Arithmetic Geometry and Automorphic L-Functions

该研究项目涉及数学中的一个基本问题：求解代数方程。关于解的信息被编码在各种数学对象中，包括算术变量和自守L函数。这项研究旨在加深对这些数学对象的理解，特别是在更高的维度上，这需要在不同领域开发新的工具和互动，并吸引新的视角，这些视角可能会为长期存在的开放问题带来新的启示。该项目还旨在推进理解椭圆曲线算术的技术，特别是Birch和Swinnerton-Dyer猜想，这是克莱数学研究所的七个千禧年奖问题之一。该研究包括几个将算术几何与自守L-函数相关的项目，围绕Gross-Zagier公式的推广和应用这一共同主题。PI将研究分歧酉群和正交群的Kudla-Rapoport猜想。PI计划扩展算术内积公式，并将其应用于椭圆曲线的对称幂动机的Beilinson-Bloch和Bloch-Kato公式。PI还将研究一种新的算术相对轨迹公式方法，用于正交志村品种的Gross-Zagier型公式。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知识价值和更广泛的影响审查标准进行评估来支持。

The research project concerns one of the basic questions in mathematics: solving algebraic equations. Information about solutions is encoded in various mathematical objects, including arithmetic varieties and automorphic L-functions. This research aims to deepen the understanding of these mathematical objects, especially in higher dimension, which requires developing new tools and interactions in diverse areas and appealing to new perspectives that may shed new light on longstanding open questions. The project also aims to advance the techniques for understanding the arithmetic of elliptic curves, particularly the Birch and Swinnerton-Dyer conjecture, one of the seven Millennium Prize Problems of the Clay Mathematics Institute.The research consists of several projects relating arithmetic geometry with automorphic L-functions, centered on the common theme of the generalization and applications of the Gross–Zagier formula. The PI will investigate the Kudla–Rapoport conjecture for ramified unitary groups and orthogonal groups. The PI plans to extend the arithmetic inner product formula and apply it to the Beilinson–Bloch and Bloch–Kato conjectures of symmetric power motives of elliptic curves. The PI will also investigate a new arithmetic relative trace formula approach towards a Gross–Zagier type formula for orthogonal Shimura varieties.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.