Galois representation of the product of two Drinfeld modules of generic characteristic

TL;DR

This paper investigates Galois representations linked to products of Drinfeld modules, demonstrating that their images are large and follow a natural determinant condition, extending classical elliptic curve results.

Contribution

It establishes a Serre-type openness result for Galois representations of product Drinfeld modules in generic characteristic, using Pink's minimal model theory and explicit reciprocity laws.

Findings

The Galois image is commensurable with a subgroup defined by a determinant condition.

The results extend classical elliptic curve Galois image theorems to Drinfeld modules.

The approach combines local and global techniques in function field arithmetic.

Abstract

In this paper, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre's classical result on the images of Galois representations associated with products of elliptic curves, we prove that for any finite set of primes, the image of the corresponding product representation is sufficiently large, in the sense that it is commensurable with a subgroup defined by a natural determinant condition. Our approach combines Pink's minimal model theory for compact subgroups of linear groups over local fields with explicit reciprocity laws for global function fields.