A uniform bound for inertially equivalent, pure $\ell$-adic representations: an extension of Faltings' theorem

TL;DR

This paper establishes a uniform bound for certain pure $ell$-adic Galois representations, extending Faltings' theorem by considering inertial equivalence and lifting properties over global fields.

Contribution

It introduces inertial equivalence for integral $ell$-adic Galois representations and proves a uniform boundedness result for representations with fixed residual and inertial data.

Findings

Uniform bound for inertially equivalent, pure $ell$-adic representations

Extension of Faltings' theorem to inertial types

Bound independent of inertial type outside finite set

Abstract

We introduce a notion of inertial equivalence for integral $\ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\ell$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.