The Arithmetic of Automorphic Forms

自守形式的算术

• 批准号: 2101888
• 负责人: Aaron Pollack
• 金额: $ 20.74万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101888&HistoricalAwards=false
• 关键词: Arithmetic; Automorphic; Forms

The area of mathematics known as number theory concerns understanding integer and rational solutions to polynomial equations. These solution sets are conjecturally connected to what are called automorphic forms, high-dimensional analogues of the trigonometric sine and cosine functions. Like the sine and cosine functions, automorphic forms are functions that satisfy certain differential equations and have infinitely many discrete symmetries, and they are objects of intense mathematical study in their own right, not just for their connection to polynomial equations. The PI will investigate topics in the study of automorphic forms, especially those automorphic forms whose system of symmetries is "exceptional." The PI will also investigate topics in the L-functions of automorphic forms. L-functions are generalizations of the Riemann zeta function, and conjecturally contain large amounts of subtle arithmetic information.In more detail, this project has two distinct areas of focus. The first concerns unexpected arithmeticity in a class of special automorphic forms on exceptional groups. There is evidence that (non-holomorphic) "modular forms" on exceptional groups behave surprisingly similarly to classical holomorphic modular forms and possess surprising arithmetic features. This project aims to further develop the theory of these modular forms on exceptional groups, such as developing the mathematics that could be used to produce a database of modular forms on the exceptional group G_2. The second focus of this project concerns work consistent with Beilinson's conjecture about the special values of L-functions of motives. Efforts in this direction involve obtaining regulator formulas for generalized Beilinson-Flach motivic classes and finding a generalization of the Kronecker limit formula. The techniques and ideas involved in the project include exceptional theta correspondences, the Rankin-Selberg method, and Deligne cohomology.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.

称为数论的数学领域涉及理解多项式方程的整数和有理解。这些解集推测与所谓的自同构形式、三角正弦和余弦函数的高维类似物相关。与正弦和余弦函数一样，自守形式是满足某些微分方程并具有无限多个离散对称性的函数，它们本身就是深入数学研究的对象，而不仅仅是因为它们与多项式方程的联系。 PI 将研究自同构形式的研究主题，特别是那些对称系统“例外”的自同构形式。 PI 还将研究自守形式的 L 函数中的主题。 L 函数是黎曼 zeta 函数的推广，据推测包含大量微妙的算术信息。更详细地说，该项目有两个不同的重点领域。第一个涉及例外群上的一类特殊自同构形式中意想不到的算术性。有证据表明，特殊群上的（非全纯）“模形式”的行为与经典全纯模形式惊人地相似，并且具有令人惊讶的算术特征。该项目旨在进一步发展异常群上这些模形式的理论，例如开发可用于生成异常群 G_2 上的模形式数据库的数学。该项目的第二个重点涉及与 Beilinson 关于动机 L 函数的特殊值的猜想相一致的工作。在这个方向上的努力包括获得广义 Beilinson-Flach 动机类的调节公式以及找到克罗内克极限公式的推广。该项目涉及的技术和想法包括特殊的 theta 对应、Rankin-Selberg 方法和德利涅上同调。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。

项目成果

Modular forms on indefinite orthogonal groups of rank three

三阶不定正交群的模形式

• DOI: 10.1016/j.jnt.2021.09.011
• 发表时间: 2022-09
• 期刊: Journal of Number Theory
• 影响因子: 0.7
• 作者: Pollack, Aaron;Savin, Gordan
• 通讯作者: Savin, Gordan

The completed standard L-function of modular forms on $$G_2$$

$$G_2$$ 上已完成的模块化形式的标准 L 函数

• DOI: 10.1007/s00209-022-03067-8
• 发表时间: 2022-09
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Çiçek, Fatma;Davidoff, Giuliana;Dijols, Sarah;Hammonds, Trajan;Pollack, Aaron;Roy, Manami
• 通讯作者: Roy, Manami