Vol. 11, No. 10, 2017

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

Stefan Patrikis

Vol. 11 (2017), No. 10, 2397–2423

In previous work, we described conditions under which a single geometric representation ΓF H( ¯) of the Galois group of a number field F lifts through  a central torus quotient H˜ H to a geometric representation. In this paper, we prove a much sharper result for systems of -adic representations, such as the -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 H˜ H).

Galois representations, Kuga–Satake construction
Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11R37
Received: 30 April 2017
Revised: 8 August 2017
Accepted: 6 September 2017
Published: 31 December 2017
Stefan Patrikis
Department of Mathematics
University of Utah
Salt Lake City, UT
United States