represented by a positive-definite quadratic form with coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This point of view is explored in some parts of the paper, where we apply results such as the classical Lagrange’s four-square theorem [HW79, §20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5], the counting of integers represented by a binary quadratic form [Ber12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 91] and the more recent Bhargava-Hanke’s 290-theorem on universal quadratic forms [BH, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2 Statement of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Given k ∈ Z≥0 we investigate whether there is a pair of sections P1, P2 ∈ E(K) such that P1 · P2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If such a pair does not exist, we say that X has a k-gap, or that k is a gap number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Our first result is a complete identification of gap numbers in some cases: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If E(K) is torsion-free with rank r = 1, we have the following characterization of gap numbers on X according to the lattice T associated to the reducible fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T k is a gap number ⇔ none of the following are perfect squares E7 k + 1, 4k + 1 A7 k+1 4 , 16k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 16k + 9 D7 k+1 2 , 8k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 8k + 4 A6 ⊕ A1 k+1 7 , 28k − 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 28k + 21 E6 ⊕ A1 k+1 3 , 12k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 12k + 9 D5 ⊕ A2 k+1 6 , 24k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 24k + 16 A4 ⊕ A3 k+1 10 , 40k − 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 40k + 25 A4 ⊕ A2 ⊕ A1 k+1 15 , 60k − 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 60k + 45 We also explore the possibility of X having no gap numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that, in fact, this is always the case if the Mordell-Weil rank is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r ≥ 5, then X has no gap numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' On the other hand, for r ≤ 2 we show that gap numbers occur with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r ≤ 2, then the set of gap numbers of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' G := {k ∈ N | k is a gap number of X} has density 1 in N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' lim n→∞ #G ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', n} n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' At last we answer the question from the original motivation, which consists in classifying the rational elliptic surfaces with a 1-gap: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' X has a 1-gap if and only if r = 0 or r = 1 and π has a III∗ fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 3 Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The text is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Section 2 introduces the main objects, namely the Mordell-Weil lattice, the bounds cmax, cmin for the contribution term, the difference ∆ = cmax −cmin and the quadratic form QX induced by the height pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In Section 3 we explain the role of torsion sections in the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The key technical results are gathered in Section 4, where we state necessary and sufficient conditions for having P1 · P2 = k for a given k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Section 5 contains the main results of the paper, namely: the description of gap numbers when E(K) is torsion-free with r = 1 (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3), the absence of gap numbers for r ≥ 5 (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1), density of gap numbers when r ≤ 2 (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2) and the classification of surfaces with a 1-gap (Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Section 6 is an appendix containing Table 8, which stores the relevant information about the Mordell-Weil lattices of rational elliptic surfaces with r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2 Preliminaries Throughout the paper X denotes a rational elliptic surface over an algebraically closed field k of any characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More precisely, X is a smooth rational projective surface with a fibration π : X → P1, with a section, whose general fiber is a smooth curve of genus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We assume moreover that π is relatively minimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' each fiber has no exceptional curve in its support) [SS19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The generic fiber of π is an elliptic curve E/K over K := k(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The set E(K) of K-points is called the Mordell-Weil group of X, whose rank is called the Mordell-Weil rank of X, denoted by r := rank E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In what follows we introduce the main objects of our investigation and stablish some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 The Mordell-Weil Lattice We give a brief description of the Mordell-Weil lattice, which is the central tool used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Although it can be defined on elliptic surfaces in general, we restrict ourselves to rational elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For more information on Mordell-Weil lattices, we refer the reader to the com- prehensive introduction by Schuett and Shioda [SS19] in addition to the original sources, namely [Elk90], [Shi89], [Shi90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We begin by noting that points in E(K) can be regarded as curves on X and by defining the lattice T and the trivial lattice Triv(X), which are needed to define the Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Sections, points on E(K) and exceptional curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The sections of π are in bijective cor- respondence with points on E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, since X is rational and relatively minimal, points on E(K) also correspond to exceptional curves on X [SS10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For this reason we identify sections of π, points on E(K) and exceptional curves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The lattice T and the trivial lattice Triv(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let O ∈ E(K) be the neutral section and R := {v ∈ P1 | π−1(v) is reducible} the set of reducible fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The components of a fiber π−1(v) are denoted by Θv,i, where Θv,0 is the only component intersected by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The Néron-Severi group NS(X) together with the intersection pairing is called the Néron-Severi lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4 We define the following sublattices of NS(X), which encode the reducible fibers of π: Tv := Z⟨Θv,i | i ̸= 0⟩ for v ∈ R, T := � v∈R Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Kodaira’s classification [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12], each Tv with v ∈ R is represented by a Dynkin diagram Am, Dm or Em for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We also define the trivial lattice of X, namely Triv(X) := Z⟨O, Θv,i | i ≥ 0, v ∈ R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Next we define the Mordell-Weil lattice and present the height formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In order to give E(K) a lattice structure, we cannot use the inter- section pairing directly, which only defines a lattice on NS(X) but not on E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is achieved by defining a Q-valued pairing, called the height pairing, given by ⟨·, ·⟩ : E(K) × E(K) → Q P, Q �→ −ϕ(P) · ϕ(Q), where ϕ : E(K) → NS(X) ⊗Z Q is defined from the orthogonal projection with respect to Triv(X) (for a detailed exposition, see [SS19, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, dividing by torsion elements we get a positive-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩) [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='20], called the Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The height formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The height pairing can be explicitly computed by the height formula [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For rational elliptic surfaces, it is given by ⟨P, Q⟩ = 1 + (P · O) + (Q · O) − (P · Q) − � v∈R contrv(P, Q), (1) h(P) := ⟨P, P⟩ = 2 + 2(P · O) − � v∈R contrv(P), (2) where contrv(P) := contrv(P, P) and contrv(P, Q) are given by Table 1 [SS19, Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1] assuming