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arXiv · 2407.14264

Galois representations are surjective for almost all Drinfeld modules

Anwesh Ray
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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.

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Anwesh Ray. 2024-07-19. Galois representations are surjective for almost all Drinfeld modules. https://doi.org/10.1112/mtk.70118

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