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Adelic Galois representations are maximal for almost all Drinfeld modules

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Contributed by Anwesh Ray ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

Let r≥3r\geq 3 be an integer, let qq be an odd prime power, and set A=Fq[T]A=\mathbb{F}_q[T] and F=Fq(T)F=\mathbb{F}_q(T). We study the Galois representations associated with rank rr Drinfeld modules of the form ϕT=T+g1τ+⋯+grτr\phi_T=T+g_1\tau+\cdots+g_r\tau^r, where gi∈Ag_i\in A and gr≠0g_r\neq 0. Ordering the coefficient tuples g⃗=(g1,…,gr)\vec{g}=(g_1,\ldots,g_r) by their maximum degree, we prove that, for a density-one set of tuples, the λ\lambda-adic Galois representation is surjective for every prime λ\lambda of AA, and the adelic Galois image contains SLr(A^)\mathrm{SL}_r(\widehat A). Moreover, for every tuple in this set, the index of the adelic image equals the index of the corresponding rank-one determinant representation and hence divides q−1q-1. These results extend Zywina's theorem in rank 22 to arbitrary ranks r≥3r\geq 3. As a further consequence, we establish the existence of rank rr Drinfeld modules with surjective adelic Galois representations for every r≥3r\geq 3, extending previous existence results which were known for prime values of rr.

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Version history

  1. v1Submitted by Anwesh RayInitial depositCurrentOct 18, 2026