Beginning in 1967 Langlands suggested an intricate web of far-reaching conjectures connecting the theories of automorphic forms and Galois representations. He also predicted some of these correspondences to be encoded in the cohomology groups of Shimura varieties and their local models. The proposed research pursues the study of the geometric implications of these conjectures, focusing on two key features: the compatibility between global and local conjectures and the functoriality principle. The PI aims to enlarge the class of Shimura varieties whose local behavior at unramified primes is well understood by eliminating two common restrictive hypotheses: the assumption of properness and the restriction to the subclass of Shimura varieties of P.E.L. type. In the first case, the PI proposes to develop the integral theory of arithmetical compactifications for Shimura varieties, following the works of Faltings-Chai and Pink. In the second case, she plans to study the bad reduction of Shimura varieties of Hodge type, following the strategy of Rapoport-Zink.
The Langlands program proposes the existence of a relation between seemingly unrelated objects in number theory and harmonic analysis. A Galois group is the collection of all symmetries existing among the roots of a given polynomial in one variable and a Galois representation is the realization of such symmetries as matrices. On the other hand, automorphic forms are complex functions possessing many self-similarities. A connection between the two theories as it is proposed by the Langlands program would provide an incredibly powerful mathematical tool translating theorems of harmonic analysis into theorems about algebra, and vice versa. In some cases, it is expected such a correspondence to be encoded in the geometry of certain algebraic objects: the Shimura varieties and their local models. This project aims to shed some light on how the geometry of these spaces could reflect some of the deepest aspects of the Langlands' conjectures.
