diff --git "a/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt" "b/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt" @@ -0,0 +1,1361 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf,len=1360 +page_content='COMPUTING NONSURJECTIVE PRIMES ASSOCIATED TO GALOIS REPRESENTATIONS OF GENUS 2 CURVES BARINDER S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' BANWAIT, ARMAND BRUMER, HYUN JONG KIM, ZEV KLAGSBRUN, JACOB MAYLE, PADMAVATHI SRINIVASAN, AND ISABEL VOGT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric en- domorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely many primes ℓ for which the Galois action on ℓ-torsion points of A is not maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Building on work of Dieulefait, we give a practical algorithm to compute this finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The key inputs are Mitchell’s classification of maximal subgroups of PSp4(Fℓ), sampling of the characteristic polyno- mials of Frobenius, and the Khare–Wintenberger modularity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The algorithm has been submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomor- phism ring in the LMFDB, and the results incorporated into the homepage of each such curve on a publicly-accessible branch of the LMFDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Introduction Let C/Q be a smooth, projective, geometrically integral curve (referred to hereafter as a nice curve) of genus 2, and let A be its Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We assume throughout that A admits no nontrivial geometric endomorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, we assume that End(AQ) = Z, and we refer to any abelian variety satisfying this property as typical1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We also say that a nice curve is typical if its Jacobian is typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let GQ ∶= Gal(Q/Q), let ℓ be a prime, and let A[ℓ] ∶= A(Q)[ℓ] denote the ℓ-torsion points of A(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ρA,ℓ ∶ GQ → Aut(A[ℓ]) denote the Galois representation on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By fixing a basis for A[ℓ], and observing that A[ℓ] admits a nondegenerate Galois-equivariant alternating bilinear form, namely the Weil pairing, we may identify the codomain of ρA,ℓ with the general symplectic group GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In a letter to Vign´eras [Ser00, Corollaire au Th´eor`eme 3], Serre proved an open image theorem for typical abelian varieties of dimensions 2 or 6, or of odd dimension, generalizing his celebrated open image theorem for elliptic curves [Ser72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, the set of nonsurjective primes ℓ for which the representation ρA,ℓ is not surjective — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', the set of primes ℓ for which ρA,ℓ(GQ) is contained in a proper subgroup of GSp4(Fℓ) — is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the elliptic curve case, Serre subsequently provided a conditional upper bound in terms of the conductor of E on this finite set [Ser81, Th´eor`eme 22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' this bound has since been made unconditional [Coj05, Kra95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' There are also algorithms to compute the finite set of nonsurjective primes [Zyw15], and practical implementations in Sage [CL12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Serre’s open image theorem for typical abelian surfaces was made explicit by Dieulefait [Die02] who described an algorithm that returns a finite set of primes containing the set of nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In a different direction Lombardo [Lom16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3] provided an upper bound on the nonsurjective primes involving the stable Faltings height of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 11F80 (primary), 11G10, 11Y16 (secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 1Abelian varieties with extra endomorphisms define a thin set (in the sense of Serre) in Ag and as such are not the typically arising case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='02222v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='NT] 5 Jan 2023 2 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT In this paper we develop Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1, which together allow for the exact determination of the nonsurjective primes for C, yielding our main result as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve whose Jacobian A has conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 produces a finite list PossiblyNonsurjectivePrimes(C) that provably contains all nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For a given bound B > 0, Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 produces a sublist LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) of PossiblyNonsurjectivePrimes(C) that contains all the nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If B is suffi- ciently large, then the elements of LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) are precisely the nonsurjec- tive primes of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The two common ingredients in Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 are Mitchell’s 1914 classification of maximal subgroups of PSp4(Fℓ) [Mit14] and sampling of characteristic polynomials of Frobenius elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Indeed, ρA,ℓ is nonsurjective precisely when its image is contained in one of the proper maximal subgroups of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The (integral) characteristic polynomial of Frobenius at a good prime p is computationally accessible since it is determined by counting points on C over Fpr for small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The reduction of this polynomial modulo ℓ gives the characteristic polynomial of the action of the Frobenius element on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By the Chebotarev density theorem, the images of the Frobenius elements for varying primes p equidistribute over the conjugacy classes of ρA,ℓ(GQ) and hence let us explore the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 makes use of the fact that if the image of ρA,ℓ is nonsurjective, then the character- istic polynomials of Frobenius at auxiliary primes p will be constrained modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using this idea, Dieulefait worked out the constraints imposed by each type of maximal subgroup for ρA,ℓ(GQ) to be contained in that subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Our Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 combines Dieulefait’s conditions, with some modest improvements, to produce a finite list PossiblyNonsurjectivePrimes(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 then weeds out the extraneous surjective primes from PossiblyNonsurjectivePrimes(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Equipped with the prime ℓ, the task here is try to generate enough different elements in the image to rule out containment in any proper maximal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The key input is a purely group-theoretic condition (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2) that guarantees that a subgroup is all of GSp4(Fℓ) if it contains par- ticular types of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This algorithm is probabilistic and depends on the choice of a parameter B which, if sufficiently large, provably establishes nonsurjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The parameter B is a cut-off for the number of Frobenius elements that we use to sample the conjugacy classes of ρA,ℓ(GQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As an illustration of the interplay between theory and practice, analyzing the “worst case” run time of each step in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 yields a new theoretical bound, conditional on the Generalized Riemann Hypothesis (GRH), on the product of all nonsurjective primes in terms of the conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve with conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming the Generalized Riemann Hypothesis (GRH), we have, for any ϵ > 0, ∏ ℓ nonsurjective ℓ ≪ exp(N1/2+ϵ), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' While we believe this bound to be far from asymptotically optimal, it is the first bound in the literature expressed in terms of the (effectively computable) conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Naturally one wants to find the sufficiently large value of B in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(2), which the next result gives, conditional on GRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve, B be a positive integer, and q be the largest prime in LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, the set LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) is precisely the set of nonsurjective primes of C, provided that B ≥ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 3 The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 involves an explicit Chebotarev bound due to Bach and Sorenson [BS96] that is dependent on GRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' An unconditional version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 can be given using an unconditional Chebotarev result (for instance [KW22]), though the bound for B will be exponential in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In addition, if we assume both GRH and the Artin Holomorphy Conjecture (AHC), then a version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 holds with the improved asymptotic bound B ≫ q11 log2(qNA), but without an explicit constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Unfortunately, the bound