November 9, 2025Open Access

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

Key Points

Explicit surjectivity achieved for products of elliptic curves over function fields, illustrating complexity in representation.
Key metric demonstrates new isogeny degree bounds applicable to various elliptic curve families.
Analysis integrates techniques from Serre and Masser–Wüstholz, enhancing understanding of Galois representations
Research indicates most elliptic curve products avoid exceptional primes, proposing implications for number theory.

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 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

Key Points

Abstract

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