Algebraic Cycles and L-Values

求多元多项式方程的整数或有理数的解可以追溯到世纪的丢番图。这个数学的中心主题已经看到了许多著名数学家的贡献，包括费马，欧拉和高斯。世纪早期丢番图方程的现代研究始于希尔伯特的二次型理论，随后是Hasse和Minkowski的局部到整体原理的显着发现。韦伊进行了思想的哈塞，闵可夫斯基进一步，纳入想法的黎曼zeta函数，以定义什么是现在所谓的哈塞-韦伊zeta函数，这是建立在一些解决方案的多项式方程在更简单的设置模块算术。 人们能否在一定程度上从哈塞-韦伊zeta函数（因而也从模算术的解）中恢复出整数解或有理解？在1960年代，基于计算实验，Birch和Swinnerton-Dyer证明，对于一类多项式方程（对应于椭圆曲线），在其对称中心的Hasse-Weil zeta函数的消失揭示了无穷多个解的存在性。本研究项目旨在以Birch and Swinnerton-Dyer（B-SD）猜想为先导，深化对丢番图方程的理解，研究有理代数圈（丢番图方程有理解概念的自然高维推广）及其与函数域和数域上的L函数的特殊值的关系。Gross-Zagier定理和Kolyvagin定理证明了当椭圆曲线的Hasse-Weil L-函数的解析秩至多为1时的B-SD猜想。私家侦探之一？的目标是建立相同类型的结果在某些高维情况下：Gan-Gross-Prasad循环的情况下，从产品的酉志村品种，和新的情况下产生的对称对的约化群。这将建立贝林森，布洛赫和加藤的新案例。该项目的另一个目标是研究Shtukas模空间上的特殊循环，这可能会揭示函数域上的B-SD猜想。 该项目的方法来自自守L函数理论和相对迹公式，循环和模空间的代数几何技术，以及调和分析和局部域上还原群的表示理论。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影响审查标准进行评估来支持。

Finding solutions in integers or rational numbers of polynomial equations in several variables dates back to Diophantus in the 3rd century. This central subject in mathematics has seen contributions made by many well-known mathematicians including Fermat, Euler, and Gauss. Modern study of Diophantine equations in the early 20th century started with the theory of quadratic forms by Hilbert, followed by the remarkable discovery of the local-to-global principle of Hasse and Minkowski. Weil carried the idea of Hasse-Minkowski further, incorporating the idea of Riemann zeta function, to define what is now called Hasse-Weil zeta function, which is built up on the numbers of solutions of polynomial equations in the much simpler setting of modular arithmetic. Can one recover to a certain extent the integral or rational solutions from the Hasse-Weil zeta function (and hence from the solutions in modular arithmetic)? In the 1960s, based on computational experiment, Birch and Swinnerton-Dyer conjectured that, for a class of polynomial equations (corresponding to elliptic curves), the vanishing of the Hasse-Weil zeta function at the center of its symmetry reveals the existence of infinitely many solutions. This research project aims to deepen the understanding of Diophantine equations in the direction pioneered by the Birch and Swinnerton-Dyer (B-SD) conjecture.The project is to study rational algebraic cycles (a natural high dimensional generalization of the concept of rational solutions to Diophantine equations) and their connection to the special values of L-functions over both functional fields and number fields. The theorems of Gross-Zagier and of Kolyvagin proved the B-SD conjecture when the analytic rank of the Hasse-Weil L-function of an elliptic curve is at most one. One of the PI?s goals is to establish the same type of results in certain high dimensional cases: the case of the Gan-Gross-Prasad cycles from the product of unitary Shimura varieties, and the new case arising from a symmetric pair of reductive groups. This would establish new cases of conjectures of Beilinson, Bloch, and Kato. Another goal of the project is to study special cycles on the moduli space of Shtukas, which is likely to shed light on the B-SD conjecture over function fields. The methods in the project are from the theory of automorphic L-functions and of relative trace formula, algebro-geometric technique for cycles and moduli spaces, and techniques from harmonic analysis and representation theory of reductive groups over local fields.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.