Generalized Kuga-Satake theory and good reduction properties of Galois representations

TL;DR

This paper proves that systems of l-adic Galois representations with good reduction properties can be lifted to geometric representations with good reduction outside a finite set of primes, providing new proofs of classical theorems.

Contribution

It establishes a sharper result for systems of l-adic representations, showing they admit geometric lifts with controlled good reduction, extending previous work on Galois representation lifting.

Findings

Systems of l-adic representations admit geometric lifts with good reduction outside a finite set of primes.

Provides new proofs of Tate's theorem on lifting projective representations.

Extends Wintenberger's results on lifting through central isogenies.

Abstract

In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a much sharper result for systems of l-adic representations, such as the l-adic realizations of a motive, having common "good reduction" properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over number fields) and Wintenberger (an analogue of our main result in the setting of lifting through a central isogeny).