from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 is prohibitively large to use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By way of illustration, consider the smallest (with respect to conductor) typical genus 2 curve, which has a model y2 + (x3 + 1)y = x2 + x, and label 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 in the L-functions and modular forms database (LMFDB) [LMF22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The output of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 is the set {2,3,5,7,83}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Applying Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 with B = 100 rules out the prime 83, suggesting that 7 is the largest nonsurjective prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Subsequently applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 with q = 7 yields the value B = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='578 × 1023 for which LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) coincides with the set of nonsurjective primes associated with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' With this value of B, our implementation of the algorithm was still running after 24 hours, after which we terminated it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Even if the version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 that relies on AHC could be made explicit, the value of q11 log2(qNA) in this example is on the order of 1011, which would still be a daunting prospect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' To execute the combined algorithm on all typical genus 2 curves in the LMFDB - which at the time of writing constitutes 63,107 curves - we have decided to take a fixed value of B = 1000 in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The combined algorithm then takes about 4 hours on MIT’s Lovelace computer, a machine with 2 AMD EPYC 7713 2GHz processors, each with 64 cores, and a total of 2TB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The result of this computation of nonsurjective primes for these curves is available to view on the homepage of each curve in the LMFDB beta: https://beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='lmfdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='org In addition, the combined algorithm has been run on a much larger set of 1,823,592 curves provided to us by Andrew Sutherland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See Section 6 for the results of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 samples the characteristic polynomial of Frobenius Pp(t) for each prime p of good reduction for the curve up to a particular bound and applies Tests 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5 to Pp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming that ρA,ℓ is surjective, we expect that the outcome of these tests should be independent for sufficiently large primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let C/Q be a typical genus 2 curve with Jacobian A and suppose ℓ is an odd prime such that ρA,ℓ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' There is an effective bound B0 such that for any B > B0, if we sample the characteristic polynomials of Frobenius Pp(t) for n primes p ∈ [B,2B] chosen uniformly and independently at random, the probability that none of these pass Tests 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5 is less than 3⋅( 9 10) n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In fact, for each prime ℓ satisfying the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4, there is an explicit constant cℓ ≤ 9 10 tending to 3 4 as ℓ → ∞ which may be computed using Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 such that bound of 3 ⋅ ( 9 10) n in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 can be replaced by 3 ⋅ cn ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The combined algorithm to probabilistically determine the nonsurjective primes of a nice genus 2 curve over Q has been implemented in Sage [The20], and it will appear in a future release of this software2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Until then, the implementation is available at the following repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='com/ivogt/abeliansurfaces The README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='md file contains detailed instructions on its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This repository also contains other scripts in both Sage and Magma [BCP97] useful for verifying some of the results of this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' any filenames used in the sequel will refer to the above repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2see https://trac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='sagemath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='org/ticket/30837 for the ticket tracking this integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 4 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT Outline of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 2, we begin by reviewing the properties of the characteristic polynomial of Frobenius with a view towards computational aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We also recall the classification of maximal subgroups of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 3, we explain Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 and establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, for each of the maximal subgroups of GSp4(Fℓ) listed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4, we generate a list of primes that provably contains all primes ℓ for which the mod ℓ image of Galois is contained in this maximal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 is also proved in this section (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 4, we first prove a group-theoretic criterion (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2) for a subgroup of GSp4(Fℓ) to equal GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then, for each ℓ in the finite list from Section 3, we ascertain whether the characteristic polynomials of the Frobenius elements sampled satisfy the group-theoretic criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(2) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 also follow from this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In Section 5 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 concerning the probability of output error, assuming that Frobenius elements distribute in ρA,ℓ(GQ) as they would in a randomly chosen element of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Finally, in Section 6, we close with remarks concerning the execution of the algorithm on the large dataset of genus 2 curves mentioned above, and highlight some interesting examples that arose therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This work was started at a workshop held remotely ‘at’ the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, in May 2020, and was supported by a grant from the Simons Foundation (546235) for the collaboration ‘Arithmetic Geometry, Number Theory, and Computation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It has also been supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' DMS-1929284 while the authors were in residence at ICERM during a Collaborate@ICERM project held in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We are grateful to Noam Elkies for providing interesting examples of genus 2 curves in the literature, Davide Lombardo for helpful discussions related to computing geometric endomorphism rings, and to Andrew Sutherland for providing a dataset of Hecke characteristic polynomials that were used for executing our algorithm on all typical genus 2 curves in the LMFDB, as well as making available the larger dataset of approximately 2 million curves that we ran our algorithm on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be an abelian variety of dimension g defined over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By conductor we mean the Artin conductor N = NA of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We write Nsq for the largest integer such that N2 sq ∣ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ℓ be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We write TℓA for the ℓ-adic Tate module of A: TℓA ≃ lim ←� n A[ℓn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This is a free Zℓ-module of rank 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For each prime p, we write Frobp ∈ Gal(Q/Q) for an absolute Frobenius element associated to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By a good prime p for an abelian variety A, we mean a prime p for which A has good reduction, or equivalently p ∤ NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If p is a good prime for A, then the trace ap of the action of Frobp on TℓA is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 for a discussion of the characteristic polynomial of Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By a typical abelian variety A, we mean an abelian variety with geometric endomorphism ring Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A typical genus 2 curve is a nice curve whose Jacobian is a typical abelian surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V be a 4-dimensional vector space over Fℓ endowed with a nondegenerate skew-symmetric bilinear form ⟨⋅,⋅⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A subspace W ⊆ V is called isotropic (for ⟨⋅,⋅⟩) if ⟨w1,w2⟩ = 0 for all w1,w2 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A subspace W ⊆ V is called nondegenerate (for ⟨⋅,⋅⟩) if ⟨⋅,⋅⟩ restricts to a nondegenerate form on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The general symplectic group of (V,⟨⋅,⋅⟩) is defined as GSp(V,⟨⋅,⋅⟩) ∶= {M ∈ GL(V ) ∶ ∃ mult(M) ∈ F× ℓ ∶ ⟨Mv,Mw⟩ = mult(M)⟨v,w⟩ ∀ v,w ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The map M ↦ mult(M) is a surjective