Number Theory Seminar

Given a smooth projective algebraic variety over a field F, one obtains, for each prime number l, a representation of the absolute Galois group of F on the l-adic cohomology of the variety. By Deligne's work on the Weil conjectures, these l-adic representations are compatible as the prime l varies; at a more basic level, they all at least bear the mark of the variety having good reduction modulo almost all primes. In many cases it is natural to regard these representations as valued in some subgroup of the linear group--for instance, the representation on even-degree cohomology will take values in an appropriate orthogonal group--and a group-theoretic perspective can then suggest new questions in both geometry and arithmetic--for instance, does an orthogonal representation on even degree cohomology lift to the corresponding spin (or spin similitude) group? Classical motivation for asking such questions comes from the Kuga-Satake construction, which carries out precisely this lifting procedure in the case of the degree 2 cohomology of a K3 surface, and finds the associated "Kuga-Satake abelian variety" as output. My talk will introduce this circle of ideas, and then discuss some recent refined Galois-theoretic evidence for a "generalized Kuga-Satake theory:" namely, when F is a number field, I'll explain when one can lift all the l-adic realizations of a motive (through some central quotient of reductive groups, such as (G)Spin to SO), with independent-of-l control of "good reduction" properties.