diff --git "a/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/load_file.txt" "b/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/load_file.txt" @@ -0,0 +1,510 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf,len=509 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='00543v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='GR] 2 Jan 2023 ON PSEUDO-REAL FINITE SUBGROUPS OF PGL3(C) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let G be a finite subgroup of PGL3(C), and let σ be the generator of Gal(C/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We say that G has a real field of moduli if σG and G are PGL3(C)-conjugates, that is, if ∃ φ ∈ PGL3(C) such that φ−1 G φ = σG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we say that R is a field of definition for G or that G is definable over R if G is PGL3(C)-conjugate to some G′ ⊂ PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this situation, we call G′ a model for G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If G has R as a field of definition but is not definable over R, then we call G pseudo-real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this paper, we first show that any finite cyclic subgroup G = Z/nZ in PGL3(C) has a real field of moduli and we provide a necessary and sufficient condition for G = Z/nZ to be definable over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We also prove that any dihedral group D2n with n ≥ 3 in PGL3(C) is definable over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we study all six classes of finite primitive subgroups of PGL3(C), and show that all of them except the icosahedral group A5 are pseudo-real;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='5, whereas A5 is definable over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='6 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Introduction The projective general linear group over the complex numbers PGL3(C) is widely studied in several branches of mathematics for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Some of these mo- tivations come from algebraic geometry, arithmetic geometry, and also from group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We give some examples of such motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (1) In complex algebraic geometry, PGL3(C) can be viewed as the automorphism group Aut(P2(C)) of the complex projective plane P2(C), see [11, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, any isomorphism between two smooth complex plane curves C and C′ of a fixed degree d ≥ 4 is induced by an element of PGL3(C), see [8, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For such a curve we have the finiteness result | Aut(C)| < +∞ due to Hurwitz [19], hence we can view Aut(C) as a finite subgroup of PGL3(C) acting on a non-singular plane model F(X, Y, Z) = 0 for C inside P2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is thus natural to classify finite subgroups G in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Based on geometrical methods, Mitchell [23] achieved such classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Recently, Harui [12] made Mitchell’s classification more precise under the assumption that G = Aut(C) for some smooth plane curves C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' However, some of these groups live in a short exact sequence, hence group extension problems arise, which can sometimes be hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Another parallel line of research is to obtain the stratification of C-isomorphism classes of smooth plane curves of a fixed degree d by their automorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Henn in his PhD dissertation [13] and Komiya-Kuribayashi [22] accomplished this task for smooth quartic curves (d = 4), Badr-Bars [3, 4, 5] for smooth quinitcs (d = 5) and for smooth sextics (d = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 20G20, 14L35, 14H37, 22F50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Projective linear groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Field of moduli;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Fields of definitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Pseudo- real;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Smooth plane curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Automorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 1 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY (2) In complex arithmetic geometry, the problem of studying fields of definition versus fields of moduli for a Riemann surface S has attracted a lot of recent research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For example, we refer to [1, 2, 7, 9, 14, 15, 16, 18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More precisely, a subfield K of C is called a field of definition for S if there exists a model of S defined by polynomials with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The field of moduli for S is the intersection of all fields of definition for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The work of Koizumi [20] guarantee the existence of a model for S over a finite extension of its field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this direction, the surface S is said to be pseudo-real if its field of moduli is a subfield of R, but S does not have R as a field of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The above aspects from algebraic geometry and arithmetic geometry are the main motivation for us to extend the notions of fields of definition, fields of moduli, pseudo-real, to the study of arithmetic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, there has been other in- stances in which it has been fruitful to translate concepts from arithmetic geometry to group theory, as we illustrate next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (3) In group theory, we can measure to which extent an infinite group Γ is similar to an abelian group by computing its Jordan constant, denoted by J(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is defined to be the smallest positive integer such that any finite subgroup of Γ has an abelian normal subgroup with index not exceeding J(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This definition originated from the theory of abelian varieties, more specifically, [24, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Concerning the Jordan constant J(PGL3(K)), where K is a field of characteristic 0, Hu [17] showed that it assumes only one of the values: 360, 168, 60, 24, 12, 6, depending on whether √ 5 or ζ3 belongs to K or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Here ζ3 denotes a primitive 3rd root of unity in K, a fixed algebraic closure of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, J(PGL3(C)) = 360, see [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2] for full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Throughout the paper, we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Norm(G, PGL3(C)) is the normalizer of G inside PGL3(C), ζn = e 2πi n , a fixed primitive nth root of unity in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We shall view C× as a subgroup of GL3(C) by identifying 0 ̸= c ∈ C with diag(c, c, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If A is in GL3(C), we let π(A) denote its image under the canonical projection onto PGL3(C), namely π(A) is the coset (or equiva- lence class) C×A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To ease notation, we occasionally continue to use A in place of π(A) when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If A = (ai,j) ∈ GL3(C), then the projective linear transformation π(A) ∈ PGL3(K) is sometimes written as [a1,1X + a1,2Y + a1,3Z : a2,1X + a2,2Y + a2,3Z : a3,1X + a3,2Y + a3,3Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Galois group Gal(C/R)- action on PGL3(C) is a left action, denoted by σφ for any φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For c ∈ C, ℜ(c) and ℑ(c) denote the real and the imaginary parts of c respectively, and |c| denotes the absolute value of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Main results Let G ⊂ PGL3(C) be cyclic of order 1 < n < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Up to PGL3(C)-conjugation, such G is generated by a diagonal element A := diag(1, ζa n, ζb n), for some 0 ≤ a < b ≤ n − 1 such that gcd(a, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, we have that (1) G always has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 3 (2) R is a field of definition for G if and only if A and A−1 are conjugates via a transformation of the shape φ σφ−1 for some φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this situation, φ−1 G φ would give a model for G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' An homology of period n is a projective linear transformation of the plane P2(C), which is PGL3(C)-conjugate to diag(1, 1, ζn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Such a transformation fixes point- wise a projective line L, its axis, and a point P ∈ P2(C) − L, its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In its canonical form, the line is L : Z = 0 and the point is P = (0 : 0 : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Otherwise, it is a non-homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, we have: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, there exists a model for G over R if and only if n = 2 or n > 2 such that a + b, a − 2b or 2a − b equals 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, any cyclic group generated by a homology of period n ≥ 3 is pseudo-real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we can get a model for G over R generated by φ−1 A φ = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 2ℑ(α β ζa n) 2|β|2 sin(2πa/n) 0 −2|α|2 sin(2πa/n) 2ℑ(α β ζ−a n ) \uf8f6 \uf8f8 for some α, β ∈ C∗ The above results can be reformulated using characteristic polynomials of lifts to B ∈ GL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If we denote the characteristic polynomial of such B by fB(t), then it is straightforward to see that for c ∈ C∗ fcB(t) = c3fB(t/c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1) So while we can not attach a single polynomial as a characteristic polynomial to an