P, Q meet π−1(v) at Θv,i, Θv,j resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' with 0 < i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P or Q meets Θv,0, then contrv(P, Q) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The minimal norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since E(K) is finitely generated, there is a minimal positive value for h(P) as P runs through E(K) with h(P) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' It is called the minimal norm, denoted by µ := min{h(P) > 0 | P ∈ E(K)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The narrow Mordell-Weil lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' An important sublattice of E(K) is the narrow Mordell-Weil lattice E(K)0, defined as E(K)0 := {P ∈ E(K) | P intersects Θv,0 for all v ∈ R} = {P ∈ E(K) | contrv(P) = 0 for all v ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As a subgroup, E(K)0 is torsion-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' as a sublattice, it is a positive-definite even integral lattice with finite index in E(K) [SS19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The importance of the narrow lattice can be explained by its considerable size as a sublattice and by the easiness to compute the height pairing on it, since all contribution terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' A complete classification of the lattices E(K) and E(K)0 on rational elliptic surfaces is found in [OS91, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5 Tv A1 E7 A2 E6 An−1 Dn+4 Type of π−1(v) III III∗ IV IV∗ In I∗ n contrv(P) 1 2 3 2 2 3 4 3 i(n−i) n � 1 (i = 1) 1 + n 4 (i > 1) contrv(P, Q) 1 3 2 3 i(n−j) n � 1 2 (i = 1) 1 2 + n 4 (i > 1) Table 1: Local contributions from reducible fibers to the height pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Gap numbers We introduce some convenient terminology to express the possibility of finding a pair of sections with a given intersection number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there are no sections P1, P2 ∈ E(K) such that P1 · P2 = k, we say that X has a k-gap or that k is a gap number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We say that X is gap-free if for every k ∈ Z≥0 there are sections P1, P2 ∈ E(K) such that P1 · P2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case the Mordell-Weil rank is r = 0, we have E(K) = E(K)tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, any two distinct sections are disjoint [SS19, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='30], hence every k ≥ 1 is a gap number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For positive rank, the description of gap numbers is less trivial, thus our focus on r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Bounds cmax, cmin for the contribution term We define the estimates cmax, cmin for the contribution term � v contrv(P) and state some simple facts about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We also provide an example to illustrate how they are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The need for these estimates comes from the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Suppose we are given a section P ∈ E(K) whose height h(P) is known and we want to determine P · O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case P ∈ E(K)0 we have a direct answer, namely P · O = h(P)/2 − 1 by the height formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' However if P /∈ E(K)0, the computation of P · O depends on the contribution term cP := � v∈R contrv(P), which by Table 1 depends on how P intersects the reducible fibers of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Usually we do not have this intersection data at hand, which is why we need estimates for cP not depending on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If the set R of reducible fibers of π is not empty, we define cmax := � v∈R max{contrv(P) | P ∈ E(K)}, cmin := min {contrv(P) > 0 | P ∈ E(K), v ∈ R} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The case R = ∅ only occurs when X has Mordell-Weil rank r = 8 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1 in Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this case E(K)0 = E(K) and � v∈R contrv(P) = 0 ∀P ∈ E(K), hence we adopt the convention cmax = cmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use cmax, cmin as bounds for cP := � v contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For our purposes it is not necessary to know whether cP actually attains one of these bounds for some P, so that cmax, cmin should be understood as hypothetical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We state some facts about cmax, cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If π admits a reducible fiber, then: i) cmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) cmax < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' iii) cmin ≤ � v∈R contrv(P) ≤ cmax ∀P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For P ∈ E(K)0, only the second inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' iv) If � v∈R contrv(P) = cmin, then contrv′(P) = cmin for some v′ and contrv(P) = 0 for v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Item i) is immediate from the definition of cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For ii) it is enough to check the values of cmax directly in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For iii), the second inequality follows from the definition of cmax and clearly holds for any P ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P /∈ E(K)0, then cP := � v contrv(P) > 0, so contrv0(P) > 0 for some v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore cP ≥ contrv0(P) ≥ cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For iv), let � v contrv(P) = cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume by contradiction that there are distinct v1, v2 such that contrvi(P) > 0 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By definition of cmin we have cmin ≤ contrvi(P) for i = 1, 2 so cmin = � v contrv(P) ≥ contrv1(P) + contrv2(P) ≥ 2cmin, which is absurd because cmin > 0 by i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore there is only one v′ with contrv′(P) > 0, while contrv(P) = 0 for all v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, contrv′(P) = cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Explicit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Once we know the lattice T associated with the reducible fibers of π (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1), the computation of cmax, cmin is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For a fixed v ∈ R, the extreme values of the local contribution contrv(P) are given in Table 2, which is derived from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We provide an example to illustrate this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Tv max{contrv(P) | P ∈ E(K)} min{contrv(P) > 0 | P ∈ E(K)} An−1 ℓ(n−ℓ) n , where ℓ := �n 2 � n−1 n Dn+4 1 + n 4 1 E6 4 3 4 3 E7 3 2 3 2 Table 2: Extreme values of contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 7 Example: Let π with fiber configuration (I4, IV, III, I1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The reducible fibers are I4, IV, III, so T = A3 ⊕ A2 ⊕ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Table 2, the maximal contributions for A3, A2, A1 are 2·2 4 = 1, 2 3, 1 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The minimal positive contributions are 1·3 4 = 3 4, 2 3, 1 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then cmax = 1 + 2 3 + 1 2 = 13 6 , cmin = min �3 4, 2 3, 1 2 � = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 The difference ∆ = cmax − cmin In this section we explain why the value of ∆ := cmax − cmin is relevant to our discussion, specially in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We also verify that ∆ < 2 in most cases and identify the exceptional ones in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As noted in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, in case P /∈ E(K)0 and h(P) is known, the difficulty of determining P ·O lies in the contribution term cP := � v∈R contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, the range of possible values for cP determines the possibilities for P · O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This range is measured by the difference ∆ := cmax − cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence a smaller ∆ means a better control over the intersection number P · O, which is why ∆ plays an important role in determining possible intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 we assume ∆ ≤ 2 and state necessary and sufficient conditions for having a pair P1, P2 such that P1 · P2 = k for a given k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If however ∆ > 2, the existence of such a pair is not guaranteed a priori, so a case-by-case treatment