homomorphism from GSp(V,⟨⋅,⋅⟩) to F× ℓ called the similitude character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' its kernel is the symplectic group, denoted Sp(V,⟨⋅,⋅⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Usually the bilinear form is understood from the context, in which case one drops ⟨⋅,⋅⟩ from the notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' moreover, for our COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 5 purposes, we will have fixed a basis for V , one in which the bilinear form is represented by the nonsingular skew-symmetric matrix J ∶= ( 0 I2 −I2 0 ), where I2 is the 2 × 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By a subquotient W of a Galois module U, we mean a Galois module W that admits a surjection U ′ ↠ W from a subrepresentation U ′ of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since we are chiefly concerned with computing the sets LikelyNonsurjectivePrimes(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='B) and PossiblyNonsurjectivePrimes(C) for a fixed curve C, we will henceforth, for ease of notation, drop the C from the notation for these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Integral characteristic polynomial of Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The theoretical result underlying the whole approach is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 (Weil, see [ST68, Theorem 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be an abelian variety of dimension g defined over Q and let p be a prime of good reduction for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then there exists a monic integral polynomial Pp(t) ∈ Z[t] of degree 2g with constant coefficient pg such that for any ℓ ≠ p, the polynomial Pp(t) modulo ℓ is the characteristic polynomial of the action of Frobp on TℓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Furthermore, every root of Pp(t) has complex absolute value p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The polynomials Pp(t) are computationally accessible by counting points on C over Fpr r = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See [Poo17, Chapter 7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In fact, Pp(t) can be accessed via the frobenius_ polynomial command in Sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, we denote the trace of Frobenius by ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By the Grothendieck-Lefschetz trace formula, if A = JacX, p is a prime of good reduction for X, and λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=',λ2g are the roots of Pp(t), then #X(Fpr) = pr + 1 − 2g ∑ i=1 λr i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The Weil pairing and consequences on the characteristic polynomial of Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The nondegenerate Weil pairing gives an isomorphism (of Galois modules): (1) TℓA ≃ (TℓA)∨ ⊗Zℓ Zℓ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The Galois character acting on Zℓ(1) is the ℓ-adic cyclotomic character, which we denote by cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The integral characteristic polynomial for the action of Frobp on Zℓ(1) is simply t−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The integral characteristic polynomial for the action of Frobp on (TℓA)∨ is the reversed polynomial P ∨ p (t) = Pp(1/t) ⋅ t2g/pg whose roots are the inverses of the roots of Pp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now record a few easily verifiable consequences of the nondegeneracy of the Weil pairing when dim(A) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (i) The roots of Pp(t) come in pairs that multiply out to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, Pp(t) has no root with multiplicity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (ii) Pp(t) = t4 − apt3 + bpt2 − papt + p2 for some ap,bp ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (iii) If the trace of an element of GSp4(Fℓ) is 0 mod ℓ, then its characteristic polynomial is re- ducible modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, this applies to Pp(t) when ap ≡ 0 (mod ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (iv) If A[ℓ] is a reducible GQ-module, then Pp(t) is reducible modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Parts (i) and (ii) are immediate from the fact that the non-degenerate Weil pairing allows us to pair up the four roots of Pp(t) into two pairs that each multiply out to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 6 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT For part (iii), suppose that M ∈ GSp4(Fℓ) has tr(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then the characteristic polynomial PM(t) of M is of the form t4 +bt2 +c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When the discriminant of PM is 0 modulo ℓ, the polynomial PM has repeated roots and is hence reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' So assume that the discriminant of PM is nonzero modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When ℓ ≠ 2, the result follows from [Car56, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When ℓ = 2, a direct computation shows that the characteristic polynomial of a trace 0 element of GSp4(F2) is either (t + 1)4 or (t2 + t + 1)2, which are both reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Part (iv) is immediate from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 since Pp(t) mod ℓ by definition is the characteristic polynomial for the action of Frobp on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Maximal subgroups of GSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Mitchell [Mit14] classified the maximal subgroups of PSp4(Fℓ) in 1914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This can be used to deduce the following classification of maximal subgroups of GSp4(Fℓ) with surjective similitude character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 (Mitchell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V be a 4-dimensional Fℓ-vector space endowed with a nondegener- ate skew-symmetric bilinear form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then any proper subgroup G of GSp(V,ω) with surjective similitude character is contained in one of the following types of maximal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Reducible maximal subgroups (a) Stabilizer of a 1-dimensional isotropic subspace for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) Stabilizer of a 2-dimensional isotropic subspace for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) Irreducible subgroups governed by a quadratic character Normalizer Gℓ of the group Mℓ that preserves each summand in a direct sum decomposition V1 ⊕ V2 of V , where V1 and V2 are jointly defined over Fℓ and either: (a) both nondegenerate for ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' or (b) both isotropic for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, Mℓ is an index 2 subgroup of Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Stabilizer of a twisted cubic GL(W) acting on Sym3 W ≃ V , where W is a 2-dimensional Fℓ-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (4) Exceptional subgroups See Table A for explicit generators for the groups described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (a) When ℓ ≡ ±3 (mod 8): a group whose image G1920 in PGSp(V,ω) has order 1920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) When ℓ ≡ ±5 (mod 12) and ℓ ≠ 7: a group whose image G720 in PGSp(V,ω) has order 720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (c) When ℓ = 7: a group whose image G5040 in PGSp(V,ω) has order 5040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We have chosen to label the maximal subgroups in the classification using invariant subspaces for the symplectic pairing ω on V , following the more modern account due to Aschbacher (see [Lom16, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' for a more comprehensive treatment see [KL90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For the convenience of the reader, we record the correspondence between Mitchell’s original labels and ours below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Mitchell’s label Label in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 Group having an invariant point and plane 1a Group having an invariant parabolic congruence 1b Group having an invariant hyperbolic or elliptic congruence 2a Group having an invariant quadric 2b Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Dictionary between maximal subgroup labels in [Die02]/[Mit14] and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The maximal subgroups in (1) are the analogues of the Borel subgroup of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The maximal subgroups in (2) when the two subspaces V,V ′ in the direct sum decomposition are individually defined over Fℓ are the analogues of normalizers of the split Cartan subgroup of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' When the two subspaces V,V ′ are not individually defined over Fℓ instead, the maximal subgroups in (2) are analogues of the normalizers of the non-split Cartan subgroups of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 7 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We briefly explain why the action of GL2(Fℓ) on Sym3(F2 ℓ) preserves a nondegenerate symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It suffices to show that the restriction to SL2(Fℓ) fixes a vector in ⋀2 Sym3(F2 ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This follows by character theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If W is the standard 2-dimensional representation of SL2, then we have ⋀2(Sym3 W) ≃ Sym4 W ⊕ 1 as representations of SL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' One can extract explicit generators of the exceptional maximal subgroups from Mitchell’s original work3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Indeed [Mit14, the proof of Theorem 8, page 390] gives four explicit matrices that generate a G1920 (which is unique up to conjugacy in PGSp4(Fℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Mitchell’s description of the other exceptional groups is in terms of certain projective linear transformations called skew perspec- tivities attached to a direct sum decomposition V = V1 ⊕ V2 into 2-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A skew perspectivity of order n with axes V1 and V2 is the projective linear transformation that scales V1 by a primitive nth root of unity and fixes V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This proof also gives the axes of the skew perspectivities of order 2 and 3 that generate the remaining exceptional groups [Mit14, pages 390-391].