Algebraic Cycles and L-Values

求整数或有理数的多元多项式方程的解可以追溯到丢番图在3世纪。这是数学中的核心课题，包括费马、欧拉和高斯在内的许多著名数学家都做出了贡献。20世纪初对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的局部到全局原理的显著发现。Weil进一步发扬了Hasse-Minkowski的思想，结合了Riemann Zeta函数的思想，定义了现在称为Hasse-Weil Zeta函数的概念，该函数建立在多项式方程的解的数目上，模运算的设置要简单得多。人们能从Hasse-Weil Zeta函数(从而从模算术中的解)恢复到一定程度的积分解或有理解吗？20世纪60年代，基于计算实验，Birch和Swinnerton-Dyer猜想，对于一类多项式方程(对应于椭圆曲线)，Hasse-Weil Zeta函数在其对称性中心的消失揭示了无穷多解的存在。本研究以Birch和Swinnerton-Dyer(B-SD)猜想为先导，旨在加深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函数的特定值之间的关系。当椭圆曲线的哈斯-韦尔-L函数的解析秩至多为1时，Gross-Zagier和Kolyvan in的定理证明了B-SD猜想。Pi？S的目标之一是在某些高维情形下建立相同类型的结果：由酉Shimura簇的乘积得到的Gan-Gross-Prasad圈的情形，以及由对称的约化群对产生的新情形。这将建立贝林森、布洛赫和加藤猜想的新案例。该项目的另一个目标是研究Shtukas模空间上的特殊循环，这可能有助于揭示函数域上的B-SD猜想。该项目中的方法来自自同构L函数理论和相对迹公式，圈和模空间的代数几何技术，以及局部域上简群的调和分析和表示理论的技术。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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.