Topics in automorphic Forms and Algebraic Cycles

自守形式和代数循环主题

• 批准号: 2401548
• 负责人: Wei Zhang
• 金额: $ 60万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401548&HistoricalAwards=false
• 关键词: Topics; automorphic; Forms; Algebraic; Cycles

This awards concern research in Number Theory. Solving polynomial equations in rational numbers dates back to Diophantus in the 3rd century and has been a central subject in mathematics for generations. The modern study of Diophantine equations has incorporated the revolutionary idea of Riemann from his use of a class of special functions called "zeta functions” or "L-functions". Such special functions are built up on counting the numbers of solutions of polynomial equations in the much simpler setting of modular arithmetic. In the 1960s, Birch and Swinnerton-Dyer came up with a remarkable conjecture revealing a relation between the zeros of L-functions and the solutions to a special class of polynomial equations in the rationals. Later Beilinson and Bloch conjectured that, for general polynomial equations in the rationals, there should always be a relation between the zeros of L-functions and algebraic cycles which are “parameter solutions to polynomial equations”. The project will study the zeros of L-functions through automorphic forms and special cycles on modular varieties. The theory of automorphic form provides a fruitful way to access the zeros of L-functions. The modular varieties are either Shimura varieties over number fields or moduli spaces of Shtukas over function fields. They play a central role in modern number theory and arithmetic geometry, and they often come with a great supply of algebraic cycles. The project aims to prove results relating zeros of L-functions and algebraic cycles on modular varieties, including new cases of the arithmetic Gan–Gross–Prasad conjecture for Shimura varieties associated to unitary groups, certain Higher Gross–Zagier formula over function fields, and the function field analog of Kudla’s program with an emphasis on the modularity of generating series of special cycles and the arithmetic Siegel—Weil formula. The project will also develop new relative trace formula, a powerful equation connecting spectral information and geometric structure, to study general automorphic period integral including the unitary Friedberg–Jacquet period. The broader impacts of this project include mentoring of graduate students and seminar organization.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.

该奖项关注的是数论中求解有理数多项式方程的研究，其历史可以追溯到三世纪的丢番图，并且一直是数代数学的中心课题，现代对丢番图方程的研究融入了黎曼的革命性思想。使用一类称为“zeta 函数”或“L 函数”的特殊函数，这些特殊函数是在更简单的模运算设置中计算多项式方程解的数量而建立的。 20 世纪 60 年代，Birch 和 Swinnerton-Dyer 提出了一个非凡的猜想，揭示了 L 函数的零点与有理数中一类特殊多项式方程的解之间的关系，后来 Beilinson 和 Bloch 猜想，对于有理数，L 函数的零点和代数循环之间应该始终存在关系，即“多项式方程的参数解”。通过自同构形式和模簇上的特殊循环来研究 L 函数的零点 自同形理论提供了一种访问 L 函数零点的有效方法，模簇要么是数域上的 Shimura 簇，要么是 Shtukas 的模空间。它们在现代数论和算术几何中发挥着核心作用，并且通常提供大量代数环，该项目旨在证明与 L 函数和代数零点相关的结果。模簇上的循环，包括与酉群相关的 Shimura 簇的算术 Gan-Gross-Prasad 猜想的新案例、函数域上的某些更高 Gross-Zagier 公式，以及 Kudla 程序的函数域模拟，重点是模块性该项目还将开发新的相对迹公式，这是一个连接光谱信息和几何结构的强大方程，以研究一般自守周期积分，包括。该项目的更广泛影响包括对研究生的指导和研讨会组织。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。