Arithmetic Geometry

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方面发挥了核心作用，如费马最后定理和关于曲线上有理点个数的莫德尔猜想。本研究项目的主要研究对象是Shimura簇，它是模曲线的高维类似物，它将Riemann曲面族参数化。Shimura簇的研究处于代数几何、数论和表示论的交界处，在许多影响深远的猜想中有着广泛的应用，本课题的主要研究课题之一就是Shimura簇的算法。其中一部分工作旨在将一类下村变种的Hasse-Weil Zeta函数--Hoge型变种--与自同构的L函数联系起来，从而建立了一个朗兰兹猜想。另一个目的是研究这些簇上特殊点的结构，并证明在某些情况下，每个mod p同源类都包含一个特殊点的约化。该项目的其他方向包括Grothendieck-Katz p-Curvature猜想，以及构造代数圈的丢番图方法。

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 Shimura varieties, which are higher dimensional analogues of modular curves that parametrize families of Riemann surfaces. The study of Shimura varieties is at the interface of algebraic geometry, number theory, and representation theory and has broad applications to a number of far-reaching and influential conjectures.One of the main topics of this research project concerns the arithmetic of Shimura varieties. One part of the work aims to relate the Hasse-Weil zeta function of a certain class of Shimura varieties -- those of Hodge type -- to automorphic L-functions, establishing a conjecture of Langlands. Another aims to study the structure of special points on these varieties and to show that, in certain situations, every mod p isogeny class contains the reduction of a special point. Other directions in the project include the Grothendieck-Katz p-curvature conjecture, as well Diophantine methods for constructing algebraic cycles.