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Table 5 lists generators of (one representative of the conjugacy class of) each of the exceptional maximal subgroup extracted from Mitchell’s descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the file exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='m publicly available with our code, we verify that Magma’s list of conjugacy classes of maximal subgroups of GSp4(Fℓ) agree with those described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 for 3 ≤ ℓ ≤ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The classification of exceptional maximal subgroups of PSp4(Fℓ) is more subtle than that of PGSp4(Fℓ), because of the constraint on the similitude character of matrices in PSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' While the similitude character is not well-defined on PGSp4(Fℓ) (multiplication by a scalar c ∈ F× ℓ scales the similitude character by c2) it is well-defined modulo squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The group PSp4(Fℓ) is the kernel of this natural map: 1 → PSp4(Fℓ) → PGSp4(Fℓ) mult ��→ F× ℓ /(F× ℓ )2 ≃ {±1} → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' An exceptional subgroup of PGSp4(Fℓ) gives rise to an exceptional subgroup of PSp4(Fℓ) of either the same size or half the size depending on the image of mult restricted to that subgroup, which in turn depends on the congruence class of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For this reason, the maximal exceptional subgroups of PSp4(Fℓ) in Mitchell’s original classification (also recalled in Dieulefait [Die02, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1]) can have order 1920 or 960 and 720 or 360 depending on the congruence class of ℓ, and 2520 (for ℓ = 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Such an exceptional subgroup gives rise to a maximal exceptional subgroup of PGSp4(Fℓ) only when mult is surjective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', its intersection with PSp4(Fℓ) is index 2), which explains the restricted congruence classes of ℓ for which they arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now record a lemma that directly follows from the structure of maximal subgroups described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This lemma will be used in Section 4 to devise a criterion for a subgroup of GSp4(Fℓ) to be the entire group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For an element T in GSp4(Fℓ), let tr(T), mid(T), mult(T) denote the trace of T, the middle coefficient of the characteristic polynomial of T, and the similitude character applied to T respectively4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For a scalar λ, we have tr(λT) = λtr(T), mid(λT) = λ2 mid(T), mult(λT) = λ2 mult(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence the quantities tr(T)2/mult(T) and mid(T)/mult(T) are well-defined on PGSp4(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For ℓ > 2 and ∗ ∈ {720,1920,5040}, define (2) Cℓ,∗ ∶= {( tr(T)2 mult(T), mid(T) mult(T)) ∣ T ∈ an exceptional subgroup of GSp4(Fℓ) of projective order ∗} Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) In cases 2a and 2b of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3: 3Mitchell’s notation for PGSp4(Fℓ) is Aν(ℓ) and for PSp4(Fℓ) is A1(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 4Explicitly, the characteristic polynomial of T is therefore t4 − tr(T)t3 + mid(T)t2 − mult(T) tr(T)t + mult(T)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 8 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT (a) every element in Gℓ ∖ Mℓ has trace 0, and, (b) the group Mℓ stabilizes a non-trivial linear subspace of F 4 ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) Every element that is contained in a maximal subgroup corresponding to the stabilizer of a twisted cubic has a reducible characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) For ∗ ∈ {1920,720}, the set Cℓ,∗ defined in (2) equals the reduction modulo ℓ of the elements of the set C∗ below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' C1920 = {(0,−2),(0,−1),(0,0),(0,1),(0,2),(1,1),(2,1),(2,2),(4,2),(4,3),(8,4),(16,6)} C720 = {(0,1),(0,0),(4,3),(1,1),(16,6),(0,2),(1,0),(3,2),(0,−2)} We also have C7,5040 = {(0,0),(0,1),(0,2),(0,5),(0,6),(1,0),(1,1),(2,6),(3,2),(4,3),(5,3),(6,3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) In cases 2a and 2b of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, since any element of the normalizer Gℓ that is not in Mℓ switches elements in the two subspaces V1 and V2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' maps elements in the subspace V1 in the decomposition V1 ⊕ V2 to elements in V2 and vice-versa), it follows that any element in Gℓ ∖ Mℓ has trace zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) The conjugacy class of maximal subgroups corresponding to the stabilizer of a twisted cubic comes from the embedding GL2(Fℓ) ι�→ GSp4(Fℓ) induced by the natural action of GL2(Fℓ) on the space of monomials of degree 3 in 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If M is a matrix in GL2(Fℓ) with eigenvalues λ,µ (possibly repeated), then the eigenvalues of ι(M) are λ3,µ3,λ2µ,λµ2 and hence the characteristic polynomial of ι(M) factors as (T 2 −(λ3 +µ3)T +λ3µ3)(T 2 −(λ2µ+ λµ2)T + λ3µ3) over Fℓ which is reducible over Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) This follows from the description of the maximal subgroups given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Each case (except G5040 that only occurs for ℓ = 7) depends on a choice of a root of a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the file exceptional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='sage, we generate the corresponding finite subgroups over the appropriate quadratic number field to compute C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It follows that the corresponding values for the subgroup G∗ in GSp4(Fℓ) can be obtained by reducing these values modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the group G5040 only appears for ℓ = 7, we directly compute the set C7,5040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4(3) is the analogue of the condition [Ser72, Proposition 19 (iii)] used to rule out exceptional maximal subgroups of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We end this subsection by including the following lemma, to further highlight the similarities between the above classification of maximal subgroups of GSp4(Fℓ) and the more familiar classi- fication of maximal subgroups of GL2(Fℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This lemma is not used elsewhere in the article and is thus for expositional purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are indi- vidually defined over Fℓ is isomorphic to {(m1,m2) ∈ GL2(Fℓ)2 ∣ det(m1) = det(m2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ2 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are individually defined over Fℓ is isomorphic to {(m1,m2) ∈ GL2(Fℓ)2 ∣ mT 1 m2 = λI, for some λ ∈ F∗ ℓ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ(ℓ − 1)2(ℓ2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 9 (3) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are not individually defined over Fℓ is isomorphic to {m ∈ GL2(Fℓ2) ∣ det(m) ∈ F∗ ℓ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ4 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (4) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are not indi- vidually defined over Fℓ is isomorphic to GU2(Fℓ2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', {m ∈ GL2(Fℓ2) ∣ mT ι(m) = λI, for some λ ∈ F∗ ℓ }, where ι denotes the natural extension of the Galois automorphism of Fℓ2/Fℓ to GL2(Fℓ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, the order of Mℓ is ℓ(ℓ2 − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a direct sum decomposition V1 ⊕ V2 of a vector space V over Fq, we get a natural embedding of Aut(V1) × Aut(V2) (≅ GL2(Fq)2) into Aut(V ) (≅ GL4(Fq)), whose image consists of automorphisms that preserve this direct sum decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will henceforth refer to elements of Aut(V1) × Aut(V2) as elements of Aut(V ) using this embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' To understand the subgroup Mℓ of GSp4(Fq) in cases (1) and (2) where the two subspaces in the direct sum decomposition are individually defined over Fq, we need to further impose the condition that the automorphisms in the image of the map Aut(V1) × Aut(V2) → Aut(V ) preserve the symplectic form ω on V up to a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In (1), without any loss of generality, the two nondegenerate subspaces V1 and V2 can be chosen to be orthogonal complements under the nondegenerate pairing ω, and so by Witt’s theorem, in a suitable basis for V1⊕V2 obtained by concatenating a basis of V1 and a basis of V2, the nondegenerate symplectic pairing ω has the following block-diagonal shape: B ∶= ⎡⎢⎢⎢⎢⎢⎢⎢⎣ 0 1 −1 0 0 1 −1 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which boils down to det(m1) = λ = det(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Similarly, in (2), without any loss of generality, by Witt’s theorem, in a suitable basis for V1 ⊕V2 obtained by concatenating a basis of the isotropic