Variations on a theorem of Tate

Abstract

Let F be a number field, with absolute Galois group G. For any homomorphism r of G valued in the l-adic points of a linear algebraic group H, we consider lifting problems through covers H' of H with central torus kernel. By a theorem of Tate, elaborated by B. Conrad, any such continuous homomorphism to H lifts to H'. Largely motivated by a question of Conrad, who asked when geometric homomorphisms (in the sense of Fontaine-Mazur) should admit geometric liftings, we address a number of Galois-theoretic, automorphic, and motivic variants of the lifting problem.