Galois representations are surjective for almost all Drinfeld modules

2024/07/19 by Anwesh Ray, Ray, Anwesh
#math.NT #math.AG

paper · pdf · doi:10.48550/arxiv.2407.14264

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 ≥ 2, the T-adic Galois representation: \widehatρ_{ϕ, T}: Gal(F^sep/F) → GL_r(\mathbbF_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.

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