Mod p geometry of Shimura varieties and applications

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Arithmetic geometry is a branch of number theory which is concerned with finding integer solutions to polynomial equations. Although the motivating problems in the field are relatively easy to state and understand, such as the statement of Fermat's Last Theorem, the solutions often require deep mathematical concepts and build on profound connections with other mathematical fields. This proposal is concerned with the study of Shimura varieties; these are geometric objects which lie on the interface between number theory, algebraic geometry and representation theory, and have far-reaching applications to a number of important and influential problems in arithmetic geometry.The proposal aims to study new problems and phenomena concerning the mod p geometry of Shimura varieties, and to apply these to study fundamental questions in arithmetic. The main geometric problems we focus on are to extend previous work of the PI on using motivic cohomology to establish geometric realizations of Jacquet-Langlands correspondences, and to study the geometry of the basic locus of Shimura varieties in the case of bad parahoric reduction. These results would then be applied to study new cases of the Bloch-Kato conjecture on special values of L-functions. Another part of the project is to study the semi-simple local zeta function of Shimura varieties of abelian type with parahoric reduction. The final part of the project will use Shimura varieties to answer a fundamental question concerning the independence of l of the system of Galois representations attached to abelian varieties. Along the way, we aim to develop new tools and techniques to study these questions.