Date: Monday, February 13, 2017
Location: 4088 East Hall (4:10 PM to 5:30 PM)
Title: Generalized KugaSatake theory and good reduction properties of Galois representations
Abstract: Given a smooth projective algebraic variety over a number field F, one obtains a compatible system of geometric representations of the absolute Galois group of F on the ladic cohomology groups of the variety; at a more basic level, the different ladic realizations 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 groupfor instance, the representation on evendegree cohomology will take values in an appropriate orthogonal groupand a grouptheoretic perspective can then suggest new questions in both geometry and arithmeticfor 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 KugaSatake construction, which carries out precisely this lifting procedure in the case of the degree 2 cohomology of a K3 surface, and finds the associated "KugaSatake abelian variety" as output.
My talk will introduce this circle of ideas, and then discuss some refined Galoistheoretic evidence for a "generalized KugaSatake theory:" namely, when F is a number field, I'll explain when one can lift all the ladic realizations of a motive over F through some central quotient of reductive groups (eg, (G)Spin to SO), with independentofl control of "good reduction" properties.
Files:
Speaker: Stefan Patrikis
Institution: University of Utah
Event Organizer: Wei Ho
