Galois representations are surjective for almost all 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 $r\ge 2$, the $T$-adic Galois representation: $\hat{\ddot{I}}{}_{\ddot{I}•,T}:Gal(F^{sep}/F)\to G{L}_r({\mathbb{F}}_q[[T]])$ 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.

Version 2: 17 pages. Expanded the proof of Proposition 4.6. Some minor corrections. Accepted for publication in Mathematika