Heegner Points, L-Functions of Elliptic Curves, and Generalizations

这个研究项目涉及数学中的一个基本问题：求解代数方程。关于解的信息编码在各种数学对象中：代数循环、自同构形式和l函数。该项目旨在加深对这些数学对象及其之间联系的理解。它还将推进理解椭圆曲线算法的技术，特别是Birch和Swinnerton-Dyer猜想，它是克莱数学研究所的七个千年奖问题之一。椭圆曲线和类似的代数方程在密码学等其他学科中有广泛的应用；研究结果也有望促进对这些领域的理解。本研究包括数论方面的几个课题：Heegner点的算法、椭圆曲线的l函数及其高维推广。本研究将探讨Heegner点之间的同余及其各种应用，包括二次扭转族椭圆曲线上的Goldfeld猜想和Rubin-Silverberg族椭圆曲线的秩。该项目还将研究在加性素数下椭圆曲线的Birch和Swinnerton-Dyer公式。对于高维的推广，研究者将探讨由算术Gan-Gross-Prasad猜想和算术基本引理引起的Rapoport-Zink空间上的算术相交问题。还计划启动一个新方案，同时推广Waldspurger和Gross-Zagier公式。该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知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project concerns one of the basic questions in mathematics: solving algebraic equations. Information about solutions is encoded in various mathematical objects: algebraic cycles, automorphic forms, and L-functions. The project aims to deepen the understanding of these mathematical objects and the connection between them. It will also 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. Elliptic curves and similar algebraic equations have wide application in other disciplines such as cryptography; results of the research are expected to advance understanding in these areas as well.The research comprises several projects in number theory: the arithmetic of Heegner points, L-functions of elliptic curves, and their higher dimensional generalizations. The work will investigate the congruences between Heegner points and their various applications, including Goldfeld's conjecture on elliptic curves in quadratic twists families and the rank of elliptic curves in Rubin-Silverberg families. The project will also investigate the Birch and Swinnerton-Dyer formula for elliptic curves at additive primes. For higher-dimensional generalizations, the investigator will explore arithmetic intersection problems on Rapoport-Zink spaces arising from arithmetic Gan-Gross-Prasad conjectures and the arithmetic fundamental lemma. It is also planned to initiate a new program for simultaneous generalization of the Waldspurger and Gross-Zagier formulas.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.