This project is in the field of arithmetic algebraic geometry, and is concerned with the study of moduli spaces of abelian varieties and p-divisible groups, and in particular with that of Shimura varieties and their local models. Questions about the geometry and cohomology of these arithmetic spaces arise naturally within the framework of the Langlands program. The proposed investigation focuses on two distinct but interralated problems. The first one is global and pursues the study of the Galois representations contributing to the cohomology of Shimura varieties focusing on ramified primes. T o do so, the PI proposes to extend the integral theory for Shimura varieties and their arithmetical compactifications to include primes of bad reduction, building on work of Kisin for Shimura varieties of Hodge type and on work of Chai, Faltings and Pink for arithmetical compactification of PEL type. The second problem is local and aims to identifying which reprensetations of a given p-adic group contribute to the cohomology of local models of Shimura varieties. In particular, the PI plans to purse new instances of a conjecture of Harris by a combination of local and global methods.
Langlands' conjectures explore the interrelation between seemingly unrelated objects in two distinct fields: automorphic forms in harmonic analysis and Galois representations in number theory. An automorphic form is an analytic function of several complex variables, possessing many self-similarities. A Galois representation is a realizations of the symmetries existing among the solutions to a polynomial equation in one variable as matrices. Langlands' original idea was to pursue a connection between these two theories by the medium of some other functions, called L-functions. On one hand, the prospected correspondences are a way of organizing the analytic objects in terms of the number theoretic ones. On the other, their existence would provide answers to many open questions in number theory. This project is aimed to study those arithmetical spaces which are expected to encoded in their geometry instances of these correspondences.
