Algebraic Cycles and L-Values

用整数或有理数求解多个变量多项式方程可以追溯到3世纪的丢芬图斯。这个数学的中心课题由许多著名的数学家做出了贡献，包括费马、欧拉和高斯。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，随后是哈塞和闵可夫斯基对局域到全局原理的重大发现。韦尔进一步发扬了哈塞-闵可夫斯基的思想，结合黎曼ζ函数的思想，定义了现在所谓的哈塞-韦尔ζ函数，它建立在多项式方程的解的数量上，在更简单的模算法的设置上。一个人能否在一定程度上从Hasse-Weil zeta函数（因此从模算法的解）中恢复到积分解或有理解？在20世纪60年代，Birch和Swinnerton-Dyer在计算实验的基础上推测，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函数的消失表明存在无穷多个解。本研究项目旨在沿着Birch和Swinnerton-Dyer （B-SD）猜想所开创的方向加深对丢番图方程的理解。本课题研究有理代数循环（丢芬图方程有理解概念的自然高维推广）及其与l函数在泛函域和数域上的特殊值的联系。Gross-Zagier定理和Kolyvagin定理证明了椭圆曲线的Hasse-Weil l -函数的解析秩不大于1时的B-SD猜想。PI中的一个？s的目标是在某些高维情况下建立相同类型的结果：由酉Shimura变积产生的Gan-Gross-Prasad环的情况，以及由对称约化群对产生的新情况。这将为贝林森、布洛赫和加藤的猜想建立新的案例。本课题的另一个目标是研究Shtukas模空间上的特殊环，这可能有助于解释函数场上的B-SD猜想。本课题采用自同构l -函数及其相关迹公式理论、环与模空间的代数-几何技术、局部域上约化群的调和分析与表示理论等方法。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。

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.