element A ∈ PGL3(C), we can attach to such an A an equivalence class of polynomials in C[t] coming from the action given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Such classes are preserved under conjugation in PGL3(C), and we can prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A finite cyclic group G of order n ≥ 3 is definable over R if there exists A ∈ GL3(C) such that π(A) (the image of A in PGL3(C) under the natural projection) generates G in PGL3(C) and the characteristic polynomial fA(t) ∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The converse is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For G = D2n, a dihedral group in PGL3(C), we prove: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any dihedral group D2n of order 2n with n ≥ 3 in PGL3(C) is conjugate to ⟨B, π(A)⟩, where B = [X : Z : Y ] and A = diag(1, ζa n, ζ−a n ) for some integer a such that gcd(n, a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, we always can descend it to R as ⟨ φ−1 B φ, φ−1 A φ⟩, where φ−1 A φ is as given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 and φ−1 B φ = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 −2ℑ(α β) −2ℑ(β2) 0 2ℑ(α2) 2ℑ(α β) \uf8f6 \uf8f8 for some α, β ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' When G is one of the finite primitive subgroup of PGL3(C), we show the follow- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any of the finite primitive subgroups namely, the Hessian groups Hess∗, for ∗ = 216, 72 and 36, the Klein group PSL(2, 7) of order 168, the icosa- hedral group A5 of order 60 and the alternating group A6 of order 360, has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, none of them descends to R except A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More concretely, 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY we always can descend A5 to R as φ−1 ⟨ A, B, C⟩ φ, such that φ−1 A φ and φ−1 B φ are as given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4 with n = 5 and a = 4, and φ−1 C φ equals \uf8eb \uf8ed 4ℑ(α β) 8ℑ(α β) ℜ(α) 8ℑ(α β) ℜ(β) 2ℑ(β) 2 � cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ) � −2 cos(2π/5)ℑ(β2) 2ℑ(α) 2 cos(2π/5)ℑ(α2) 2 � cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ) � \uf8f6 \uf8f8 , for some α, β ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A connection with these notions in arithmetic geometry is described by the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let C : F(X, Y, Z) = 0 be a smooth plane curve over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If C has a real field of moduli in the Arithmetic Geometry sense, then its automorphism group Aut(C) has a real field of moduli in the Group Theory sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The converse of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='6 is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Below is a counter example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' There are infinitely many smooth plane quintic curves defined over C by an equation of the form Cα,β : X5 + Y 5 + Z5 + αX(Y Z)2 + βX3(Y Z) = 0, such that the automorphism group Aut(Cα,β) = D10 has a real field of moduli, but Cα,β does not have a real field of moduli as its field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The case when G is cyclic Suppose that G = ⟨diag(1, ζa n, ζb n)⟩ in PGL3(C) such that 0 ≤ a < b ≤ n − 1 and gcd(a, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since the complex conjugation automorphism σ : C → C sends ζn �→ ζ−1 n , then σG = ⟨diag(1, ζ−a n , ζ−b n )⟩ = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, G has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This proves Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1-(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1-(2), we assume that G descends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' That is, there exists φ ∈ PGL3(C) satisfying φ−1 A φ ∈ PGL3(R), where A = diag(1, ζa n, ζb n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This holds if and only if φ−1 A φ = σ � φ−1 A φ � = σφ−1 A−1 σφ, which we can read in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' First as � φ σφ−1�−1 A � φ σφ−1� = A−1, which shows that A and A−1 are conjugates via φ σφ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Second as φ−1 A φ = σ � φ−1 A φ � , which shows that φ−1 A φ ∈ PGL3(R) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We need the following lemma to discuss Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Assume A and B are matrices in GL3(C) such that π(A) and π(B) are PGL3(C)-conjugates (where π denotes the natural projection from GL3(C) to PGL3(C)), then there is a constant c ∈ C∗ such that the eigenvalues of B are precisely cν1, cν2, cν3, where ν1, ν2, ν3 are the eigenvalues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Suppose that there is an ψ ∈ PGL3(C) such that ψ−1 π(A) ψ = π(B) in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, this equation corresponds to ψ−1 A ψ = (1/c)B in GL3(C) for some c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hence, A and (1/c)B are similar matrices in GL3(C), so by elementary linear algebra, we guarantee