Geometric Langlands Program and Arithmetic Algebraic Geometry

朗兰兹计划是一个数学框架，它统一了数学中许多不同领域的问题，特别是数论和线性代数。传统的算术朗兰兹程序已经研究了50多年，并在求解经典的丢番图方程中得到了重要的应用，例如费马最后定理的求解。几何朗兰兹课程相对较新，正在迅速发展，因为它还与几何和物理等其他学科联系在一起，人们可以从这些学科中得出直觉。在这个项目中，首席研究员将通过应用几何方法研究算术问题来探索朗兰兹哲学的这两个不同方面之间的联系。该奖项还将为研究生学习与这一令人兴奋但要求严格的研究领域相关的主题提供支持。更详细地，首席研究员将推广他之前关于下村品种mod p纤维之间上同调对应的结果，并将它们应用于算术水平上升问题和Beilinson-Bloch-Kato猜想的研究。在此过程中，他将发展必要的元素，例如p-adic群的驯服局部几何langaland对应。首席研究人员还将继续他对上同调理论中的p-进方面的研究，并将它们与复杂方面和L-进方面进行比较和联系。这一奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Langlands Program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and linear algebra. The traditional arithmetic Langlands program has been studied for more than fifty years, and has found significant applications to solving classical Diophantine equations, such as the solution of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development as it is also connected with other subjects such as geometry and physics, from where one can draw intuitions. In this project, the principal investigator will explore connections between these two different facets of the Langlands philosophy by applying geometric methods to study arithmetic problems. This award will also provide support for graduate students studying topics related to this exciting but demanding area of research.In more detail, the principal investigator will generalize his previous results on cohomological correspondences between mod p fibers of Shimura varieties, and apply them to the study of arithmetic level rising problems and the Beilinson-Bloch-Kato conjecture. Along the way, he will develop necessary ingredients such as the tame local geometric Langalands correspondence for p-adic groups. The principal investigator will also continue his study of p-adic aspects of cohomology theory with coefficients, and compare and relate them with the complex aspects and l-adic aspects.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.