Arithmetic of automorphic forms: cycles, periods and p-adic L-functions

自守形式的算术：循环、周期和 p 进 L 函数

• 批准号: 1160720
• 负责人: Kartik Prasanna
• 金额: $ 40万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160720&HistoricalAwards=false
• 关键词: Arithmetic; automorphic; forms; cycles; periods

The PI will study various problems in the arithmetic theory of automorphic forms suggested by conjectures on algebraic cycles, notably the Tate conjecture and the Bloch-Beilinson conjecture. One of the problems involves proving integral period relations for arithmetic automorphic forms on quaternionic Shimura varieties. These relations are known up to algebraic factors, due to previous work of Michael Harris. The PI proposes to prove much more precise relations that identify more or less exactly the missing algebraic factors. Such relations would have applications to the theory of special values of L-functions. In addition, the methods used to study this problem are expected to yield new constructions of algebraic cycles. Another project involves studying the relations between cycles and p-adic L-functions, especially for unitary groups. This develops and generalizes a theme studied in the PI's previous work with Bertolini and Darmon. The general area of this proposal is algebraic number theory. More specifically, it deals with the study of algebraic cycles over number fields which may be thought of as a higher dimensional generalization of the solutions in rational numbers or integers to a given polynomial equation. The study of integer solutions to polynomial equations (also called Diophantine equations) is a problem that has interested people for the last two thousand years. It is such a basic problem in mathematics that finding new insights into it is likely to have many applications, not just to other parts of mathematics but also practical in nature. Some of the key objects that will be studied, namely elliptic curves, have many practical applications to coding theory and cryptography. The projects in the proposal will lead to not just a better theoretical understanding of such objects, but also develop new computational tools to study them.

PI将研究代数圈猜想所提出的自同构算术理论中的各种问题，特别是泰特猜想和布洛赫-贝林森猜想。问题之一涉及证明四元数 Shimura 簇上算术自守形式的积分周期关系。由于迈克尔·哈里斯 (Michael Harris) 之前的工作，这些关系由代数因素可知。 PI 提议证明更精确的关系，或多或少准确地识别缺失的代数因子。这种关系可以应用于 L 函数特殊值的理论。此外，用于研究这个问题的方法预计会产生代数循环的新结构。另一个项目涉及研究循环和 p 进 L 函数之间的关系，特别是酉群。这发展并概括了 PI 之前与 Bertolini 和 Darmon 合作研究的主题。 该提案的总体领域是代数数论。更具体地说，它涉及数域上的代数循环的研究，这可以被认为是给定多项式方程的有理数或整数解的高维推广。多项式方程（也称为丢番图方程）整数解的研究是过去两千多年来人们一直感兴趣的一个问题。这是数学中的一个基本问题，因此对其进行新的见解可能有很多应用，不仅适用于数学的其他部分，而且本质上也很实用。将要研究的一些关键对象，即椭圆曲线，在编码理论和密码学中有许多实际应用。提案中的项目不仅会带来对这些物体更好的理论理解，还会开发新的计算工具来研究它们。