Cohomology of Locally Symmetric Spaces and Applications to Number Theory

Grant Number: DMS-05-02821    
Funding Agency: National Science Foundation
PI: Leslie D. Saper
Effective Dates: 2005/07-2008/06
Amount: $105,000

Description: The PI will study the structure of cohomology groups associated to locally symmetric spaces under the action of Hecke operators and Frobenius operators. Such a study will have applications to Langlands's program, specifically the prediction that the Hasse-Weil zeta function of a Shimura variety may be expressed in terms of automorphic L-functions. In order to perform this study, the PI will generalize and further develop the tools of L-modules and micro-support developed in the PI's previous work. As an application the PI will prove a decomposition theorem for the ordinary homology of the reductive Borel-Serre compactification and use this to construct cycles in intersection cohomology corresponding to sub-Shimura varieties. The PI will also apply these tools to explicitly compute the local intersection cohomology of the Baily-Borel-Satake compactification in some cases. The PI will compute a trace formula for the action of Hecke operators on the intersection cohomology of the reductive Borel-Serre compactification and compare this with the formula of Goresky and MacPherson. The PI will show that this cohomology group admits a Galois action and has a meaning "mod p" despite the fact that the reductive Borel-Serre compactification is not an algebraic variety..