Bounded-conductor finiteness for mod p Galois representations over number fields

Establish that, for every number field K, positive integer n, and nonzero ideal N of the ring of integers of K, only finitely many isomorphism classes of continuous semisimple representations of the absolute Galois group GK into GLn(Fp) have prime-to-p Artin conductor bounded by N.

Background

The paper introduces the bounded-conductor finiteness conjecture as a number-field analogue of finiteness questions motivated by Serre’s modularity conjecture and the Fontaine–Mazur finiteness conjectures. The conjecture concerns continuous semisimple mod-p Galois representations of fixed dimension whose prime-to-p Artin conductors are bounded by a prescribed ideal.

The authors discuss known cases, including representations over Q arising from odd modular forms and certain two-dimensional representations over totally real fields, but do not resolve the conjecture for arbitrary number fields, dimensions, and conductors. The paper instead studies the corresponding function-field conjecture, particularly in equal characteristic.