Algebraic cycles, L-functions and rational points on elliptic curves

本文旨在研究L函数的特殊值与周期、周期之间的关系，并对一些公开的关于L函数的猜想，如Birch-Swinnerton-Dyer和Bloch-Kato-Beilinson的猜想有重要的应用。具体地说，作者和他的合作者将(I)研究与Rankin-Selberg L-函数及其在Abel-Jacobi映射下的映象有关的代数圈，并应用这些结果给出CM椭圆曲线上有理点的新构造：(Ii)研究p-进L-函数和关于CM Hida变形的岩泽主猜想；(3)探索了四元数模形式与伴随L-值相关周期的一个猜想的证明方法，并应用于Bloch-Kato猜想的某些情况；(4)研究了亚普勒群表示的三重积上不变线性形式的构造和计数问题，从而将与整权模形式相关的三重积L-函数的结果推广到半整权模形式的设置上。这项提议的总体焦点是数论领域。数论与素数和丢番图方程等对象有关。除了可能是最古老的数学分支外，它在当今世界具有重要意义，因为许多密码协议(在互联网上安全传输所需的)和纠错码(光盘、硬盘等所需的)是基于数论方法的。这些实际应用实际上涉及到相当复杂的几何对象，如椭圆曲线。在过去的半个世纪里，我们已经意识到，通过研究某些被称为L函数的函数，可以更好地理解这些几何对象。人们猜测，关于几何对象的非常有趣的信息被编码在相关的L函数在某些特定点的行为中。调查员希望通过目前提案中要做的工作，加深我们对这一联系的理解。这个项目的具体成果之一将是找到某些三次方程的有理数解的新方法，这是数论中的一个中心问题。

This proposal aims to investigate relations between special values of L-functions, cycles and periods with important applications to some open conjectures about L-functions such as those of Birch-Swinnerton-Dyer and Bloch-Kato-Beilinson. Specifically, the investigator and his collaborators will (i) Study the algebraic cycles associated to Rankin-Selberg L-functions and their images under Abel-Jacobi maps, and apply these results to give new constructions of rational points on CM elliptic curves; (ii) Study p-adic L-functions and the Iwasawa main conjecture for CM Hida deformations; (iii) Explore methods to prove a conjecture of his relating periods of quaternionic modular forms to adjoint L-values, with applications to some cases of the Bloch-Kato conjecture (iv) Study the problem of constructing and counting invariant linear forms on a triple product of representations of the metaplectic group, thus generalizing results on triple product L-functions associated to modular forms of integral weight to the setting of modular forms of half-integral weight. The general focus of this proposal is the area of number theory. Number theory has to do with such objects as prime numbers and diophantine equations. Other than being perhaps the oldest branch of mathematics, it is of great significance in today's world, since many cryptographic protocols (needed for secure transmissions over the internet) and error correcting codes (needed for compact discs, hard discs and the like) are based on number theoretic methods. These practical applications in fact involve rather sophisticated geometrical objects such as elliptic curves. Over the last half-century, we have realized that one can gain a better understanding of these geometric objects by studying certain functions, called L-functions. Conjecturally one expects that very interesting information about the geometric object is encoded in the behavior of the associated L-function at certain special points. The investigator hopes to deepen our understanding of this connection through the work to be done in the current proposal. One of the concrete consequences of this project will be a new method to find solutions in rational numbers to certain cubic equations, a central problem in number theory.