Kai-Wen Lan ( U. Minnesota)

Local systems over Shimura varieties: a comparison between two constructions

Abstract: Given a Shimura variety X associated with some algebraic group G, and some algebraic representation V of G (satisfying some conditions when restricted to the center), we can define two kinds of filtered vector bundles with integrable connections, over X. The first one is based on the classical complex analytic construction using double quotients, while the second one is a new p-adic analytic construction based on the p-adic Riemann-Hilbert correspondence in the recent work by Ruochuan Liu and Xinwen Zhu. We know how to relate these two when X is of Hodge type, using the relative cohomology of some family of abelian varieties over X. But what should we do when X is a general Shimura variety, in which case no convenient family of algebraic varieties (or, rather, "motives") are available? In this talk, we shall review the background materials and formulate the problem more precisely, and give an answer.