is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Fortunately by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 the case ∆ > 2 is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The only cases with ∆ = 2 and ∆ > 2 are in Table 3 and 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular we have ∆ < 2 whenever E(K) is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T E(K) cmax cmin 24 A⊕5 1 A∗ 1 ⊕3 ⊕ Z/2Z 5 2 1 2 38 A3 ⊕ A⊕3 1 A∗ 1 ⊕ ⟨1/4⟩ ⊕ Z/2Z 5 2 1 2 53 A5 ⊕ A⊕2 1 ⟨1/6⟩ ⊕ Z/2Z 5 2 1 2 57 D4 ⊕ A⊕3 1 A∗ 1 ⊕ (Z/2Z)⊕2 5 2 1 2 58 A⊕2 3 ⊕ A1 A∗ 1 ⊕ Z/4Z 5 2 1 2 61 A⊕3 2 ⊕ A1 ⟨1/6⟩ ⊕ Z/3Z 5 2 1 2 Table 3: Cases with ∆ = 2 8 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T E(K) cmax cmin ∆ 41 A2 ⊕ A⊕4 1 1 6 � 2 1 1 2 � ⊕ Z/2Z 8 3 1 2 13 6 42 A⊕6 1 A∗ 1 ⊕2 ⊕ (Z/2Z)⊕2 3 1 2 5 2 59 A3 ⊕ A2 ⊕ A⊕2 1 ⟨1/12⟩ ⊕ Z/2Z 8 3 1 2 13 6 60 A3 ⊕ A⊕4 1 ⟨1/4⟩ ⊕ (Z/2Z)⊕2 3 1 2 5 2 Table 4: Cases with ∆ > 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By searching Table 8 for all cases with ∆ = 2 and ∆ > 2, we obtain Table 3 and Table 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Notice in particular that in both tables the torsion part of E(K) is always nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Consequently, if E(K) is torsion-free, then ∆ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 The quadratic form QX We define the positive-definite quadratic form with integer coefficients QX derived from the height pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The relevance of QX is due to the fact that some conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) can be stated in terms of what integers can be represented by QX (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The definition of QX consists in clearing denominators of the rational quadratic form induced by the height pairing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' the only question is how to find a scale factor that works in every case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More precisely, if E(K) has rank r ≥ 1 and P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr are generators of its free part, then q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) := h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr) is a quadratic form with coefficients in Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' we define QX by multiplying q by some integer d > 0 so as to produce coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We show that d may always be chosen as the determinant of the narrow lattice E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X with r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr be generators of the free part of E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Define QX(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) := (det E(K)0) · h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We check that the matrix representing QX has entries in Z, therefore QX has coefficients in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let A be the matrix representing the quadratic form QX, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) = xtAx, where x := (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then A has integer entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, QX has integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr be generators of the free part of E(K) and let L := E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The free part of E(K) is isomorphic to the dual lattice L∗ [OS91, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ], so we may find generators P 0 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', P 0 r of L such that the Gram matrix B0 := (⟨P 0 i , P 0 j ⟩)i,j of L is the inverse of the Gram matrix B := (⟨Pi, Pj⟩)i,j of L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 9 We claim that QX is represented by the adjugate matrix of B0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' the matrix adj(B0) such that B0 · adj(B0) = (det B0) · Ir, where Ir is the r × r identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, by construction B represents the quadratic form h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr), therefore QX(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', xr) = (det E(K)0) · h(x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + xrPr) = (det B0) · xtBx = (det B0) · xt(B0)−1x = xtadj(B0)x, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' To prove that A := adj(B0) has integer coefficients, notice that the Gram matrix B0 of L = E(K)0 has integer coefficients (as E(K)0 is an even lattice), then so does A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ We close this subsection with a simple consequence of the definition of QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If h(P) = m for some P ∈ E(K), then QX represents d · m, where d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr be generators for the free part of E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P = a1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + arPr + Q, where ai ∈ Z and Q is a torsion element (possibly zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since torsion sections do not contribute to the height pairing, then h(P − Q) = h(P) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) = d · h(a1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + arPr) = d · h(P − Q) = d · m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 3 Intersection with a torsion section Before dealing with more technical details in Section 4, we explain how torsion sections can be of help in our investigation, specially in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We first note some general properties of torsion sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As the height pairing is positive- definite on E(K)/E(K)tor, torsion sections are inert in the sense that for each Q ∈ E(K)tor we have ⟨Q, P⟩ = 0 for all P ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, in the case of rational elliptic surfaces, torsion sections also happen to be mutually disjoint: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [MP89, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1] On a rational elliptic surface, Q1 · Q2 = 0 for any distinct Q1, Q2 ∈ E(K)tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, if O is the neutral section, then Q·O = 0 for all Q ∈ E(K)tor\\{O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As stated in [MP89, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 holds for elliptic surfaces over C even without assuming X is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' However, for an arbitrary algebraically closed field the rationality hypothesis is needed, and a proof can be found in [SS19, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By taking advantage of the properties above, we use torsion sections to help us find P1, P2 ∈ E(K) such that P1 · P2 = k for a given k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is particularly useful when ∆ ≥ 2, in which case E(K)tor is not trivial by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Given k ∈ Z≥0, suppose we can find P ∈ E(K)0 with height h(P) = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), P · O = k − 1 < k, which is not yet what we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the next lemma we show that replacing O with a torsion section Q ̸= O gives P · Q = k, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 10 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P ∈ E(K)0 such that h(P) = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then P · Q = k for all Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume there is some Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1, Q · O = 0 and by the height formula (2), 2k = 2 + 2(P · O) − 0, hence P · O = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We use the height formula (1) for ⟨P, Q⟩ in order to conclude that P · Q = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since P ∈ E(K)0, it intersects the neutral component Θv,0 of every reducible fiber π−1(v), so contrv(P, Q) = 0 for all v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence 0 = ⟨P, Q⟩ = 1 + P · O + Q · O − P · Q − � v∈R contrv(P, Q) = 1 + (k − 1) + 0 − P · Q − 0 = k − P · Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 4 Existence of a pair of sections with a given intersection number Given k ∈ Z≥0, we state necessary and (in most cases) sufficient conditions for having P1 ·P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Necessary conditions are stated in generality in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1, while sufficient ones depend on the value of ∆ and are treated separately in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4, we collect all sufficient conditions proven in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 Necessary Conditions If k ∈ Z≥0, we state necessary conditions for having P1·P2 = k for some sections P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We note that the value of ∆ is not relevant in this subsection, although it plays a decisive role for sufficient conditions in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 · P2 = k for some P1, P2 ∈ E(K), then one of the following holds: i) h(P) = 2 + 2k for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Without loss of generality we may assume P2 is the neutral section, so that P1 · O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P1) = 2 + 2k − c, where c := � v contrv(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 ∈ E(K)0, then c = 0 and h(P1) = 2 + 2k, hence i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 /∈ E(K)0, then cmin ≤ c ≤ cmax by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But h(P1) = 2 + 2k − c, therefore 2 + 2k − cmax ≤ h(P1) ≤ 2 + 2k − cmin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 · P2 = k for some P1, P2 ∈ E(K), then QX represents some integer in [d · (2 + 2k − cmax), d · (2 + 2k)], where d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 and rephrase it in terms of QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If i) holds, then QX represents d · (2 + 2k) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But if ii) holds, then h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11, QX represents d · h(P) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In both i) and ii), QX represents some integer in [d · (2 + 2k − cmax), d · (2 + 2k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Sufficient conditions when ∆ ≤ 2 In this subsection we state sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) under the assumption that ∆ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, this covers almost all cases (more precisely, all but No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 41, 42, 59, 60 in Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We treat ∆ < 2 and ∆ = 2 separately, as the latter needs more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 The case ∆ < 2 We first prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, which gives sufficient conditions assuming ∆ < 2, then Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, which states sufficient conditions in terms of integers represented by QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is followed by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6, which is a simplified version of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ < 2 and let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let O ∈ E(K) be the neutral section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P) = 2 + 2(P · O) − c, where c := � v contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin], then 2 + 2k − cmax ≤ 2 + 2(P · O) − c ≤ 2 + 2k − cmin ⇒ c − cmax 2 ≤ P · O − k ≤ c − cmin 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore P · O − k is an integer in I := � c−cmax 2 , c−cmin 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that 0 is the only integer in I, so that P · O − k = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P · O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First notice that c ̸= 0, as P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 iii), cmin ≤ c ≤ cmax, consequently c−cmax 2 ≤ 0 ≤ c−cmin 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover ∆ < 2 implies that I has length cmax−cmin 2 = ∆ 2 < 1, so I contains no integer except 0 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 also applies when cmax = cmin, in which case the closed interval degen- erates into a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The following corollary of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 states a sufficient condition in terms of integers represented by the quadratic form QX (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ < 2 and let d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If QX represents an integer not divisible by d in the interval [d · (2+ 2k − cmax), d · (2+ 2k − cmin)], then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar ∈ Z such that QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)] with d ∤ QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P := a1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + arPr, where P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', Pr are generators of the free part of E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then d ∤ QX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) = d · h(P), which implies that h(P) /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular P /∈ E(K)0 since E(K)0 is an integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover h(P) = 1 dQX(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', ar) ∈ [2 + 2k − cmax, 2 + 2k − cmin] and we are done by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 12 The next corollary, although weaker than Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, is more practical for concrete examples and is frequently used in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' It does not involve finding integers represented by QX, but only finding perfect squares in an interval depending on the minimal norm µ (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � such that n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Take P ∈ E(K) such that h(P) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since h(nP) = n2µ /∈ Z, we must have nP /∈ E(K)0 as E(K)0 is an integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin] and we are done by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 The case ∆ = 2 The statement of sufficient conditions for ∆ = 2 is almost identical to the one for ∆ < 2: the only difference is that the closed interval Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 is substituted by a right half-open interval in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This change, however, is associated with a technical difficulty in the case when a section has minimal contribution term, thus the separate treatment for ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The results are presented in the following order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, which is a statement about sections whose contribution term is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Next we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, which states sufficient conditions for ∆ = 2, then Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there is P ∈ E(K) such that � v∈R contrv(P) = cmin, then P · Q = P · O + 1 for every Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If Q ∈ E(K)tor \\ {O}, then Q · O = 0 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, by the height formula (1), 0 = ⟨P, Q⟩ = 1 + P · O + 0 − P · Q − � v∈R contrv(P, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' (∗) Hence it suffices to show that contrv(P, Q) = 0 ∀v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 iv), contrv′(P) = cmin for some v′ and contrv(P) = 0 for all v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular P meets Θv,0, hence contrv(P, Q) = 0 for all v ̸= v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus from (∗) we see that contrv′(P, Q) is an integer, which we prove is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We claim that Tv′ = A1, so that contrv′(P, Q) = 0 or 1 2 by Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this case, as contrv′(P, Q) is an integer, it must be 0, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' To see that Tv′ = A1 we analyse contrv′(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since ∆ = 2, then cmin = 1 2 by Table 3 and contrv′(P) = cmin = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Table 1, this only happens if Tv′ = An−1 and 1 2 = i(n−i) n for some 0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The only possibility is i = 1, n = 2 and Tv′ = A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ With the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 we are able to state sufficient conditions for ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2 and let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let O ∈ E(K) be the neutral section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P) = 2 + 2(P · O) − c, where c := � v contrv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We repeat the arguments from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, in this case with the right half-open interval, so that the hypothesis that h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin), implies that P · O − k is an integer in I′ := � c−cmax 2 , c−cmin 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since I′ is half-open with length cmax−cmin 2 = ∆ 2 = 1, then I′ contains exactly one integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If 0 ∈ I′, then P · O − k = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P · O = k and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence