Integral models and endoscopy for Shimura varieties with deeper level structure

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The Langlands program seeks to illuminate currently-mysterious relations between different kinds of symmetries, thereby connecting the mathematical realms of arithmetic, algebraic geometry, and harmonic analysis on Lie groups. This program has been fundamental in answering many longstanding problems in number theory over the last few decades; to cite one example, it played a key role in the solution of Fermat's Last Theorem by Andrew Wiles in the 1990s. Variants of the classical Langlands program, namely the geometric Langlands program initiated by Drinfeld and Laumon, have been inspirational for developments in theoretical physics, for instance in the work of Kapustin and Witten on Quantum Field Theory. This project relates to Shimura varieties, arithmetical objects which have been indispensable for establishing local and global Langlands correspondences between representations of Galois groups on the one hand, and automorphic representations on the other hand.The PI seeks to build new relations between the study of Shimura varieties and the geometric Langlands program. The Shimura varieties of interest are moduli spaces of abelian varieties with additional structure, and that additional structure can give rise to singularities when the equations are reduced modulo various prime numbers. The PI will introduce new local models to study those singularities when the structure is pro-p Iwahori level; the local models will be generalizations of the Rapoport-Zink local models which have been so effective for studying parahoric level structure. He will relate these local models to 'pro-p' affine flag varieties and their deformations, bringing in ideas used in the geometric Langlands program in the setting of tame ramification. This connection will permit an understanding of nearby cycles and singularities, and also of endoscopy and Hasse-Weil zeta functions, for interesting new classes of Shimura varieties. In order to put these constructions into a very general context, the PI will work out a Tannakian theory of Bruhat-Tits buildings.