Bad Reduction of Shimura varieties

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The principal investigator proposes to study problems in arithmetic algebraic geometry and the representation theory of p-adic groups and their Hecke algebras, which arise in the study of Shimura varieties with bad reduction. A classical problem due to Langlands and others which motivates much of this research is the description of the Hasse-Weil zeta function of a Shimura variety in terms of automorphic L-functions. The former is an object defined using cohomology which is expected to contain deep arithmetic and geometric information about the Shimura variety (e.g., the Bloch-Kato and Beilinson conjectures). The automorphic L-functions are functions of a complex variable with amazing symmetry properties, whose analytic behavior is better understood than that of the zeta functions, and which can therefore be used (via the hoped-for description) to uncover mysterious properties of the latter. A complete description of the zeta function requires one to consider primes of bad reduction.Shimura varieties have historically played an important role in establishing "higher reciprocity laws" which are at the heart of the Langlands program, which in general posits an important link between arithmetic (Galois representations) and analysis (automorphic forms). The study of their bad reduction is little understood in general, but is often crucial: for example, it played a key role in the proof of the local Langlands conjecture for the general linear group over a p-adic field, due to M. Harris and R. Taylor. The PI proposes to study bad reduction of parahoric type, where many of the issues which complicate the general picture (describing the reduction modulo p, problems with combinatorics and endoscopy, monodromy questions) arise, but where progress is possible by relating the singularities to those in affine flag varieties, and by applying the theory of perverse sheaves and other techniques useful in geometric representation theory.