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

Abstract

We prove an effective surjectivity result for Galois representations of products of non-isotrivial, non-isogenous elliptic curves over certain function fields of characteristic $0$. This is by way of an isogeny degree bound in this setting, generated from bounds for elliptic curves by Griffon--Pazuki and from the function field analogue of the Frey--Mazur conjecture, by employing techniques originating in work by Serre and Masser--W{ü}stholz in the number field setting.

20 pages. Comments are welcome! This version adds an appendix, written with Ben Bakker, that generalizes a function field Frey--Mazur theorem of Bakker--Tsimerman to composite level. This is used to correct an error in a prior version of Corollary 4.5, and this version includes major revisions in accordance with this fix