Shimura Varieties and the Bernstein center

首席研究员将研究志村变体的坏约简与相关p元群的伯恩斯坦中心之间的新联系。志村变数是求解自同构l -函数上代数变数的Hasse-Weil zeta函数的主要论据之一。函数被定义为局部函数的乘积，在定义域的所有素数理想上。在好的约简素数上，Langlands和Kottwitz的方法已经完成，这部分归功于Ngo、Laumon、Waldspurger等人最近对Langlands- shelstad猜想（“基本引理”）的证明。当素数理想的约化模中存在奇点时，会出现几何和表示理论上的困难，PI将对此进行研究。一些新的“基本引理”将发挥重要作用，这些引理是Langlands没有预测到的，它们是用Bernstein对p进群的光滑表示范畴的分解来表述的。另一个组成部分将是用与Bernstein分解平行的方式分解附近的环，以及显示这些附近环的几何论证，从而在相关p进群的Bernstein中心产生测试函数。数论中许多深奥的结果都涉及到用纯解析对象（如l函数）表示纯算术对象（如zeta函数）。后者是起源于经典模形式理论的复变量函数，它是复数上半平面上满足非常严格的对称条件的函数。几个著名的猜想（例如Birch和Swinnerton-Dyer猜想，Clay数学基础千年问题）假设了zeta函数值和算术不变量之间的关系。志村变异体形成了一类重要的对象，其中算术和分析之间的联系可以充分地进行。他们的研究进展经常对自同构形式中的其他中心问题产生影响：最近的一个例子是M. Harris和R. Taylor对p进域上一般线性群的局部朗兰兹猜想的证明。PI将寻求进一步了解志村品种和朗兰兹计划的其他方面。

The principal investigator will study an emerging connection between the bad reduction of Shimura varieties and the Bernstein center of an associated p-adic group. Shimura varieties form one of the main testing grounds for conjectures of Langlands on the calculation of Hasse-Weil zeta functions of algebraic varieties over number fields in terms of automorphic L-functions. The zeta function is defined as a product of local zeta functions, over all prime ideals in the field of definition. At primes of good reduction, the approach of Langlands and Kottwitz has been completed, thanks in part to the recent proof of the Langlands-Shelstad conjecture ("fundamental lemma") due to Ngo, Laumon, Waldspurger, and others. When singularities exist in the reduction modulo a prime ideal, geometric and representation-theoretic difficulties arise, which the PI will investigate. An important role will be played by some new "fundamental lemmas" which were not predicted by Langlands and which are formulated using Bernstein's decomposition of the category of smooth representations of a p-adic group. Another ingredient will be a decomposition of nearby cycles in a manner parallel to Bernstein's decomposition, and geometric arguments showing these nearby cycles give rise to test functions in the Bernstein center of the associated p-adic group.Many deep results in number theory involve the expression of a purely arithmetic object (such as a zeta function) in terms of a purely analytic object (such as an L-function). The latter are functions of a complex variable originating in the classical theory of modular forms, which are functions on the complex upper-half plane satisfying very stringent symmetry conditions. Several famous conjectures (e.g. the Birch and Swinnerton-Dyer conjecture, a Clay Math foundation Millennium Problem) postulate relations between values of zeta functions and arithmetic invariants. Shimura varieties form an important class of objects where links between arithmetic and analysis can be fully carried out. Progress in their study often has impact on other central questions in automorphic forms: a recent example is the proof due to M. Harris and R. Taylor of the local Langlands conjecture for general linear groups over p-adic fields. The PI will seek to further our understanding of Shimura varieties and other aspects of the Langlands program.