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

TL;DR

This paper establishes an explicit surjectivity result for Galois representations of products of non-isotrivial, non-isogenous elliptic curves over function fields, extending known bounds and applying to families over rationals.

Contribution

It provides a new explicit isogeny degree bound for such elliptic curve products over function fields, generalizing previous number field results.

Findings

Most elliptic curve products over $\\mathbb{Q}$ have no exceptional primes above a certain constant.

The result applies to non-isotrivial, non-isogenous elliptic curves over arbitrary characteristic.

The method combines bounds from Griffon--Pazuki with classical techniques from Serre and Masser--Wüstholz.

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--W\"{u}stholz 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.