Local Langlands Correspondences and Vanishing Cycles on Shimura Varieties

Abstract

We report on the results and techniques of the author’s recent joint work with Richard Taylor, which analyzes in detail the bad reduction of certain Shimura varieties in order to prove the compatibility of local and global Langlands correspondences, obtaining as a consequence the local Langlands conjecture for GL(n) of a p-adic field. These Shimura varieties have natural models over p-adic integer rings, as moduli spaces for abelian varieties with additional structure. The starting point of the work with Taylor is the stratification of the special fiber of an integral model in minimal level, according to the isogeny type of the universal family of p-divisible groups attached to these abelian varieties Similar stratifications can conjecturally be constructed for any Shimura variety, and indeed are known to exist for most PEL types. We discuss a series of conjectures regarding the behavior of vanishing cycles along these strata, with the aim of extending Kottwitz’ conjectures on the cohomology of Shimura varieties to the case of bad reduction.KeywordsParabolic SubgroupSpecial FiberLevi FactorShimura VarietyDiscrete Series RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.