we assume 0 /∈ I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We claim that P ·O = k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First, notice that if c > cmin, then the inequalities cmin < c ≤ cmax give c−cmax 2 ≤ 0 < c−cmin 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 ∈ I′, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence c = cmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since ∆ = 2, then I′ = [−1, 0), whose only integer is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus P · O − k = −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P · O = k − 1, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Finally, let Q ∈ E(K)tor \\ {O}, so that P · Q = P · O + 1 = k by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We remark that E(K)tor is not trivial by Table 3, therefore such Q exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ The following corollaries are analogues to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 adapted to ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Similarly to the case ∆ < 2, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 is stronger than Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10, although the latter is more practical for concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We remind the reader that µ denotes the minimal norm (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2 and let d := det E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If QX represents an integer not divisible by d in the interval [d·(2+2k −cmax), d·(2+2k −cmin)), then P1 ·P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We repeat the arguments in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, in this case with the half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If there is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � such that n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We repeat the arguments in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6, in this case with the half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Necessary and sufficient conditions for ∆ ≤ 2 For completeness, we present a unified statement of necessary and sufficient conditions assuming ∆ ≤ 2, which follows naturally from results in Subsections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume ∆ ≤ 2 and let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then P1 · P2 = k for some P1, P2 ∈ E(K) if and only if one of the following holds: i) h(P) = 2 + 2k for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some P /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' iii) h(P) = 2 + 2k − cmin and � v∈R contrv(P) = cmin for some P ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If i) or iii) holds, then P · O = k directly by the height formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But if ii) holds, it suffices to to apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 when ∆ < 2 and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 when ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, let P1·P2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Without loss of generality, we may assume P2 = O, so that P1·O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), h(P1) = 2 + 2k − c, where c := � v contrv(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If c = 0, then P1 ∈ E(K)0 and h(P1) = 2+2k, so i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence we let c ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' P1 /∈ E(K)0, so that cmin ≤ c ≤ cmax by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case c = cmin, then h(P1) = 2 + 2k − cmin and iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Otherwise cmin < c ≤ cmax, which implies 2 + 2k − cmax ≤ h(P1) < 2 + 2k − cmin, so ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 Summary of sufficient conditions For the sake of clarity, we summarize in a single proposition all sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) proven in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If one of the following holds, then P1 · P2 = k for some P1, P2 ∈ E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1) h(P) = 2 + 2k for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 2) h(P) = 2k for some P ∈ E(K)0 and E(K)tor is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 3) ∆ < 2 and there is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � with n2µ /∈ Z, where µ is the minimal norm (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case ∆ = 2, consider the right half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4) ∆ < 2 and the quadratic form QX represents an integer not divisible by d := det E(K)0 in the interval [d · (2 + 2k − cmax), d · (2 + 2k − cmin)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In case ∆ = 2, consider the right half-open interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In 1) a height calculation gives 2 + 2k = h(P) = 2 + 2(P · O) − 0, so P · O = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For 2), we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 to conclude that P · Q = k for any Q ∈ E(K)tor \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In 3) we use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 when ∆ < 2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10 when ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In 4), we apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 if ∆ < 2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 if ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 5 Main Results We prove the four main theorems of this paper, which are independent applications of the results from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The first two are general attempts to describe when and how gap numbers occur: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 tells us that large Mordell-Weil groups prevent the existence of gaps numbers, more precisely for Mordell-Weil rank r ≥ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 we show that for small Mordell-Weil rank, more precisely when r ≤ 2, then gap numbers occur with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The last two theorems, on the other hand, deal with explicit values of gap numbers: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7 provides a complete description of gap numbers in certain cases, while Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 is a classification of cases with a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 No gap numbers in rank r ≥ 5 We show that if E(K) has rank r ≥ 5, then X is gap-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Our strategy is to prove that for every k ∈ Z≥0 there is some P ∈ E(K)0 such that h(P) = 2+2k, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 1) we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We accomplish this in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we show that this holds when there is an embedding of A⊕ 1 or of A4 in E(K)0 (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Second, we show that if r ≥ 5, then such embedding exists, hence X is gap-free (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 15 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume E(K)0 has a sublattice isomorphic to A⊕4 1 or A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then for every ℓ ∈ Z≥0 there is P ∈ E(K)0 such that h(P) = 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First assume A⊕4 1 ⊂ E(K)0 and let P1, P2, P3, P4 be generators for each factor A1 in A⊕4 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then h(Pi) = 2 and ⟨Pi, Pj⟩ = 0 for distinct i, j = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lagrange’s four-square theorem [HW79, §20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5] there are integers a1, a2, a3, a4 such that a2 1 + a2 2 + a2 3 + a2 4 = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Defining P := a1P1 + a2P2 + a3P3 + a4P4 ∈ A⊕4 1 ⊂ E(K)0, we have h(P) = 2a2 1 + 2a2 2 + 2a2 3 + 2a2 4 = 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Now let A4 ⊂ E(K)0 with generators P1, P2, P3, P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then h(Pi) = 2 for i = 1, 2, 3, 4 and ⟨Pi, Pi+1⟩ = −1 for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We need to find integers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', x4 such that h(P) = 2ℓ, where P := x1P1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' + x4P4 ∈ A4 ⊂ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Equivalently, we need that ℓ = 1 2⟨P, P⟩ = x2 1 + x2 2 + x2 3 + x2 4 − x1x2 − x2x3 − x3x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Therefore ℓ must be represented by q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', x4) := x2 1 + x2 2 + x2 3 + x2 4 − x1x2 − x2x3 − x3x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that q represents all positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Notice