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

Patrikis, Stefan

doi:10.2140/ant.2017.11.2397

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

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

Stefan Patrikis

Vol. 11 (2017), No. 10, 2397–2423

DOI: 10.2140/ant.2017.11.2397

Abstract

In previous work, we described conditions under which a single geometric representation $Γ_{F} \to H ({\bar{ℚ}}_{ℓ})$ 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 $ℓ$ -adic representations, such as the $ℓ$ -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 $\tilde{H} \to H$ ).

Keywords

Galois representations, Kuga–Satake construction

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11R37

Milestones

Received: 30 April 2017

Revised: 8 August 2017

Accepted: 6 September 2017

Published: 31 December 2017

Authors

Stefan Patrikis
Department of Mathematics University of Utah Salt Lake City, UT United States

Vol. 11, No. 10, 2017