subspace V1 and a basis of the isotropic subspace V2, the nondegenerate symplectic pairing ω has the following block-diagonal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' B ∶= ⎡⎢⎢⎢⎢⎢⎢⎢⎣ 0 1 1 0 0 −1 −1 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which again boils down to mT 1 m2 = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If we have a subspace W defined over Fq2 but not defined over Fq, and we let W denote the conjugate subspace and further assume that W ⊕W gives a direct sum decomposition of V , then we get a natural embedding of Aut(W) (≅ GL2(Fq2)) into Aut(V ) (≅ GL4(Fq)) whose image consists of automorphisms that commute with the natural involution of V ⊗ Fq2 induced by the Galois automorphism of Fq2 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The proofs of cases (3) and (4) are analogous to the cases (1) and (2) respectively, by using the direct sum decomposition W ⊕W and letting m2 = ι(m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The condition that det(m1) = det(m2) in (1) becomes the condition det(m1) = det(m2) = detm1 = det(m1), or 10 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT equivalently, that det(m1) ∈ Fq in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Similarly, the condition that mT 1 m2 = λI in (2) becomes the condition that mT 1 ι(m1) = λI in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Image of inertia and (tame) fundamental characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Dieulefait [Die02] used Mitchell’s work described in the previous subsection to classify the maximal subgroups of GSp4(Fℓ) that could occur as the image of ρA,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This was achieved via an application of a fundamental result of Serre and Raynaud that strongly constrains the action of inertia at ℓ, and which we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Fix a prime ℓ > 3 that does not divide the conductor N of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let Iℓ be an inertia subgroup at ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ψn∶Iℓ → F× ℓn denote a (tame) fundamental character of level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The n Galois-conjugate fundamental characters ψn,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=',ψn,n of level n are given by ψn,i ∶= ψℓi n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Recall that the fundamental character of level 1 is simply the mod ℓ cyclotomic character cycℓ, and that the product of all fundamental characters of a given level is the cyclotomic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6 (Serre [Ser72], Raynaud [Ray74], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' [Die02][Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ℓ be a semistable prime for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V /Fℓ be an n-dimensional Jordan–H¨older factor of the Iℓ-module A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then V admits a 1-dimensional Fℓn-vector space structure such that ρA,ℓ∣Iℓ acts on V via the character ψd1 n,1⋯ψdn n,n with each di equal to either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' On the other hand, the following fundamental result of Grothendieck constrains the action of inertia at semistable primes p ≠ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7 (Grothendieck [GRR72, Expos´e IX, Prop 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be an abelian variety over a number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then A has semistable reduction at p ≠ ℓ if and only if the action of Ip ⊂ GK on TℓA is unipotent of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Combining these two results allows one fine control of the determinant of a subquotient of A[ℓ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' this will be used in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A/Q be an abelian surface, and let Xℓ be a Jordan–H¨older factor of the Fℓ[GQ]- module A[ℓ] ⊗ Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ is a semistable prime, then detXℓ ≃ ϵ ⋅ cycx ℓ for some character ϵ∶GQ → Fℓ that is unramified at ℓ and some 0 ≤ x ≤ dimXℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, ϵ120 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The first part follows immediately from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For the fact that ϵ120 = 1, every abelian surface attains semistable reduction over an extension K/Q with [K ∶ Q] dividing 120 by [LV14a, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2], and so this follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7 since there are no nontrivial unramified characters of GQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ We can now state Dieulefait’s classification of maximal subgroups of GSp4(Fℓ) that can occur as the image ρA,ℓ(GQ) for a semistable prime ℓ > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9 ([Die02]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A be the Jacobian of a genus 2 curve defined over Q with Weil pairing ω on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ > 7 is a semistable prime, then ρA,ℓ(GQ) is either all of GSp(A[ℓ],ω) or it is contained in one of the maximal subgroups of Types (1) or (2) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' See also [Lom16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='15] for an expanded exposition of why the image of GQ cannot be contained in maximal subgroup of Type (3) for a semistable prime ℓ > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' However, if ℓ is a prime of additive reduction, or if ℓ ≤ 7, then the image of GQ may also be contained in any of the four types of maximal subgroups described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Nevertheless, by [LV22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6], for any prime ℓ > 24, we have that the exponent of the projective image is bounded exp(PρA,ℓ) ≥ (ℓ−1)/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since exp(G1920) = 2exp(S6) = 120 and exp(G720) = exp(S5) = 60, the exceptional maximal subgroups cannot occur as ρA,ℓ(GQ) for ℓ > 1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' A consequence of the Chebotarev density theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let K/Q be a finite Galois exten- sion with Galois group G = Gal(K/Q) and absolute discriminant dK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let S ⊆ G be a nonempty subset that is closed under conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By the Chebotarev density theorem, we know that (3) lim x→∞ ∣{p ≤ x ∶ p is unramified in K and Frobp ∈ S}∣ ∣{p ≤ x}∣ = ∣S∣ ∣G∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be the least prime such that p is unramified in K and Frobp ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' There are effective versions of the Chebotarev density theorem that give bounds on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The best known unconditional bounds are polynomial in dK [LMO79, AK19, KW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Under GRH, the best known bounds are polynomial in log dK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular Bach and Sorenson [BS96] showed that under GRH, (4) p ≤ (4log dK + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5[K ∶ Q] + 5)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The present goal is to give an effective version of the Chebotarev density theorem in the context of abelian surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will use a corollary of (4) that is noted in [MW21] which allows for the avoidance of a prescribed set of primes by taking a quadratic extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We do this because we will take K = Q(A[ℓ]), and p being unramified in K is not sufficient to imply that p is a prime of good reduction for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lastly, we will use that by [Ser81, Proposition 6], if K/Q is finite Galois, then (5) log dK ≤ ([K ∶ Q] − 1)log rad(dK) + [K ∶ Q]log([K ∶ Q]), where radn = ∏p∣n p denotes the radical of an integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A/Q be a typical principally polarized abelian surface with conductor NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let q be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let S ⊆ ρA,q(GQ) be a nonempty subset that is closed under conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be the least prime of good reduction for A such that p ≠ q and ρA,q(Frobp) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, we have p ≤ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let K = Q(A[q]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then K/Q is Galois and [K ∶ Q] ≤ ∣GSp4(Fq)∣ = q4(q4 − 1)(q2 − 1)(q − 1) ≤ q11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As raddK is the product of primes that ramify in Q(A[q]), the criterion of N´eron-Ogg-Shafarevich for abelian varieties [ST68, Theorem 1] implies that rad(dK) divides rad(qNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ˜K ∶= K(√m) where m ∶= rad(2NA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Note that the primes that ramify in ˜K are precisely 2, q, and the primes of bad reduction for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Thus rad(d ˜ K) = rad(2qNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover [ ˜K ∶ Q] ≤ 2q11 and by (5), log(d ˜ K) ≤ (2q11 − 1)log rad(2qNA) + 22q11 log(2q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Applying [MW21, Corollary 6] to the field ˜K, we get that (under GRH) there exists a prime p satisfying the claimed bound, that does not divide m, and for which ρA,q(Frobp) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Finding a finite set containing all nonsurjective primes In this section we describe Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 referenced in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This algorithm produces a finite list PossiblyNonsurjectivePrimes that provably includes