Shimura varieties and their local models

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ABSTRACT for award DMS-0701310 of Mantovan Beginning in 1967 Langlands suggested an intricate web offar-reaching conjectures connecting the theories of automorphicforms and Galois representations. He also predicted some of thesecorrespondences to be encoded in the cohomology groups of Shimuravarieties and their local models. The proposed research pursues thestudy of the geometric implications of these conjectures, focusingon two key features: the compatibility between global and localconjectures and the functoriality principle. The PI aims to enlargethe class of Shimura varieties whose local behavior at unramifiedprimes is well understood by eliminating two common restrictivehypotheses: the assumption of properness and the restriction to thesubclass of Shimura varieties of P.E.L. type. In the first case, thePI proposes to develop the integral theory of arithmeticalcompactifications for Shimura varieties, following the works ofFaltings-Chai and Pink. In the second case, she plans to study thebad reduction of Shimura varieties of Hodge type, following thestrategy of Rapoport-Zink. The Langlands program proposes the existence of a relation betweenseemingly unrelated objects in number theory and harmonic analysis.A Galois group is the collection of all symmetries existing amongthe roots of a given polynomial in one variable and a Galoisrepresentation is the realization of such symmetries as matrices. Onthe other hand, automorphic forms are complex functions possessingmany self-similarities. A connection between the two theories as itis proposed by the Langlands program would provide an incrediblypowerful mathematical tool translating theorems of harmonic analysisinto theorems about algebra, and vice versa. In some cases, it isexpected such a correspondence to be encoded in the geometry ofcertain algebraic objects: the Shimura varieties and their localmodels. This project aims to shed some light on how the geometry ofthese spaces could reflect some of the deepest aspects of theLanglands' conjectures.