Variations on a theorem of Tate

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 $Gal(\bar{F}/F)\to PG{L}_n(C)$ lift to $G{L}_n(C)$. We take special interest in the interaction of this result with algebraicity (on the automorphic side) 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.

minor revisions. some exposition expanded

Cited by in corpus (5)

• Automorphy and irreducibility of some l-adic representations
• Anabelian geometry and descent obstructions on moduli spaces
• Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves
• Independence of Algebraic Monodromy Groups in Compatible Systems
• The Chabauty-Kim Method for Relative Completions