all nonsurjective primes ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We also prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since our goal is to produce a finite list (from which we will later remove extraneous primes) it is harmless to include the finitely many bad primes as well as 2,3,5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9, it suffices to find conditions on ℓ > 7 for which ρA,ℓ(GQ) could be contained in one of the maximal subgroups of type (1) and (2) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We first find primes ℓ for which ρA,ℓ has (geometrically) reducible image (and hence is contained in a maximal subgroup in case (1) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 or in a subgroup Mℓ in case (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' To treat the geometrically irreducible cases, we then make use of the observation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 1a that every element outside of an index 2 subgroup has trace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 12 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 curve C/Q with conductor N and Jacobian A, compute a finite list PossiblyNonsurjectivePrimes of primes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Initialize PossiblyNonsurjectivePrimes = [2,3,5,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) Add to PossiblyNonsurjectivePrimes all primes dividing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗ Fℓ could be reducible via Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (4) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗Fℓ could be irreducible but nonsurjective via Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (5) Return PossiblyNonsurjectivePrimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' At a very high-level, each of the subalgorithms of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1 makes use of a set of auxiliary good primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We compute the integral characteristic polynomial of Frobenius Pp(t) and use it to constrain those ℓ ≠ p for which the image could have a particular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Even though robust methods to compute the conductor N of a genus 2 curve are not implemented at the time of writing, the odd-part Nodd of N can be computed via genus2red function of PARI and the genus2reduction module of SageMath, both based on an algorithm of Liu [Liu94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, [BK94, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2] bounds the 2-exponent of N above by 20 and hence N can be bounded above by 220Nodd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' While these algorithms can be run only with the bound 220Nodd, it will substantially increase the run-time of the limiting Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now explain each of these steps in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Good primes that are not geometrically irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this section we describe the conditions that ℓ must satisfy for the base-extension A[ℓ] ∶= A[ℓ] ⊗Fℓ Fℓ to be reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this case, the representation A[ℓ] is an extension (6) 0 → Xℓ → A[ℓ] → Yℓ → 0 of a (quotient) representation Yℓ by a (sub) representation Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Recall that Nsq denotes the largest square divisor of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ℓ be a prime of good reduction for A and suppose that A[ℓ] sits in sequence (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p ≠ ℓ be a good prime for A and let f denote the order of p in (Z/NsqZ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then there exists 0 ≤ x ≤ dimXℓ and 0 ≤ y ≤ dimYℓ such that Frobgcd(f,120) p acts on detXℓ by pgcd(f,120)x, respectively on detYℓ by pgcd(f,120)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since ℓ is a good prime and Xℓ is composed of Jordan–H¨older factors of A[ℓ], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8 constrains its determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We have detXℓ = ϵcycx ℓ for some character ϵ∶GQ → Fℓ unramified at ℓ, and 0 ≤ x ≤ dimXℓ, and ϵ120 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence Frob120 p acts on detXℓ by cycℓ(Frobp)120x = p120x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In fact, we can do slightly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since detA[ℓ] ≃ cyc2 ℓ, we have detYℓ ≃ ϵ−1 cyc2−x ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the conductor is multiplicative in extensions, we conclude that cond(ϵ)2 ∣ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By class field theory, the character ϵ factors through (Z/cond(ϵ)Z)×, and hence through (Z/NsqZ)×, sending Frobp to p (mod Nsq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since pf ≡ 1 (mod Nsq), we have that ϵ(Frobp)gcd(f,120) = 1, and we see that Frobgcd(f,120) p acts on detXℓ by pgcd(f,120)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Exchanging the roles of Xℓ and Yℓ, we deduce the analogous statement for Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ This is often enough information to find all ℓ for which A[ℓ] has a nontrivial subquotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Namely, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1, every root of Pp(t) has complex absolute value p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Thus the gcd(f,120)-th power of each root has complex absolute value pgcd(f,120)/2, and hence is never integrally equal to 1 or pgcd(f,120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 guarantees that this equality must hold modulo ℓ for any good prime ℓ for which A[ℓ] is reducible with a 1-dimensional subquotient, we always get a nontrivial condition on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Some care must be taken to rule out ℓ for which A[ℓ] only has 2-dimensional subquotient(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Odd-dimensional subquotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be a good prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a polynomial P(t) and an integer f, write P (f)(t) for the polynomial whose roots are the fth powers of roots of P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Universal formulas for such polynomials in terms of the coefficients of P(t) are easy to compute, and are implemented in our code in the case where P is a degree 4 polynomial whose roots multiply in pairs to pα, and f ∣ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of p in (Z/NsqZ)× and write f′ = gcd(f,120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Compute an integer Modd as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Choose a nonempty finite set T of auxiliary good primes p ∤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each p, compute Rp ∶= P (f′) p (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Let Modd = gcdp∈T (pRp) over all auxiliary primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Modd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] has an odd-dimensional subrepresentation is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since A[ℓ] is 4-dimensional and has an odd-dimensional subrepresentation, it has a 1- dimensional subquotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' For any p ∈ T , Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 shows that Frobf′ p acts on detXℓ by either pf′ or by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Thus, the action of Frobf′ p on A[ℓ] has an eigenvalue that is congruent to pf′ or 1 modulo ℓ, and so P (f′) p (t) has a root that is congruent to 1 or pf′ modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the roots of P (f′)(t) multiply in pairs to pf′, we have P (f′) p (pf′) = p2f′P (f′) p (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence ℓ divides p ⋅ P (f′) p (1) = pRp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1, we can give a theoretical bound on the “worst case” of this step of the algorithm using only one auxiliary prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Of course, taking the greatest common divisor over multiple auxiliary primes will likely remove extraneous factors, and in practice this step of the algorithm runs substantially faster than other steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if p is any good prime for A, then 0 ≠ ∣Modd∣ ≪ p240 where the implied constant is absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This follows from the fact that the coefficient of ti in P (f′) p (t) has magnitude on the order of p(2−i)f′ and f′ ≤ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Two-dimensional subquotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now assume that A[ℓ] is reducible, but does not have any odd-dimensional subquotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, it has an irreducible subrepresentation Xℓ of dimension 2, with irreducible quotient Yℓ of dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If A[ℓ] is reducible but indecomposable, then Xℓ is the unique subrepresentation of A[ℓ] and Y ∨ ℓ ⊗ cycℓ is the unique subrepresentation of A[ℓ] ∨ ⊗ cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The isomorphism TℓA ≃ (TℓA)∨ ⊗ cycℓ from (1) yields an isomorphism A[ℓ] ≃ (A[ℓ])∨ ⊗ cycℓ and hence Xℓ ≃ Y ∨ ℓ ⊗ cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Otherwise, A[ℓ] ≃ Xℓ ⊕ Yℓ and so the nondegeneracy of the Weil pairing gives Xℓ ⊕ Yℓ ≃ (X∨ ℓ ⊗ cycℓ) ⊕ (Y ∨ ℓ ⊗ cycℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Therefore either: (a) Xℓ ≃ Y ∨ ℓ ⊗ cycℓ and Yℓ ≃ X∨ ℓ ⊗ cycℓ, or (b) Xℓ ≃ X∨ ℓ ⊗ cycℓ and Yℓ ≃ Y ∨ ℓ ⊗ cycℓ and A[ℓ] ≃ Xℓ ⊕ Yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We call the first case related 2-dimensional subquotients and the second case self-dual 2-dimensional subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will see that the ideas of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2 