An explicit bound on reducibility of mod \mathfrakl Galois image for Drinfeld modules of arbitrary rank and its application on the uniformity problem

2021/08/28 by Chien‐Hua Chen, Chen, Chien-Hua
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2108.12660

openalex publication_date 2021/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose we are given a Drinfeld Module ϕ over \mathbbF_q(t) of rank r and a prime ideal \mathfrakl of \mathbbF_q[T]. In this paper, we prove that the reducibility of mod \mathfrakl Galois representation \rmGal(\mathbbF_q(T)^\rmsep/\mathbbF_q(T))→ \rmAut(ϕ[\mathfrakl])≅ \rmGL_r(\mathbbF_\mathfrakl) gives a bound on the degree of \mathfrakl which depends only on the rank r of Drinfeld module ϕ and the minimal degree of place P where ϕ has good reduction at P. Then, we apply this reducibility bound to study the Drinfeld module analogue of Serre's uniformity problem.

Citations

• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Adelic openness for Drinfeld modules in generic characteristic
• The Galois image of twisted Carlitz modules

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