that q is positive-definite, since it is induced by ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Bhargava-Hanke’s 290-theorem [BH][Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1], q represents all positive integers if and only if it represents the following integers: 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The representation for each of the above is found in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ We now prove the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r ≥ 5, then X is gap-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We show that for every k ≥ 0 there is P ∈ E(K)0 such that h(P) = 2 + 2k, so that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 1) we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 it suffices to prove that E(K)0 has a sublattice isomorphic to A⊕4 1 or A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The cases with r ≥ 5 are No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-7 (Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-6, E(K)0 = E8, E7, E6, D6, D5, A5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Each of these admit an A4 sublattice [Nis96, Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 7 we claim that E(K)0 = D4 ⊕ A1 has an A⊕4 1 sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This is the case because D4 admits an A⊕4 1 sublattice [Nis96, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 16 n x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' x4 with x2 1 + x2 2 + x2 3 + x2 4 − x1x2 − x2x3 − x3x4 = n 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' −9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 8 290 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 0 Table 5: Representation of the critical integers in Bhargava-Hanke’s 290-theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 Gaps with probability 1 in rank r ≤ 2 Fix a rational elliptic surface π : X → P1 with Mordell-Weil rank r ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that if k is a uniformly random natural number, then k is a gap number with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' More precisely, if G := {k ∈ N | k is a gap number of X} is the set of gap numbers, then G ⊂ N has density 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' d(G) := lim n→∞ #G ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', n} n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 17 We adopt the following strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If k ∈ N \\ G, then P1 · P2 = k for some P1, P2 ∈ E(K) and by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 the quadratic form QX represents some integer t depending on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This defines a function N\\G → T, where T is the set of integers represented by QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since QX is a quadratic form on r ≤ 2 variables, T has density 0 in N by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By analyzing the pre-images of N\\G → T, in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 we conclude that d(N \\ G) = d(T) = 0, hence d(G) = 1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let Q be a positive-definite quadratic form on r = 1, 2 variables with integer coeffi- cients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then the set of integers represented by Q has density 0 in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let S be the set of integers represented by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If d is the greatest common divisor of the coefficients of Q, let S′ be the set of integers representable by the primitive form Q′ := 1 d · Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By construction S′ is a rescaling of S, so d(S) = 0 if and only if d(S′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r = 1, then Q′(x1) = x2 1 and S′ is the set of perfect squares, so clearly d(S′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r = 2, then Q′ is a binary quadratic form and the number of elements in S′ bounded from above by x > 0 is given by C · x √log x + o(x) with C > 0 a constant and limx→∞ o(x) x = 0 [Ber12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus d(S′) = lim x→∞ C √log x + o(x) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ We now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let π : X → P1 be a rational elliptic surface with Mordell-Weil rank r ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then the set G := {k ∈ N | k is a gap number of X} of gap numbers of X has density 1 in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If r = 0, then the claim is trivial by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3, hence we may assume r = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that S := N \\ G has density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If S is finite, there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Otherwise, let k1 < k2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' be the increasing sequence of all elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2, for each n there is some tn ∈ Jkn := [d · (2 + 2kn − cmax), d · (2 + 2kn)] represented by the quadratic form QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let T be the set of integers represented by QX and define the function f : N \\ G → T by kn �→ tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since QX has r = 1, 2 variables, T has density 0 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For N > 0, let SN := S ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', N} and TN := T ∩ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since T has density zero, #TN = o(N), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' #TN N → 0 when N → ∞ and we need to prove that #SN = o(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We analyze the function f restricted to SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Notice that as tn ∈ Jkn, then kn ≤ N implies tn ≤ d · (2 + 2kn) ≤ d · (2 + 2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence the restriction g := f|SN can be regarded as a function g : SN → Td·(2+2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We claim that #g−1(t) ≤ 2 for all t ∈ Td·(2+2N), in which case #SN ≤ 2 · #Td·(2+2N) = o(N) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume by contradiction that g−1(t) contains three distinct elements, say kℓ1 < kℓ2 < kℓ3 with t = tℓ1 = tℓ2 = tℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since tℓi ∈ Jkℓi for each i = 1, 2, 3, then t ∈ Jkℓ1 ∩ Jkℓ2 ∩ Jkℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We prove that Jkℓ1 and Jkℓ3 are disjoint, which yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, since kℓ1 < kℓ2 < kℓ3, in particular kℓ3 − kℓ1 ≥ 2, therefore d · (2 + 2kℓ1) ≤ d · (2 + 2kℓ3 − 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' But cmax < 4 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, so d · (2 + 2kℓ1) < d · (2 + 2kℓ3 − cmax), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' max Jkℓ1 < min Jkℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus Jkℓ1 ∩ Jkℓ3 = ∅, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 Identification of gaps when E(K) is torsion-free with rank r = 1 The results in Subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 concern the existence and the distribution of gap num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the following subsections we turn our attention to finding gap numbers explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this subsection we give a complete description of gap numbers assuming E(K) is torsion-free with rank r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Such descriptions are difficult in the general case, but our assumption guarantees that each 18 E(K), E(K)0 is generated by a single element and that ∆ < 2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, which makes the problem more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We organize this subsection as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we point out some trivial facts about generators of E(K), E(K)0 when r = 1 in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Next we state necessary and sufficient conditions for having P1 · P2 = k when E(K) is torsion-free with r = 1 in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' As an application of the latter, we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, which is the main result of the subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface with Mordell-Weil rank r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P generates the free part of E(K), then a) h(P) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' b) 1/µ is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' c) E(K)0 is generated by P0 := (1/µ)P and h(P0) = 1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Item a) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Items b), c) follow