that their characteristic polynomials have the same roots, say ν1, ν2, ν3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, the eigenvalues of B are cν1, cν2, cν3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ We now present the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (of the necessity direction) First, assume that G is generated by a homology A = diag(1, 1, ζn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since {c, c, c ζn} ̸= {1, 1, ζ−1 n } for any c ∈ C∗ unless n = 2, then A and A−1 are never PGL3(C)-conjugates for n ≥ 3 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, G does not have a model over R by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Secondly, assume that G is generated by a non-homology A = diag(1, ζa n, ζb n) such that {c, c ζa n, c ζb n} = {1, ζ−a n , ζ−b n } for some c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, c is either 1, ζ−a n or ζ−b n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, if c = 1, then ζa n = ζ−a n , ζb n = ζ−b n or ζa n = ζ−b n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' That is, 2a = 2b = 0 mod n or a + b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We discard the case 2a = 2b = 0 mod n as it implies that n or n/2 would divide gcd(a, b) = 1, a contradiction because n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This leaves us with a + b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' if c = ζ−a n , then ζb−a n = ζ−b n , and n | a − 2b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' if c = ζ−b n , then ζa−b n = ζ−a n , and 2a − b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This completes the necessity part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (of the sufficiency direction) If G is cyclic generated by a homology of period 2, then G is PGL3(C)-conjugate to ⟨diag(1, 1, −1)⟩ in PGL3(R), and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Otherwise, G is generated by a non-homology A = diag(1, ζa n, ζb n) of order n ≥ 3 such that a + b, a − 2b or 2a − b equals 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' First, we show that any of the last two situation can be reduced to the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, if A = diag(1, ζ2b n , ζb n), then one can take ψ = [Y : Z : X] so that ψ−1 A ψ = diag(ζb n, 1, ζ2b n ) = diag(1, ζ−b n , ζb n) = diag(1, ζa′ n , ζ−a′ n ) in PGL3(C), where a′ := −b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Similarly, if A = diag(1, ζa n, ζ2a n ), then take ψ = [Z : X : Y ] to get ψ−1 A ψ = diag(ζa n, ζ2a n , 1) = diag(1, ζa n, ζ−a n ) in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now we are going to handle the situation when n divides a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Take φ = \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' One easily verifies that φ σφ−1 = [X : Z : Y ] ∈ Norm(G, PGL3(C)) such that [X : Z : Y ] A [X : Z : Y ] = A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, we deduce by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1 that φ−1 G φ ≤ PGL3(R) is a model of G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More specifically, φ−1 A φ = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 β −β 0 −α α \uf8f6 \uf8f8 diag(1, ζa n, ζ−a n ) \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 ζa n β −ζ−a n β 0 −ζa n α ζ−a n α \uf8f6 \uf8f8 \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 2ℑ(α β ζa n) i 2|β|2 sin(2πa/n) i 0 −2|α|2 sin(2πa/n) i 2ℑ(α β ζ−a n ) i \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 2ℑ(α β ζa n) 2|β|2 sin(2πa/n) 0 −2|α|2 sin(2πa/n) 2ℑ(α β ζ−a n ) \uf8f6 \uf8f8 ∈ PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Next, assume that G is generated by a non-homology π(A) ∈ PGL3(C) of order n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' As a consequence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, we can say that fA(t) ∈ R[t] is a sufficient (rather than necessary) condition for G to descend to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1, there exists c ∈ C∗ such that fA(t) = (t − c)(t − cζa n)(t − cζb n) ∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, the roots c, c ζa n, c ζb n of fA(t) are pairwise distinct, since π(A) is a non- homology in PGL3(C) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, the coefficients c3ζa+b n , c(1+ζa n +ζb n), c2(ζa+b n +ζa n +ζb n) belong to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus there are r, r′ ∈ R such that ζa+b n = r/c3 and ζa n + ζb n = r′/c− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Consequently, the last condition becomes c2(r/c3+r′/c−1) ∈ R, in other words, c3−r′c2+r′′c−r = 0 for some r, r′, r′′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This means that c ∈ C is algebraic over R of degree dividing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since C/R is a field extension of degree 2, then c must be algebraic over R of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, c ∈ R, which in turns implies that ζa+b n , ζa n + ζb n ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Clearly, ζa+b n ∈ R only if a+b = k( n 2 ) with k = 1, 2 or 3, since 3 ≤ a+b ≤ 2n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If