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

PI将研究由代数循环上的猜想提出的自同构形算术理论中的各种问题，特别是Tate猜想和Bloch-Beilinson猜想。其中一个问题涉及证明四元数下村簇上算术自同构型的积分周期关系。由于迈克尔·哈里斯之前的工作，这些关系一直到代数因素都是已知的。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。这种关系将在L函数的特殊值论中得到应用。此外，用来研究这个问题的方法有望产生新的代数圈的构造。另一个项目是研究圈和p-进L函数之间的关系，特别是对于酉群。这是对《少年派》先前与贝托里尼和达蒙合作研究的一个主题的发展和概括。这一建议的一般领域是代数数论。更具体地说，它研究数域上的代数循环，它可以被认为是给定多项式方程的有理数或整数的解的高维推广。多项式方程(又称丢番图方程)的整数解的研究是近两千年来人们一直感兴趣的问题。这是数学中的一个如此基本的问题，对它找到新的见解可能会有很多应用，不仅适用于数学的其他部分，而且在本质上也是实用的。其中一些关键的研究对象，即椭圆曲线，在编码理论和密码学中有着广泛的实际应用。该提案中的项目不仅将导致对这些物体的更好的理论理解，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.