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

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.