k = 1 or 3, then ζa+b n = −1 and ζa n + ζb n = ζa n − ζ−a n = 2 sin(2π a/n) i /∈ R, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hence k = 1 and a + b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 we deduce that G descends to R, which was to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To see that the converse does not hold in general, take A = diag(ζ3 5, ζ4 5, ζ2 5) in GL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Clearly, fA(t) /∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' However, G = ⟨π(A)⟩ is definable over R by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, since π(A) = diag(1, ζ5, ζ−1 5 ) = diag(1, ζa n, ζb n) with n | a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The case when G is a Dihedral group Suppose that G = ⟨A, B : An = B2 = 1, BAB = A−1⟩ is a dihedral group D2n in PGL3(C) with n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' There is no loss of generality to take A = diag(1, ζa n, ζb n) up to conjugation and projective equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since A and A−1 are PGL3(C)- conjugates via B, then, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, A must be a non-homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, we can always reduce to the case b = −a modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we can assume by [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7] that B belongs to PBD(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since BAB = A−1, we obtain B = [X : νZ : ν−1Y ] for some ν ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Through a projective transformation ψ = diag(1, λν, λ), which is in Norm (⟨A⟩, PGL3(C)), we can further reduce to ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Eventually, we conclude: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For each fixed integer n ≥ 3, there is, up to PGL3(C)-conjugation, a unique dihedral group D2n of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More precisely, any such group is conjugate to the group generated by B = [X : Z : Y ] and A = diag(1, ζn, ζ−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, we will prove that a dihedral group G = ⟨ A, B⟩ as above has a real field of moduli, moreover, it descends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since σ A = A−1 and σ B = B−1, then σG = G and G has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' On the other hand, we have seen in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 that φ−1 A φ ∈ PGL3(R) through a projective transformation φ of the shape: φ = \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It remains to see that φ−1 B φ ∈ PGL3(R) so that φ−1 G φ is a model of G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, we have ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 7 φ−1 B φ = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 β −β 0 −α α \uf8f6 \uf8f8 [X : Z : Y ] \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 −β β 0 α −α \uf8f6 \uf8f8 \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 −2ℑ(α β) i −2ℑ(β2) i 0 2ℑ(α2) i 2ℑ(α β) i \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 −2ℑ(α β) −2ℑ(β2) 0 2ℑ(α2) 2ℑ(α β) \uf8f6 \uf8f8 ∈ PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The cases when G is a finite primitive subgroup of PGL3(C) Recall that the finite primitive subgroups PGL3(C) are the Hessian groups Hess∗, for ∗ = 216, 72, 36, the alternating groups A∗, for ∗ = 5, 6, and the Klein group PSL(2, 7) of order 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We study their definability over R in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Hessian groups Hess∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Hessian group of order 216, denoted by Hess216, is unique up to conjugation in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' See [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 217] or [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For instance, we fix Hess216 = ⟨S, T, U, V ⟩ where S = diag(1, ζ3, ζ−1 3 ), U = diag(1, 1, ζ3), V = \uf8eb \uf8ed 1 1 1 1 ζ3 ζ−1 3 1 ζ−1 3 ζ3 \uf8f6 \uf8f8 , T = [Y : Z : X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Also, we consider the Hessian subgroup of order 72, Hess72 = ⟨S, T, V, UV U −1⟩, and the Hessian subgroup of order 36, Hess36 = ⟨S, T, V ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Concerning the Hessian groups Hess∗, for ∗ ∈ {36, 72, 216}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We first show Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any of the Hessian groups Hess∗ has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is easy to see that σS = S−1, σU = U −1, and σT = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore σV = 3V −1 in GL3(C), hence we also have σV = V −1 in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It follows that σ Hess∗ = Hess∗ if ∗ = 216 or 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' So Hess216 and Hess36 indeed have a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For Hess72, we get σ Hess72 = ⟨S, T, V, U −1V −1U⟩ ⊂ Hess216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' another copy of Hess72 