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Variations on a Theorem of Tate
About this Title
Stefan Patrikis, Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 258, Number 1238
ISBNs: 978-1-4704-3540-0 (print); 978-1-4704-5067-0 (online)
DOI: https://doi.org/10.1090/memo/1238
Published electronically: February 7, 2019
Keywords: Galois representations,
algebraic automorphic representations,
motives for motivated cycles,
monodromy,
Kuga-Satake construction,
hyperkähler varieties
MSC: Primary 11R39, 11F80, 14C15
Table of Contents
Chapters
- 1. Introduction
- 2. Foundations & examples
- 3. Galois and automorphic lifting
- 4. Motivic lifting
Abstract
Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate’s basic result that continuous projective representations $\mathrm {Gal}(\bar {F}/F) \to \mathrm {PGL}_n(\mathbb {C})$ lift to $\mathrm {GL}_n(\mathbb {C})$. We take special interest in the interaction of this result with algebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison of the automorphic and Galois “Tannakian formalisms”; monodromy (independence-of-$\ell$) questions for abstract Galois representations.- James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
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