from the fact that E(K)0 is an even lattice and that E(K) ≃ L∗ ⊕ E(K)tor, where L := E(K)0 [OS91, Main Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ In what follows we use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 and results from Section 4 to state necessary and sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) in case E(K) is torsion-free with r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume E(K) is torsion-free with rank r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then P1 · P2 = k for some P1, P2 ∈ E(K) if and only if one of the following holds: i) µ · (2 + 2k) is a perfect square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ii) There is a perfect square n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � such that µ · n /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5, E(K) is generated by some P with h(P) = µ and E(K)0 is generated by P0 := n0P, where n0 := 1 µ ∈ 2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First assume that P1·P2 = k for some P1, P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Without loss of generality we may assume P2 = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let P1 = nP for some n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We show that P1 ∈ E(K)0 implies i) while P1 /∈ E(K)0 implies ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 ∈ E(K)0, then n0 | n, hence P1 = nP = mP0, where m := n n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By the height formula (2), 2 + 2k = h(P1) = h(mP0) = m2 · 1 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence µ · (2 + 2k) = m2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P1 /∈ E(K)0, then n0 ∤ n, hence µ · n = n n0 /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover, h(P1) = n2h(P) = n2µ and by the height formula (2), n2µ = h(P) = 2 + 2k − c, where c := � v contrv(P1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The inequalities cmin ≤ c ≤ cmax then give 2+2k−cmax µ ≤ n2 ≤ 2+2k−cmin µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hence ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, assume i) or ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since E(K) is torsion-free, ∆ < 2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8, so we may apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If i) holds, then µ · (2 + 2k) = m2 for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since mP0 ∈ E(K)0 and h(mP0) = m2 µ = 2 + 2k, we are done by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If ii) holds, the condition µ · n /∈ Z is equivalent to n0 ∤ n, hence nP /∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Moreover n2 ∈ � 2+2k−cmax µ , 2+2k−cmin µ � , implies h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 ii), we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ By applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 to all possible cases where E(K) is torsion-free with rank r = 1, we obtain the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 19 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If E(K) is torsion-free with rank r = 1, then all the gap numbers of X are described in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T k is a gap number ⇔ none of the following are perfect squares first gap numbers 43 E7 k + 1, 4k + 1 1, 4 45 A7 k+1 4 , 16k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 16k + 9 8, 11 46 D7 k+1 2 , 8k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 8k + 4 2, 5 47 A6 ⊕ A1 k+1 7 , 28k − 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 28k + 21 12, 16 49 E6 ⊕ A1 k+1 3 , 12k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 12k + 9 3, 7 50 D5 ⊕ A2 k+1 6 , 24k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 24k + 16 6, 11 55 A4 ⊕ A3 k+1 10 , 40k − 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 40k + 25 16, 20 56 A4 ⊕ A2 ⊕ A1 k+1 15 , 60k − 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 60k + 45 22, 27 Table 6: Description of gap numbers when E(K) is torsion-free with r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For the sake of brevity we restrict ourselves to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The other cases are treated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Here cmax = 2·3 5 + 2·2 4 = 11 5 , cmin = min � 4 5, 3 4 � = 3 4 and µ = 1/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6, k is a gap number if and only if neither i) nor ii) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Condition i) is that 2+2k 20 = k+1 10 is a perfect square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Condition ii) is that � 2+2k−cmax µ , 2+2k−cmin µ � = [40k − 4, 40k + 25] contains some n2 with 20 ∤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We check that 20 ∤ n for every n such that n2 = 40k + ℓ, with ℓ = −4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=', 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Indeed, if 20 | n, then 400 | n2 and in particular 40 | n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then 40 | (n2 − 40k) = ℓ, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='4 Surfaces with a 1-gap In Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 we take each case in Table 6 and describe all its gap numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In this subsection we do the opposite, which is to fix a number and describe all cases having it as a gap number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We remind the reader that our motivating problem (Section 1) was to determine when there are sections P1, P2 such that P1 · P2 = 1, which induce a conic bundle having P1 + P2 as a reducible fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The answer for this question is the main theorem of this subsection: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let X be a rational elliptic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then X has a 1-gap if and only if r = 0 or r = 1 and π has a III∗ fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20 Our strategy for the proof is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We already know that a 1-gap exists whenever r = 0 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1) or when r = 1 and π has a III∗ fiber (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Conversely, we need to find P1, P2 with P1 · P2 = 1 in all cases with r ≥ 1 and T ̸= E7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' First we introduce two lemmas, which solve most cases with little computation, and leave the remaining ones for the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In both Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11 our goal is to analyze the narrow lattice E(K)0 and apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 to detect cases without a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If one of the following holds, then h(P) = 4 for some P ∈ E(K)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' a) The Gram matrix of E(K)0 has a 4 in its main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' b) There is an embedding of An ⊕ Am in E(K)0 for some n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' c) There is an embedding of An, Dn or En in E(K)0 for some n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Case a) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assuming b), we take generators P1, P2 from An, Am respectively with h(P1) = h(P2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Since An, Am are in direct sum, ⟨P1, P2⟩ = 0, hence h(P1 + P2) = 4, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If c) holds, then the fact that n ≥ 3 allows us to choose two elements P1, P2 among the generators of L1 = An, Dn or En such that h(P1) = h(P2) = 2 and ⟨P1, P2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Thus h(P1 + P2) = 4 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In the following cases, X does not have a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' r ≥ 3 : all cases except possibly No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' r = 1, 2 : cases No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 25, 26, 30, 32-36, 38, 41, 42, 46, 52, 54, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We look at column E(K)0 in Table 8 to find which cases satisfy one of the conditions a), b), c) from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' a) Applies to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12, 17, 19, 22, 23, 25, 30, 32, 33, 36, 38, 41, 46, 52, 54, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' b) Applies to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 10, 11, 14, 15, 18, 24, 26, 34, 35, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' c) Applies to No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-10, 13, 16, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In particular, this covers all cases with r ≥ 3 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-24) except