Modular symbols and applications

在现代数论中，要深入了解复杂的算术问题，最有效的方法之一是通过一种称为椭圆曲线的曲线将它们可视化。这些曲线具有自然的形式和简单的定义方程，但它们的结构具有深远的影响，其中著名的包括怀尔斯等人对费马大定理的证明。椭圆曲线的一些最深奥的性质，反过来，可以用复平面上的函数来表述，称为l函数（这是怀尔斯证明的核心事实）。例如，如果相应的l函数在1处求值时消失，则定义椭圆曲线的方程有无限多个有理解。这是“千年问题”中的一个特例，其解决方案可获得100万美元的奖金。更好地理解某个l函数在1处的消失，也应该回答最近一百年来最杰出的两位数论家Mazur和Rubin提出的一个问题。考虑一个所谓的数字字段，而不是有理数的集合，这是一个更大的数字集合，其属性反映了有理数的属性。他们问的问题是：如果我们要求椭圆曲线方程的解属于不同数量的域，那么这些解会发生什么变化？他们注意到，这个问题背后的不消失问题可以用一个叫做模符号的基本不变量来表达。这促使他们研究模块化符号的统计，特别是它们的“均值”和“方差”。在分析了数值数据后，他们与W. Stein一起，提出了一些推测，精确地描述了这些均值和方差的大规模行为。在最近的一篇论文中，我们完全证明了这两个猜想中的一个。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”的猜想的完整证明。2 .对上述猜想中所包含对象的重要代数对应物θ系数和θ元，提出并证明类似的猜想。利用目标1和目标2的结果推导l函数消失的信息。这些目标的意义超出了它们最初的几何背景，因为关于l函数消失的问题对解析数论来说是一个非常重要的领域。该项目不会直接涉及算术和几何问题的应用，但所介绍的方法将对算术和几何以及分析方面的进展至关重要。同样，尽管数学以外的应用不是提案的一部分，但项目的结果将间接地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.