Freddy Saia
University of Illinois Chicago
Title
Explicit surjectivity for Galois representations of products of elliptic curves over function fields
Abstract
It is natural to study the torsion of an elliptic curve via its Galois representations, which encode the action of Galois on torsion points. Serre’s Open Image Theorem tells us that for an elliptic curve E over a number field K, the mod-$\ell$ Galois representation is surjective for all sufficiently large primes $\ell$. Since Serre’s work, considerable effort has gone towards obtaining extensions to other classes of abelian varieties and also towards making “sufficiently large” effective. I will discuss joint work with Alina Cojocaru, in which we obtain an explicit surjectivity result for products of elliptic curves over function fields. This comes from careful use of techniques of Masser—W{\"u}stholz from the number field setting, in combination with a result of Cojocaru—Hall for elliptic curves over function fields and recent isogeny bounds for elliptic curves over function fields due to Griffon—Pazuki. As an application, we prove that most specializations of certain families of products of elliptic curves over the rationals have no large exceptional primes.
