Bad Reduction of Shimura varieties

主要研究者建议研究算术代数几何和p进群及其Hecke代数的表示理论中的问题，这些问题出现在具有不良还原的志村簇的研究中。 一个经典的问题，由于朗兰兹和其他人的动机大部分这项研究是描述的Hasse-Weil zeta函数的志村品种的自守L-函数。 前者是使用上同调定义的对象，期望其包含关于Shimura簇的深层算术和几何信息（例如，Bloch-Kato和Beilinson模型）。 自守L-函数是具有惊人对称性的复变函数，其分析行为比zeta函数更容易理解，因此可以用来（通过希望的描述）揭示后者的神秘性质。 一个完整的zeta函数的描述需要一个考虑坏的减少素数。志村品种在历史上发挥了重要作用，建立“更高的互惠法律”，这是在朗兰兹计划的核心，这在一般假设之间的重要联系算术（伽罗瓦表示）和分析（自守形式）。 对它们的坏约化的研究一般很少被理解，但通常是至关重要的：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，由于M。Harris和R. Taylor. PI建议研究parahoric类型的不良约简，其中许多问题使一般情况复杂化（描述约简模p，组合学和内窥镜问题，单值问题），但通过将奇点与仿射旗簇中的奇点联系起来，并通过应用反常层理论和几何表示理论中有用的其他技术，可以取得进展。

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.