## Algebra & Number Theory

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

Stefan Patrikis

#### Abstract

In previous work, we described conditions under which a single geometric representation $Γ F → H ( ℚ ¯ ℓ )$ 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 $ℓ$-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 $H ˜ → H$).

#### Article information

Source
Algebra Number Theory, Volume 11, Number 10 (2017), 2397-2423.

Dates
Revised: 8 August 2017
Accepted: 6 September 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.ant/1517454189

Digital Object Identifier
doi:10.2140/ant.2017.11.2397

Mathematical Reviews number (MathSciNet)
MR3744361

Zentralblatt MATH identifier
06825455

Subjects
Primary: 11F80: Galois representations
Secondary: 11R37: Class field theory

#### Citation

Patrikis, Stefan. Generalized Kuga–Satake theory and good reduction properties of Galois representations. Algebra Number Theory 11 (2017), no. 10, 2397--2423. doi:10.2140/ant.2017.11.2397. https://projecteuclid.org/euclid.ant/1517454189

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