No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='9 in each of these cases there is P ∈ E(K)0 with h(P) = 4 and we are done by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ In the next lemma we also analyze E(K)0 to detect surfaces without a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Assume E(K)0 ≃ An for some n ≥ 1 and that E(K) has nontrivial torsion part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Then X does not have a 1-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This applies to cases No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 28, 39, 44, 48, 51, 57, 58 in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Take a generator P of E(K)0 with h(P) = 2 and apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 21 We are ready to prove the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We need to show that in all cases where r ≥ 1 and T ̸= E7 there are P1, P2 ∈ E(K) such that P1 · P2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' This corresponds to cases No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-61 except 43 in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The cases where r = 1 and E(K) is torsion-free can be solved by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10, namely No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 45-47, 49, 50, 55, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Adding these cases to the ones treated in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='10 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='11, we have therefore solved the following: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 1-19, 21-26, 28, 30, 32-36, 38, 39, 41-52, 54-58, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' For the remaining cases, we apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='12 3), which involves finding perfect squares in the interval � 4−cmax µ , 4−cmin µ � (see Table 7), considering the half-open interval in the cases with ∆ = 2 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 53, 61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' T E(K) µ I n2 ∈ I 20 A⊕2 2 ⊕ A1 A∗ 2 ⊕ ⟨1/6⟩ 1 6 [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 23] 42 27 E6 A∗ 2 2 3 [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4] 22 29 A5 ⊕ A1 A∗ 1 ⊕ ⟨1/6⟩ 1 6 [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 21] 42 31 A4 ⊕ A2 1 15 � 2 1 1 8 � 2 15 [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 21] 42 37 A3 ⊕ A2 ⊕ A1 A∗ 1 ⊕ ⟨1/12⟩ 1 12 [22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 28] 52 40 A⊕2 2 ⊕ A⊕2 1 ⟨1/6⟩⊕2 1 6 [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 21] 42 53 A5 ⊕ A⊕2 1 ⟨1/6⟩ ⊕ Z/2Z 1 6 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12] 32 59 A3 ⊕ A2 ⊕ A⊕2 1 ⟨1/12⟩ ⊕ Z/2Z 1 12 [16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 42] 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 52,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 62 61 A⊕3 2 ⊕ A1 ⟨1/6⟩ ⊕ Z/3Z 1 6 [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 12] 32 Table 7: Perfect squares in the interval I := � 4−cmax µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 4−cmin µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' In No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 59 we have ∆ > 2, so a particular treatment is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Let T = Tv1 ⊕ Tv2 ⊕ Tv3 ⊕ Tv4 = A3 ⊕ A2 ⊕ A1 ⊕ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' If P generates the free part of E(K) and Q generates its torsion part, then h(P) = 1 12 and 4P + Q meets the reducible fibers at Θv1,2, Θv2,1, Θv3,1, Θv4,1 [Kur14][Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' By Table 1 and the height formula (2), 42 12 = h(4P + Q) = 2 + 2(4P + Q) · O − 2 · 2 4 − 1 · 2 3 − 1 2 − 1 2, hence (4P + Q) · O = 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ■ 22 6 Appendix We reproduce part of the table in [OS91, Main Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='] with data on Mordell-Weil lattices of rational elliptic surfaces with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' We only add columns cmax, cmin, ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E(K)0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E(K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='cmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='cmin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='E8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ⊕ Z/4Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='A3 ⊕ A2 ⊕ A⊕2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨12⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨1/12⟩ ⊕ Z/2Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='A3 ⊕ A⊕4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨4⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨1/4⟩ ⊕ Z/2Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='A⊕3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⊕ A1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨6⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='⟨1/6⟩ ⊕ Z/3Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='Table 8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='Mordell-Weil lattices of rational elliptic surfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='with Mordell-Weil rank r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 25 References [Ber12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Bernays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Über die Darstellung von positiven, ganzen Zahlen durch die primitive, binären quadratischen Formen einer nicht-quadratischen Diskriminante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' PhD thesis, Göttingen, 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [BH] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Bhargava and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Hanke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Universal quadratic forms and the 290-Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Preprint at http://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='edu/~vakil/files/290-Theorem-preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [Cos] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Costa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Classification of fibers of conic bundles on rational elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content='03549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [Elk90] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Elkies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The Mordell-Weil lattice of a rational elliptic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Clarendon Press, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [Kur14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Kurumadani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Pencil of cubic curves and rational elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Master’s thesis, Kyoto University, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [MP89] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Miranda and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Persson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Torsion groups of elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Compositio Mathematica, 72(3):249–267, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [Nis96] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Nishiyama.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Com- mentarii Mathematici Universitatis Sancti Pauli, 40, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [Shi89] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Shioda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' The Mordell-Weil lattice and Galois representation, I, II, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Proceedings of the Japan Academy, 65(7), 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [Shi90] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Shioda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' On the Mordell-Weil lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Commentarii Mathematici Universitatis Sancti Pauli, 39(7), 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [SS10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Schuett and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Shioda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Elliptic surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Advanced Studies in Pure Mathematics, 60:51–160, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' [SS19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Schuett and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Shioda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Mordell-Weil Lattices, volume 70 of Ergebnisse der Mathematik und ihrer Grenzgebiete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE1T4oBgHgl3EQfYQS_/content/2301.03137v1.pdf'}