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

本文旨在研究l -函数的特殊值、周期和周期之间的关系，并将其应用于一些关于l -函数的开放猜想，如Birch-Swinnerton-Dyer和Bloch-Kato-Beilinson的猜想。具体而言，研究者和他的合作者将(i)研究与Rankin-Selberg l-函数相关的代数循环及其在Abel-Jacobi映射下的图像，并应用这些结果给出CM椭圆曲线上有理点的新结构；（ii）研究CM - Hida变形的p进l函数和Iwasawa主猜想；（iii）探索证明四元数模形式周期与伴随l值有关的一个猜想的方法，并将其应用于Bloch-Kato猜想的某些情况。（iv）研究在元群表示的三重积上构造和计数不变线性形式的问题，从而将与积分权的模形式有关的三重积l函数的结果推广到半积分权的模形式的设置。这个建议的总体重点是数论领域。数论与质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之外，它在当今世界具有重要意义，因为许多加密协议（需要在互联网上安全传输）和纠错码（需要在光盘，硬盘等）都是基于数论方法的。这些实际应用实际上涉及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，人们可以通过研究某些函数来更好地理解这些几何物体，这些函数被称为l函数。从推测上讲，人们期望几何对象的有趣信息被编码在相关的l函数在某些特殊点的行为中。研究者希望通过当前提案中要做的工作加深我们对这一联系的理解。这个项目的一个具体成果将是为某些三次方程找到有理数解的新方法，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.