Modular symbols and applications

在现代数论中，洞察困难的算术问题的最有效方法之一是通过某种类型的曲线（称为椭圆曲线）来可视化它们。这些曲线具有自然形式和简单的定义方程，但它们的结构具有深远的影响，着名的包括Wiles等人对费马大定理的证明。反过来，椭圆曲线的一些最深的性质可以用复平面上的函数来表示，称为L-函数（Wiles证明的核心事实）。例如，定义椭圆曲线的方程有无穷多个有理解，如果相应的L函数在1处求值时为零。这是一个特殊的情况下，'千年问题'之一，其解决方案是一个100万美元的奖金。更好地理解某个L-函数在1消失也应该回答一个问题，最近提出的马祖尔和鲁宾，两个最杰出的数论在过去的百年。代替有理数的集合，考虑一个所谓的数域，一个更大的数的集合，其属性反映了有理数的属性。他们问的问题是：如果我们要求定义椭圆曲线的方程的解属于不同的数域，那么这些解是如何变化的？他们注意到这个问题背后的非零问题可以用一个叫做模符号的基本不变量来表达。这促使他们研究模符号的统计，特别是它们的“均值”和“方差”。在分析数值数据的基础上，他们与W. Stein提出了精确描述这些均值和方差的大尺度行为的公式，在最近的一篇论文中，我们充分证明了这两个公式中的一个。随着这一突破的实现，拟议项目的三个主要目标是：1。建立Mazur等人关于模符号“方差”猜想的充分证明.建立并证明了Theta系数和Theta元的类似公式，它们是上述公式中所描述的对象的重要代数对应.利用目标1和目标2的结果推导出关于L-函数消失的信息。这些目标的意义超越了它们起源的几何背景，因为L-函数的消失问题是解析数论中一个具有巨大意义的领域。该项目将不直接处理应用算术和几何问题，但介绍的方法将是关键的进展算术和几何以及分析方面。同样，虽然数学以外的应用不是提案的一部分，但该项目的结果将间接对椭圆曲线密码学产生潜在影响。

One of the most effective ways to gain insight into difficult arithmetic questions in modern Number Theory is by visualising them through a certain type of curves, called elliptic curves. These are curves with a natural form and a simple defining equation but their structure has far reaching consequences, famously including the proof of Fermat's Last Theorem by Wiles et al. Some of the deepest properties of elliptic curves, in turn, can be formulated in terms of functions on the complex plane called L-functions (a fact at the heart of Wiles' proof). For example, there are infinitely many rational solutions of the equation defining an elliptic curve if the corresponding L-function vanishes upon evaluation at 1. This is a special case of one of the 'Millennium Problems' for whose solution a 1 million USD prize is available.Understanding better the vanishing of a certain L-function at 1 should also answer a question recently asked by Mazur and Rubin, two of the most eminent number theorists in the last hundred years. Instead of the set of rational numbers, consider a so-called number field, a larger set of numbers with properties mirroring those of the rationals. The question they asked is: How do the solutions of the equation defining an elliptic curve change if we ask for those solutions to belong to varying number fields? They noticed that the non-vanishing problem behind this question can be expressed in terms of a fundamental invariant called modular symbol. This motivated them to study the statistics of modular symbols and especially their "mean" and "variance". Upon analysing numerical data, they, together with W. Stein, formulated conjectures describing precisely the large scale behaviour of those means and variances.In a recent paper, we fully proved one of those two conjectures. With that breakthrough achieved, the three main aims of the proposed project then are:1. To establish the full proof of the conjecture by Mazur et al. that deals with the "variance" of modular symbols.2. To formulate and prove analogous conjectures for Theta coefficients and Theta elements, which are important algebraic counterparts of the objects featuring in the above conjectures.3. To use the outcomes of Aims 1 and 2 to derive information about vanishing of L-functions. The significance of these aims goes beyond the geometric setting they originated in, because questions about vanishing of L-functions is an area of huge significance for analytic number theory. The project will not deal directly with applications to arithmetic and geometric problems but the methods introduced will be pivotal for progress in arithmetic and geometric as well as analytic aspects. Likewise, although applications outside Mathematics is not part of the proposal, results from the project will indirectly have potential impact on Elliptic Curve Cryptography.