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

PI将研究各种问题的算术理论的自守形式所建议的代数周期，特别是泰特猜想和布洛赫-贝林森猜想。其中一个问题涉及到四元数志村簇上算术自守形式的整周期关系的证明。这些关系是已知的代数因素，由于以前的工作迈克尔哈里斯。PI建议证明更精确的关系，以确定或多或少准确地丢失的代数因子。这种关系可以应用于L-函数的特殊值理论。此外，用于研究这个问题的方法预计会产生新的代数圈的结构。另一个项目涉及研究循环和p-adic L-函数之间的关系，特别是对于酉群。这发展和概括了PI以前与Bertolini和Darmon的工作中研究的主题。 这个建议的一般领域是代数数论。更具体地说，它涉及研究代数循环的一些领域，可以被认为是一个高维的推广的解决方案，在有理数或整数到一个给定的多项式方程。研究多项式方程（也称为丢番图方程）的整数解是一个在过去两千年里引起人们兴趣的问题。这是数学中的一个基本问题，发现新的见解很可能有许多应用，不仅适用于数学的其他部分，而且在本质上也很实用。将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。该提案中的项目不仅将导致对此类对象的更好的理论理解，而且还将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.