Bad Reduction of Shimura varieties

主要研究算术代数几何中的问题，以及在研究不良约简的Shimura变分中出现的p进群及其Hecke代数的表示理论。Langlands等人提出的一个经典问题是用自同构l函数描述Shimura变量的Hasse-Weil zeta函数。前者是一个使用上同调定义的对象，它被期望包含关于Shimura变量的深度算术和几何信息（例如Bloch-Kato和Beilinson猜想）。自同构l -函数是具有惊人对称性的复变量的函数，其解析行为比zeta函数的解析行为更容易理解，因此可以使用（通过期望的描述）来揭示后者的神秘性质。对ζ函数的完整描述需要考虑不良约简的素数。志村变分在历史上对建立“更高互易律”发挥了重要作用，这是朗兰兹程序的核心，它通常在算术（伽罗瓦表示）和分析（自同构形式）之间建立了重要的联系。一般来说，对它们的坏约简的研究很少被理解，但往往是至关重要的：例如，它在证明p进域上一般线性群的局部朗兰兹猜想中发挥了关键作用，这是由于M. Harris和R. Taylor的贡献。PI建议研究抛物线型的坏约简，其中出现了许多使一般情况复杂化的问题（描述约简模p，组合学和内窥镜问题，单一性问题），但通过将奇点与仿射标志变体中的奇点联系起来，并通过应用反常束理论和几何表示理论中有用的其他技术，可能会取得进展。

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.