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

PI将研究代数循环猜想所提出的自同构形式的算术理论中的各种问题，特别是Tate猜想和Bloch-Beilinson猜想。其中一个问题是证明四元数Shimura变数上算术自同构形式的积分周期关系。由于迈克尔·哈里斯先前的工作，这些关系被称为代数因子。PI建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函数的特殊值理论。此外，用于研究这一问题的方法有望产生代数循环的新结构。另一个项目涉及研究环和p进l函数之间的关系，特别是对于酉群。这发展和概括了PI之前与贝托里尼和达蒙合作研究的主题。这个建议的一般领域是代数数论。更具体地说，它涉及数域上代数循环的研究，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（也称为丢芬图方程）的整数解的研究是一个近两千年来一直引起人们兴趣的问题。这是数学中的一个基本问题，找到对它的新见解可能会有很多应用，不仅是数学的其他部分，而且在自然界中也是实用的。一些将要研究的关键对象，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导致对这些物体更好的理论理解，而且还会开发新的计算工具来研究它们。

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.