inside Hess216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Group structure of Hess216 [10] assures that all copies of Hess72 are Hess216-conjugates, that is to say, there is a projective transformation ψ ∈ Hess216 such that ψ−1 Hess72 ψ = σ Hess72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' From this we obtain that Hess72 has a real field of moduli as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ As a consequence, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Hessian groups Hess∗ for ∗ = 216, 72 and 36 are all pseudo- real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is easy to see that ST = T S, so ⟨S, T �� is isomorphic to C3 × C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' By [17, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2] (see also [25, Section 4]), C3 ×C3 is a subgroup of PGL3(K) if and only if the field K contains a nontrivial cube root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since ζ3 /∈ R, we can’t reduce ⟨S, T ⟩ to a subgroup of PGL3(R) as ζ3 /∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, φ−1 Hess∗ φ ⊈ PGL3(R) for any φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Combining with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1, we conclude that Hess∗ is pseudo-real for ∗ = 216, 72 and 36 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The alternating groups A5 and A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We first note that PGL3(C) possesses a single conjugacy class isomorphic to each of A5 and A6, see [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 224, 225] or [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, for i ∈ {5, 6} Ai and σ Ai must be PGL3(C)-conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In other words, Ai has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since A6 contains C3 × C3 as a subgroup, then we can use the same argument as in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 to deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The alternating group A6 is pseudo-real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For the icosahedral group A5, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To study it we fix the copy G := ⟨A, B, C⟩ in PGL3(C), where A = diag(1, ζ−1 5 , ζ5), B = [X : Z : Y ], C = \uf8eb \uf8ed 2 2 2 1 cos(4π/5) cos(2π/5) 1 cos(2π/5) cos(4π/5) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' According to [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7 ], G is PGL3(C)-conjugate to A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any subgroup of PGL3(C) isomorphic to A5 is PGL3(C) conjugate to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, we are going to construct an explicit model for G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Recall, from our study above of the Dihedral group in §4, that ⟨ A, B⟩ descends to R via a transformation of the shape φ = \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, one can check that φ−1 C φ equals \uf8eb \uf8ed 4ℑ(α β) 8ℑ(α β) ℜ(α) 8ℑ(α β) ℜ(β) 2ℑ(β) 2 � cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ) � −2 cos(2π/5)ℑ(β2) 2ℑ(α) 2 cos(2π/5)ℑ(α2) 2 � cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ) � \uf8f6 \uf8f8 , in PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus all generators of G when conjugated by the same φ become in PGL3(R), hence the same is true for the whole group and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Klein group PSL(2, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Again, there is a single conjugacy class of PSL(2, 7) in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus it has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Also, we know by [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7] that a representative of such a class contains the element diag(1, ζ7, ζ3 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 applies to n = 7, a = 1, b = 3 to conclude that PSL(2, 7) is not defin- able over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Connection to Arithmetic Geometry Let C : F(X, Y, Z) = 0 be a non-singular plane curve defined over C with non- trivial automorphism group Aut(C) in PGL3(C), Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We have Aut(σC) = σ Aut(C) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For any φ ∈ Aut(C), φF(X, Y, Z) = cF(X, Y, Z) for some c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Applying σ to both sides yields σ(c) σF(X, Y, Z) = σ �φF(X, Y, Z) � = σφ (σF(X, Y, Z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' That is, σφ leaves invariant σC : σF(X, Y, Z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Equivalently, σφ ∈ Aut(σC), hence σ Aut(C) ⊆ Aut(σC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' By a similar argument we can show the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let C : F(X, Y, Z) = 0 be a smooth plane curve over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If C has a real field of moduli in the Arithmetic Geometry sense, then Aut(C) has a real field of moduli in the Group Theory sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The converse need not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since