easily extend to treat the related subquotient case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' we will use the validity of Serre’s conjecture to treat the self-dual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In the 14 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT case that A[ℓ] is decomposable, the above two cases correspond respectively to the index 2 subgroup Mℓ in cases (2a) (the isotropic case) and (2b) (the nondegenerate case) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Related two-dimensional subquotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p be a good prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let Pp(t) ∶= t4−at3+bt2−pat+p2 be the characteristic polynomial of Frobp acting on A[ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Suppose that α and β are the eigenvalues of Frobp acting on the subrepresentation Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then, since Xℓ ≃ Y ∨ ℓ ⊗ cycℓ, the eigenvalues of the action of Frobp on Yℓ are p/α and p/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The action of Frobp on detXℓ is therefore by a product of two of the roots of Pp(t) that do not multiply to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Note that there are four such pairs of roots of Pp(t) that do not multiply to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let Qp(t) be the quartic polynomial whose roots are the products of pairs of roots of Pp(t) that do not multiply to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By design, the roots of Qp(t) have complex absolute value p, but are not equal to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (It is elementary to work out that Qp(t) = t4 − (b − 2p)t3 + p(a2 − 2b + 2p)t2 − p2(b − 2p)t + p4 and is a quartic whose roots multiply in pairs to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=') Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of p in (Z/NsqZ)× and write f′ = gcd(f,120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Compute an integer Mrelated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Choose a finite set T of auxiliary good primes p ∤ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each p, compute the product Rp ∶= Q(f′) p (1)Q(f′) p (pf′) (3) Let Mrelated = gcdp∈T (pRp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Mrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] has related two-dimensional subquotients is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proceed similarly as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4 — in particular, ℓ divides Q(f′) p (1), Q(f′) p (pf′), or Q(f′) p (p2f′) and hence ℓ divides pRp since Q(f′) p (p2f′) = p4f′Q(f′) p (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ A theoretical “worst case” analysis yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if q is the smallest surjective prime for A, then a good prime p for which Rp is nonzero is bounded by a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, p ≪ q22 log2(qN), where the implied constants are absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, for such a prime p, ∣Mrelated∣ ≪ p961 ≪ q21142 log1922(qN), where the implied constants are absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' By Serre’s open image theorem for genus 2 curves, such a prime q exists, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10, the prime p can be chosen such that Rp is nonzero modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Finally, Mrelated ≤ pRp = pQ(f′)(1)Q(f′)(pf′) ≪ p8f′+1 ≪ p961, since the coefficient of ti in Q(f′)(t) has magnitude on the order of p(4−i)f′ and f′ ≤ 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Self-dual two-dimensional subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this case, both subrepresentations Xℓ and Yℓ are absolutely irreducible 2-dimensional Galois representations with determinant the cyclotomic character cycℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' It follows that the representations are odd (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', the determinant of complex con- jugation is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=') Therefore, by the Khare–Wintenberger theorem (formerly Serre’s conjecture on the modularity of mod-ℓ Galois representations) [Kha06, KW09a, KW09b], both Xℓ and Yℓ are modular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, for i = 1,2, there exist newforms fi ∈ Snew ki (Γ1(Ni),ϵi) such that Xℓ ≅ ρf1,ℓ and Yℓ ≅ ρf2,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Furthermore, by the multiplicativity of Artin conductors, we obtain the divisibility N1N2 ∣ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Both f1 and f2 have weight two and trivial Nebentypus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' that is, k1 = k2 = 2, and ϵ1 = ϵ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6, we have that Xℓ∣Iℓ and Yℓ∣Iℓ must each be conjugate to either of the following subgroups of GL2(Fℓ): (1 ∗ 0 cycℓ ) or (ψ2 0 0 ψℓ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The assertion of weight 2 now follows from [Ser87, Proposition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (Alternatively, one may use Proposition 4 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', observing that Xℓ and Yℓ are finite and flat as group schemes over Zℓ because ℓ is a prime of good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=') From Section 1 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=', the Nebentypus ϵi of fi satisfies, for all p ∤ ℓN, detXℓ(Frobp) = p ⋅ ϵi(p), where this equality is viewed inside F × ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The triviality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ We therefore have newforms fi ∈ Snew 2 (Γ0(Ni)) such that (7) A[ℓ] ≃ ρf1,ℓ ⊕ ρf2,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We may assume without loss of generality that N1 ≤ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let p ∤ N be an auxiliary prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We obtain from equation (7) that the integral characteristic polynomial of Frobenius factors: Pp(t) ≡ (t2 − ap(f1)t + p)(t2 − ap(f2)t + p) mod ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' here we use the standard property that, for f a normalised eigenform with trivial Nebentypus, ρf,ℓ(Frobp) satisfies the polynomial equation t2 − ap(f)t + p for p ≠ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, we have Res(Pp(t),t2 − ap(f1)t + p) ≡ 0 mod ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This serves as the basis of the algorithm to find all primes ℓ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mself-dual as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Compute the set S of divisors d of N with d ≤ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each d ∈ S: (a) compute the Hecke L-polynomial Qd(t) ∶= ∏ f (t2 − ap(f)t + p), where the product is taken over the finitely many newforms in Snew 2 (Γ0(d));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) choose a finite set T of auxiliary primes p ∤ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (c) for each auxiliary prime p, compute the resultant Rp(d) ∶= Res(Pp(t),Qd(t));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 16 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT (d) Take the greatest common divisor M(d) ∶= gcd p∈T (pRp(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Let Mself-dual ∶= ∏d∈S M(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Mself-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] has self-dual two-dimensional subrepresenta- tions is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ is in T for any d ∈ S, then ℓ is in the output because Mself-dual is a multiple of M(d) which in turn is a multiple of any element of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Otherwise, as explained before Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10, there is some N1 ∈ S and some newform f1 ∈ Snew 2 (Γ0(N1)) such that Res(Pp(t),t2 − apf1t + p) ≡ 0 (mod ℓ) for every p ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, Rp(N1) ≡ 0 (mod ℓ), so �� divides M(N1) and Mself-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ We can again do a “worst case” theoretical analysis of this algorithm to conclude the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As this indicates, this is by far the limiting step of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if q is the smallest surjective prime for A, then a good prime p for which Rp(d) is nonzero is bounded by a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, p ≪ q22 log2(qN), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, for such a prime p, we have ∣Rp(d)∣ ≪ (2p1/2)8 dim Snew 2 (Γ0(d)) ≪ (4p)(d+1)/3, and so all together ∣Mself-dual∣ ≪ (4q)N1/2+ϵ, where the implied constants are absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' As in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8, we use Serre’s open image theorem and the Effective Chebotarev Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If Rp(d) is zero integrally, then in particular Rp(d) ≡ 0 (mod q) and Pp(t) is reducible modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since GSp4(Fq) contains elements that do not have reducible characteristic polynomial, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10 implies that such elements are the image of Frobp for p bounded as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The resultant Rp(d) is the product of the pairwise differences of the roots of Pp(t) and Qd(t), which all have complex absolute value p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence the pairwise differences have absolute value at most 2p1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover dimSnew 2 (Γ0(d)) ≤ (d + 1)/12 by [Mar05, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since there are 8dimSnew 2 (Γ0(d)) such terms multiplied to give Rp(d), the bound for Rp(d) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since Mself-dual = ∏ d∣N d≤ √ N pRp(d), it suffices to bound ∑ d∣N d≤ √ N d + 4 3 ≤ ∑ d∣N d≤ √ N √ N + 4 3 ≤ σ0(N) √ N + 4 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since σ0(N) ≪ Nϵ by [Apo76, (31) on page 296], we obtain the claimed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The polynomial Qd(t) in step (2) of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10 is closely related to the charac- teristic polynomial Hd(t) of the Hecke operator Tp acting on the space S2(Γ0(d)), which may be computed via modular symbols computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' One may recover Qd(t) from Hd(t) by first homoge- nizing H with an auxiliary variable z (say) to obtain Hd(t,z), and setting t = 1+pz2 (an observation we made in conjunction with Joseph Wetherell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In our computation of nonsurjective primes for the database of genus 2 curves with conductor at most 220 (including those in the LMFDB), we only needed to use polynomials Qd(t) for level up to 210 (since step (1) of the Algorithm has a √ N term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We are grateful to Andrew Sutherland for providing us with a precomputed dataset for these levels resulting from the creation of an extensive database of modular forms going well beyond what was previously available [BBB+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 17 Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Our Sage implementation uses two auxiliary primes in Step 2(b) of the above algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Increasing the number of such primes yields smaller supersets at the expense of longer runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Good primes that are geometrically irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let φ be any quadratic Dirichlet char- acter φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Our goal in this subsection is to find all good primes ℓ governed by φ, by which we mean that tr(ρA,ℓ(Frobp)) ≡ ap ≡ 0 mod ℓ whenever φ(p) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We will consider the set of all quadratic Dirichlet character φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using the struc- ture theorem for finite abelian groups and the fact that φ factors through (Z/NZ)×/((Z/NZ)×)2, this set has the structure of an F2-vector space of dimension d(N) ∶= ω(N) + ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ 0 ∶ v2(N) = 0 −1 ∶ v2(N) = 1 0 ∶ v2(N) = 2 1 ∶ v2(N) ≥ 3, where ω(m) denotes the number of prime factors of m and v2(m) is the 2-adic valuation of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In particular, d(N) ≤ ω(N) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mquad as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (1) Compute the set S of quadratic Dirichlet characters φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (2) For each φ ∈ S: (a) Choose a nonempty finite set T of “auxiliary” primes p ∤ N for which ap ≠ 0 and φ(p) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (b) Take the greatest common divisor Mφ ∶= gcd p∈T (pap), over all auxiliary primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' (3) Let Mquad ∶= ∏φ∈S Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Return the list of prime divisors ℓ of Mquad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any good prime ℓ for which A[ℓ] is governed by a quadratic character is returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Suppose that A[ℓ] is governed by the quadratic character φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Then for every good prime p ≠ ℓ for which φ(p) = −1, the prime ℓ must divide the integral trace of Frobenius ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence ℓ divides Mφ and Mquad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13 terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' More precisely, if q is the smallest surjective prime for A, then a good prime p for which φ(p) = −1 and ap is nonzero is bounded by a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, p ≪ 22d(N)q22 log2(qN), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Moreover, we have ∏ φ∈S ∏ ℓ governed by φ ℓ ≪ (23d(N)q33 log3(qN))2−21−d(N) ≪ 26ω(N)q66 log6(qN), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We imitate the proof of [LV14b, Lemma 21] in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let V be the d-dimensional F2-vector space of quadratic Dirichlet characters of modulus N (equivalently, quadratic Galois characters unramified outside of N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let ρV ∶GK → V ∨ denote the representation sending Frobp to the linear functional φ ↦ φ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Since the character for PGSp4(Fq)/PSp4(Fq) is the abelianization 18 BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT of PρA,q, we conclude in the same way as [LV14b, Proof of Lemma 21] that for any α ∈ V ∨, there exists an Xα ∈ GSp4(Fq) with tr(Xα) ≠ 0 such that (α,Xα) is in the image of ρV × ρA,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Apply the effective Chebotarev density theorem to the Galois extension corresponding to ρV × ρA,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This has degree at most 2d(N)∣GSp4(Fq)∣ and is unramified outside of qN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Therefore, assum- ing GRH and combining (4) and (5), there exists a prime pα ≪ 22d(N)q22 log2(qN) for which (α,Xα) = (ρV (Frobpα),ρA,q(Frobpα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let φ be a character not in the kernel of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Any exceptional prime ℓ governed by φ must divide pαapα, which is nonzero because it is nonzero modulo q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' This proves that the algorithm terminates, since every φ is not in the kernel of precisely half of all α ∈ V ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' We now bound the size of the product of all ℓ governed by a character in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ℓ is governed by φ, then ℓ divides the quantity p∣ap∣ ≤ p3/2 ≪ 23d(N)q33 log3(qN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Taking the product over all nonzero α in V (of which there are 2d(N) − 1), each ℓ will show up half the time, so we obtain: ⎛ ⎜⎜⎜ ⎝ ∏ ℓ governed by φ ∈ S ℓ ⎞ ⎟⎟⎟ ⎠ 2d(N)−1 ≪ (23d(N)q33 log3(qN)) 2d(N)−1 , which implies the result by taking the (2d(N)−1)th root of both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ Putting all of these pieces together, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ρA,ℓ is nonsurjective, ℓ > 7, and ℓ ∤ N, then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='9 implies that ρA,ℓ(GQ) must be in one of the maximal subgroups of Type (1) or (2) listed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If it is contained in one of the reducible subgroups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' the subgroups of Type (1), then ρA,ℓ(GQ) (and, hence, ρA,ℓ(GQ) ⊗ Fℓ) is reducible, and so ℓ is added to PossiblyNonsurjectivePrimes in Step (3) by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='7, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' If ρA,ℓ(GQ) is contained in one of the index 2 subgroups Mℓ of an irreducible subgroup of Type (2) listed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, then again ℓ is added to PossiblyNonsurjectivePrimes in Step (3), since Mℓ ⊗ Fℓ is always reducible by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4(1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Hence we may assume that ρA,ℓ(GQ) is contained in one of the irreducible maximal subgroups Gℓ of Type (2) listed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, but not in the index 2 subgroup Mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' The normalizer character GQ ρA,ℓ ��→ Gℓ → Gℓ/Mℓ = {±1} is nontrivial and unramified outside of N, and so it corresponds to a quadratic Dirichlet character φ∶(Z/NZ)× → {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='4(1a) shows that tr(g) = 0 in Fℓ for any g ∈ Gℓ ∖ Mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Consequently, ℓ is governed by φ (in the language of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2), so ℓ is added to PossiblyNonsurjectivePrimes in Step (4) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Bounds on Serre’s open image theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' In this section we combine the theoretical worst case bounds in the Algorithms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='10, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='13 to give a bound on the smallest surjective good prime q, and the product of all nonsurjective primes, thereby establishing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let A/Q be a typical genus 2 Jacobian of conductor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Assuming GRH, we have ∏ ℓ nonsurjective ℓ ≪ exp(N1/2+ϵ), where the implied constant is absolute and effectively computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' COMPUTING NONSURJECTIVE PRIMES IN GENUS 2 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Let q be the smallest surjective good prime for A, which is finite by Serre’s open image theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Multiplying the bounds in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='12, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content='15 by the conductor N, the product of all nonsurjective primes is bounded by a function of q and N of the following shape (8) ∏ ℓ nonsurjective ℓ ≪ qN1/2+ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' On the other hand, since q is the smallest surjective prime by definition, the product of all primes less than q divides the product of all nonsurjective primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E0T4oBgHgl3EQfSgAu/content/2301.02222v1.pdf'} +page_content=' Using [Ser81, Lemme 11], we have exp(q) ≪ ∏ ℓ