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.

该奖项涉及数字理论的研究。解决有理数的多项式方程的历史可以追溯到3世纪的Diophantus，并且几代人一直是数学中的核心主题。对二芬太汀方程式的现代研究结合了Riemann的革命性思想，他使用了一类称为“ Zeta函数”或“ L-Functions”的特殊功能。这种特殊功能建立在计算更简单的模块化算术设置中的多项式方程解决方案的数量上。在1960年代，Birch和Swinnerton-Dyer提出了一个显着的猜想，揭示了L功能的零与理性中多项式方程的解决方案之间的关系。后来，贝林森（Beilinson）和布洛克（Bloch）猜想，对于理由中的一般多项式方程式，L功能的零和代数循环的零应该始终存在关系，这是“多项式方程的参数解决方案”。该项目将通过模块化品种上的自动形式和特殊周期来研究L功能的零。自动形式的理论为进入L功能的零提供了一种富有成果的方式。模块化品种是数字字段上的shimura品种，要么是shtukas的模量空间。它们在现代数字理论和算术几何形状中发挥着核心作用，并且通常具有大量代数周期。 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算术siegel - 韦尔公式。项目还将开发新的相对痕迹公式，这是一个连接光谱信息和几何结构的强大方程式，以研究一般的自动型时期积分，包括统一的Friedberg – Jacquet时期。该项目的广播公司影响包括研究生的心理和开创性组织。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被视为通过评估而被视为珍贵的支持。