Analytic Number Theory on Groups

9800048 Goldenstein这个项目涉及离散群的研究及其在解析数论中的应用。 重点是两个具体的应用。 同余子群的模符号是在微分1-形式的闭圈上的积分，根据志村定理，这些闭圈是与同余子群相关联的雅可比簇的周期的积分线性组合。 PI先前已经证实，这种积分系数在水平上具有多项式增长，这意味着abc-收敛性。 引入新的自守形式（由模符号扭曲）来研究模符号的矩。 特别是，对二阶矩的研究可能会导致在abc猜想的方向上取得进展。 第二个应用涉及著名的Gross-Zagier公式的临界值的L-函数相关的模形式。 最近，PI和S. Zhang等人找到了Gross-Zagier L-值计算的一个更简单、更直接的证明。 其核心思想是计算全纯Poincare级数的基变。 本文将这一证明推广到全真实的域上Hilbert模L-级数的情形。 最终目标是在Birch-Swinnerton-Dyer猜想的方向上获得新的结果，这一研究福尔斯数论的一般数学领域。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。 它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

9800048GoldfeldThis project involves the study of discrete groups with applications to analytic number theory. The focus is on two specific applications. Modular symbols for congruence subgroups are integrals over closed cycles of a differential one-form, which are, by a theorem of Shimura, integral linear combinations of periods of the Jacobian variety associated to the congruence subgroup. The PI has previously conjectured that such integral coefficients have a polynomial growth in the level and that this implies the abc-conjecure. New types of automorphic forms (twisted by modular symbols) are introduced to study the moments of modular symbols. In particular, the investigation of the second moment may lead to progress in the direction of the abc-conjecture. The second application involves the celebrated Gross-Zagier formula for critical values of L-functions associated to modular forms. Recently, the PI and S. Zhang have found a much simpler and very direct proof of the L-value computation of Gross-Zagier. The central idea is to compute the base change of a holomorphic Poincare series. It is proposed to generalize this proof to the case of Hilbert modular L-series over totally real fields. The ultimate goal is to obtain new results in the direction of the Birch-Swinnerton-Dyer conjecture.This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.