A finiteness theorem for mod p Galois representations over global function fields

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发表日期：

2026年7月1日

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Yufan Luo

摘要

Let p be an odd prime number and let $\overline{\mathbb {F}}_p$ ${\bar{F}}_{p}$ be a fixed algebraic closure of the finite field of order p. Let K be a global function field of characteristic different from p and let $G_{K}$ $G_{K}$ be the absolute Galois group of K. We prove that there are only finitely many isomorphism classes of continuous geometric semisimple representations $\rho :G_{K}\rightarrow \textrm{GL}_{n}(\overline{\mathbb {F}}_{p})$ $ρ : G_{K} \to {GL}_{n} ({\bar{F}}_{p})$ such that their Artin conductors are bounded. It is worth emphasizing that we do not need to assume that p does not divide n.