Galois representations are surjective for almost all Drinfeld modules

TL;DR

This paper extends Duke's results on the surjectivity of Galois representations from elliptic curves to Drinfeld modules over function fields, showing that for most such modules, the associated Galois representations are surjective.

Contribution

It proves that for a large class of Drinfeld modules over function fields, the T-adic Galois representations are surjective, generalizing known results from elliptic curves.

Findings

Galois representations are surjective for a density 1 set of Drinfeld modules

Utilizes Hilbert irreducibility, Drinfeld's uniformization, and sieve methods

Extends Duke's results to the context of Drinfeld modules

Abstract

This article advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let $F$ be the rational function field over a finite field. I establish that for Drinfeld modules of rank $r \geq 2$, the $T$-adic Galois representation: $\widehat{\rho}_{\phi, T}: Gal(F^{sep}/F) \rightarrow GL_r(\mathbb{F}_q[[T]])$ is surjective for a density $1$ set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods.