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

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.

arXiv comment12 pages, Revised version, to appear in Proc. Amer. Math. Soc

References in corpus (1)

• Locally potentially equivalent Galois representations