C : F(X, Y, Z) = 0 has a real field of moduli, then it must be the case that σC : σF(X, Y, Z) = 0 and C : F(X, Y, Z) = 0 are C-projectively equivalent (isomorphic over C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, any isomorphism between complex non-singular plane curves C and C′ is always given by a projective transformation φ ∈ PGL3(C) such that their automorphism groups are conjugates via this φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' As a consequence, we obtain that φ−1 Aut(C) φ = Aut(σC), which equals σ Aut(C) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus Aut(C) has a real field of moduli as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To complete the argument, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3 below provides infinitely many counter examples that Aut(C) can descend R, but C : F(X, Y, Z) = 0 does not even have a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Consider the two-dimensional family Ca,b of smooth plane quintic curves given by Ca,b : X5 + Y 5 + Z5 + iaX(Y Z)2 + ibX3(Y Z), where a, b ∈ R∗ such that a/b ̸= (c5 − 3)c2 2c5 − 1 ζm 10 for any c ∈ C∗ and m ∈ {±1, ±3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Non-singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We first note that no singular points lie over Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, if C has singularity at (α : 0 : β), then α and β must be 0 in order to satisfy FX = FZ = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Second, the resultant of f1(X, Z) := FY (X, 1, Z) and f2(X, Z) := FZ(X, 1, Z) with respect to X is given by ResX(f1, f2) = −125 i b3 (Z5 − 1)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Using MATHEMATICA, one can verify that we have singular points over Z5 = 1 only if a/b = (c5 − 3)c2 2c5 − 1 ζm 10 for some c ∈ C∗ and m ∈ {±1, ±3, 5}, which is absurd by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The stratification of smooth plane quintics by their automorphism groups in [3, 6] assures that the group D10 gener- ated by ρ1 = diag(1, ζ5, ζ−1 5 ) and ρ2 = [X : Z : Y ] is a always a sub- group of automorphisms for Ca,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, if Ca,b admits a larger auto- morphism group, then it should be GAP(150, 5) = (Z/5Z)2 ⋊ S3, where in this situation Ca,b is K-isomorphic to the Fermat quintic curve F5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' the most symmetric smooth quintic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, there must be an extra automorphism ρ3 /∈ ⟨ρ1⟩ of order 5 that commutes with ρ1 as any Z/5Z inside (Z/5Z)2 ⋊ S3 is contained in a (Z/5Z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' See Group Structure of (Z/5Z)2 ⋊ S3 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Straightforward calculations in PGL3(C) lead to ρ3 = diag(1, α, β) with α5 = β5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Checking the action of such an automorphism on the defining equation of Ca,b tells us that a = b = 0 or ρ3 ∈ ⟨ρ1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, Aut(Ca,b) = D10 = ⟨ρ1, ρ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, we conclude by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4 that Aut(Ca,b) descends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Ca,b does not have a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Suppose that C is a member of the family Ca,b such that C has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hence C and σC are C-projectively equivalent via some φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since C and σC belong to the same family Ca,b, we have σ Aut(C) = Aut(C) = ⟨ρ1, ρ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, φ should be in the normalizer of ⟨ρ1, ρ2⟩ in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We reduce to the case φ−1ρ1φ = ρ1 or ρ−1 as {c, cζ5, cζ−1 5 } ̸= {1, ζ2 5, ζ−2 5 } or {1, ζ3 5, ζ−3 5 } for any c ∈ C∗ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Consequently, φ = diag(1, α, β) or [X : αZ : βY ] for some α, β ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Because φC = σC, we must have α5 = β5 = 1 and αβ = (αβ)2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The last condition is 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Proceedings and Lecture Notes, 54 (2011), 289-311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Yasinsky, The Jordan constant for Cremona group of rank 2, Korean Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5 (2017), 1859-1871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Eslam Badr Mathematics Department, Faculty of Science, Cairo University, Giza-Egypt Email address: eslam@sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='eg Mathematics and Actuarial Science Department (MACT), American University in Cairo (AUC), New Cairo-Egypt Email address: eslammath@aucegypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='edu Ahmad El-Guindy ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 11 Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt Email address: aelguindy@sci.' metadata={'source': 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