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

Abstract

In previous work, we described conditions under which a single geometric representation ${\mathrm{\Gamma }}_F\to H\left({\overline{\mathbb{Q}}}_{\ell }\right)$ of the Galois group of a number field $F$ lifts through a central torus quotient $\tilde{H}\to H$ to a geometric representation. In this paper, we prove a much sharper result for systems of $\ell$-adic representations, such as the $\ell$-adic realizations of a motive over $F$, 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 case of a central isogeny $\tilde{H}\to H$).