Drinfeld modules with maximal Galois action

2025/02/03 by David Zywina, Zywina, David
Mathematics · #11G09 (Primary) 11F80 #11R58 (Secondary) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.01030

openalex publication_date 2025/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

With a fixed prime power q>1, define the ring of polynomials A=\mathbbF_q[t] and its fraction field F=\mathbbF_q(t). For each pair a=(a₁,a₂) ∈ A² with a₂ nonzero, let ϕ(a)\colon A→ F\τ\ be the Drinfeld A-module of rank 2 satisfying t↦ t+a₁τ+a₂τ². The Galois action on the torsion of ϕ(a) gives rise to a Galois representation ρ_ϕ(a)\colon Gal(F^sep/F)→ GL₂(\widehatA), where \widehatA is the profinite completion of A. We show that the image of ρ_ϕ(a) is large for random a. More precisely, for all a∈ A² away from a set of density 0, we prove that the index [GL₂(\widehatA):ρ_ϕ(a)(Gal(F^sep/F))] divides q-1 when q>2 and divides 4 when q=2. We also show that the representation ρ_ϕ(a) is surjective for a positive density set of a∈ A².

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