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Explicit open image theorems for abelian varieties with trivial endomorphism ring

Matthew Bisatt, Davide Lombardo

TL;DR

This work proves an explicit open-image theorem for abelian varieties with trivial endomorphism ring by combining an explicit isogeny bound with a detailed analysis of maximal subgroups of finite symplectic groups via Aschbacher's classification. The main result provides an explicit bound $\ell_0(A/K)$, depending on the stable Faltings height $h(A)$ and a Frobenius-residue $q_v$, such that $G_{\ell^ abla}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all $\ell>\ell_0(A/K)$ when a suitable Frobenius polynomial at a place $v$ has Galois group $\left(\mathbb{Z}/2\mathbb{Z}\right) \wr \operatorname{Sym}_g$. The paper then develops a comprehensive, algorithmic framework to bound non-surjective primes by classifying and excluding all potential maximal subgroups (C1–C4, C6–C7 and S-type, including those of Lie type) and provides an explicit, practical example in genus $3$ where the adelic Galois action is maximal except for a pair of small primes. The results yield both explicit, uniform bounds and a concrete path to effective computations for specific abelian varieties, with broad implications for Mumford–Tate-type phenomena and open image questions in arithmetic geometry. The approach also highlights the interplay between polarisation theory, Frobenius eigenvalues, and representation theory of finite groups of Lie type.

Abstract

Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$. Assuming that the image $G_{\ell^\infty}$ of the natural Galois representation attached to the Tate module $T_\ell(A)$ is $\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all sufficiently large primes $\ell$, we provide a semi-effective bound $\ell_0(A/K)$ such that $G_{\ell^\infty}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ for all primes $\ell > \ell_0(A/K)$. The bound is given in terms of the Faltings height of $A$ and of the cardinality of the residue field at a suitably generic place of $K$. We also describe an algorithmic approach to obtain better bounds for abelian threefolds over $\mathbb{Q}$.

Explicit open image theorems for abelian varieties with trivial endomorphism ring

TL;DR

This work proves an explicit open-image theorem for abelian varieties with trivial endomorphism ring by combining an explicit isogeny bound with a detailed analysis of maximal subgroups of finite symplectic groups via Aschbacher's classification. The main result provides an explicit bound , depending on the stable Faltings height and a Frobenius-residue , such that for all when a suitable Frobenius polynomial at a place has Galois group . The paper then develops a comprehensive, algorithmic framework to bound non-surjective primes by classifying and excluding all potential maximal subgroups (C1–C4, C6–C7 and S-type, including those of Lie type) and provides an explicit, practical example in genus where the adelic Galois action is maximal except for a pair of small primes. The results yield both explicit, uniform bounds and a concrete path to effective computations for specific abelian varieties, with broad implications for Mumford–Tate-type phenomena and open image questions in arithmetic geometry. The approach also highlights the interplay between polarisation theory, Frobenius eigenvalues, and representation theory of finite groups of Lie type.

Abstract

Let be a number field and be an abelian variety of dimension . Assuming that the image of the natural Galois representation attached to the Tate module is for all sufficiently large primes , we provide a semi-effective bound such that for all primes . The bound is given in terms of the Faltings height of and of the cardinality of the residue field at a suitably generic place of . We also describe an algorithmic approach to obtain better bounds for abelian threefolds over .

Paper Structure

This paper contains 34 sections, 52 theorems, 41 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The isogeny theorem and the Weil pairing
  4. Frobenius elements and their eigenvalues
  5. Maximal subgroups of finite symplectic groups
  6. Group theoretical preliminaries
  7. Classical definitions
  8. Maximal subgroups of $\operatorname{GSp}_{2n}(\mathbb{F}_\ell)$
  9. Reducible, imprimitive and field extension cases
  10. The tensor product case
  11. Tensor induced groups
  12. Constant groups in class S
  13. Class S subgroups of Lie type
  14. Preliminaries on algebraic groups and root systems
  15. Representation theory of finite simple groups of Lie type
  16. ...and 19 more sections

Key Result

Theorem 1.2

Let $A/K$ be an abelian variety of dimension $g \geq 2$ and $G_{\ell^\infty}$ be the image of the natural representation $\rho_{\ell^\infty} : \operatorname{Gal}\left(\overline{K}/K\right) \to \operatorname{Aut} T_\ell A$. Suppose that: Then the equality $G_{\ell^\infty}=\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$ holds for every prime $\ell$ unramified in $K$ and such that

Theorems & Definitions (130)

  • Definition 1.1
  • Theorem 1.2: =\ref{['thm_MainProof']}
  • Remark 1.3
  • Theorem 1.4: Pink Pink, Serre Serre_resum8586LettreVigneras
  • Remark 1.5
  • Theorem 2.1: Isogeny Theorem, MR4592752
  • Lemma 2.2
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 120 more