Arithmetic Geometry and Applications

该奖项支持首席研究员在算术几何方面的研究。算术几何是数学的一个分支，研究多项式方程的整数解，在解决数论中的许多突出问题中发挥了核心作用，如费马大定理和关于曲线上有理点的数量的莫德尔猜想。本研究项目的主要研究对象被称为“志村簇”，其研究处于代数几何、数论和表示论的界面，并在一些意义深远和有影响力的领域中具有广泛的应用。在项目的第一部分，PI计划研究Shimura变种的上同调结构，以及它们的模p点的结构。具体地说，有一个猜想，证明了PI在某些情况下，即每一个mod p点的islogic类包含一个特殊点的减少。这些结果可以用来研究Hasse-Weil zeta函数的Shimura品种，以下Langlands的程序。该项目的另一部分旨在研究晶体上同调中的Mumford-Tate猜想的类似物;这相当于Shimura品种上某些形式循环的代数化问题。最后，PI计划应用有限旗群方案变形理论的技术来研究希尔伯特第13问题，该奖项要求最小的m，对于该最小的m，一般n次多项式的解可以写成m个变量的函数的复合。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的评估来支持。影响审查标准。

The award supports the principal investigator's research in arithmetic geometry. Arithmetic geometry is a branch of mathematics that studies integer solutions of polynomial equations and has played a central role in solving many outstanding problems in number theory, such as Fermat's Last Theorem and the Mordell conjecture concerning the number of rational points on a curve. The main objects of study in this research project are called "Shimura varieties," the study of which is at the interface of algebraic geometry, number theory, and representation theory and has broad applications to a number of far-reaching and influential conjecturesThe project concerns problems in and applications of the arithmetic of Shimura varieties, which are generalizations of the moduli space of abelian varieties. In the first part of the project the PI plans to study the structure of the cohomology of Shimura varieties, and the structure of their mod p points. Specifically, there is a conjecture, proved by the PI in some cases, that the isogeny class of every mod p point contains the reduction of a special point. These results can be used to study the Hasse-Weil zeta function of a Shimura variety, following a program of Langlands. Another part of the project aims to study an analogue of the Mumford-Tate conjecture in crystalline cohomology; this amounts to an algebraisation problem for certain formal cycles on a Shimura variety. Finally, the PI plans to apply techniques from the deformation theory of finite flag group schemes to study Hilbert's 13th problem, which asks for the minimal m for which the solution of a general polynomial of degree n can be written as a composite of functions of m variables.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.