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Explicit surjectivity of Galois representations of products of elliptic curves over function fields

Alina Cojocaru, Frederick Saia

TL;DR

The work delivers an explicit open-image result for the mod-$\ell$ Galois representations of product abelian varieties formed from non-isotrivial, non-isogenous elliptic curves over function fields, extending Igusa and Cojocaru–Hall to higher dimension. It combines an explicit isogeny-degree bound with the Masser–Wüstholz framework to bootstrap surjectivity for large primes $\ell$, including uniform genus-based and modular-height refinements. An average-case result for families over $\mathbb{Q}$ shows that, for most specializations, there are no exceptional primes above a fixed constant, highlighting practical uniformity in open-image phenomena across families. The methods yield explicit constants depending on genus (and modular heights in refined variants), and the results advance understanding of open-image phenomena for higher-dimensional abelian varieties over function fields.

Abstract

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for elliptic curves by Griffon--Pazuki, and techniques originated by Serre and Masser--Wüstholz in the number field setting. We apply our result to prove that most members of a family of products of elliptic curves over $\mathbb{Q}$ with no extra endomorphisms have no exceptional primes above a specified constant which depends neither on the elliptic curve factors nor on the dimension of the product.

Explicit surjectivity of Galois representations of products of elliptic curves over function fields

TL;DR

The work delivers an explicit open-image result for the mod- Galois representations of product abelian varieties formed from non-isotrivial, non-isogenous elliptic curves over function fields, extending Igusa and Cojocaru–Hall to higher dimension. It combines an explicit isogeny-degree bound with the Masser–Wüstholz framework to bootstrap surjectivity for large primes , including uniform genus-based and modular-height refinements. An average-case result for families over shows that, for most specializations, there are no exceptional primes above a fixed constant, highlighting practical uniformity in open-image phenomena across families. The methods yield explicit constants depending on genus (and modular heights in refined variants), and the results advance understanding of open-image phenomena for higher-dimensional abelian varieties over function fields.

Abstract

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for elliptic curves by Griffon--Pazuki, and techniques originated by Serre and Masser--Wüstholz in the number field setting. We apply our result to prove that most members of a family of products of elliptic curves over with no extra endomorphisms have no exceptional primes above a specified constant which depends neither on the elliptic curve factors nor on the dimension of the product.
Paper Structure (9 sections, 25 theorems, 90 equations)

This paper contains 9 sections, 25 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. Function fields and heights
  4. Isogenies
  5. Factorizations
  6. Separability and biseparability
  7. An isogeny degree bound
  8. Effective surjectivity for products
  9. An average result over families of products

Key Result

Theorem 1.1

Let $K$ be the function field of a smooth, projective, and geometrically integral curve $C$ of genus $g$ over a perfect field $k$. Let $E_1, \ldots, E_n$ be pairwise non-isogenous elliptic curves over $K$ with $j(E_i) \not \in \overline{k}$ for each $1 \leq i \leq n$, and let Set and finally For all primes $\ell$ not equal to $\textnormal{char}(K)$ satisfying $\ell > \widetilde{C}(g)$, we have

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • ...and 35 more