Finiteness theorems for potentially equivalent Galois representations: extension of Faltings' finiteness criteria

TL;DR

This paper extends Faltings' finiteness criteria to potential equivalence of Galois representations, linking potential equivalence to character theory and establishing finiteness results for certain unramified representations.

Contribution

It introduces a novel criterion for potential equivalence of Galois representations based on character power equality, extending existing finiteness theorems.

Findings

Potential equivalence determined by equality of m-power characters.

Extended Faltings' finiteness criteria to potential equivalence.

Finiteness results for twist unramified representations.

Abstract

We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $\rho$ is determined by an equality of an $m$-power character $g\mapsto Tr(\rho(g^m))$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.