Mathematical Sciences: Homogeneous Space Properties of Moduli Spaces: With Applications to Theta Functions and Finite Fields

模曲线是椭圆曲线上结构的模空间。模函数来自于模曲线的仿射子集上的齐次空间结构--通过SL(2，R)作用。这一奖项为一项研究提供了资金，该研究将模曲线思维应用于没有齐次空间结构的模空间。关键的定义是附加于来自任何有限群G的数据的模塔和划分G的阶数的素数p。本研究将涵盖以下主题的应用。(A)附加到特定类型曲线的theta函数的不变量。(B)有限域上Zeta函数零点之间具有特殊关系的曲线。(C)关于伽罗瓦反问题的结构化结论。(D)使用实点在模塔上的射影系统分析Grothendieck-Teichmuller群的性质的程序。其目的是展示模块化塔楼对经典问题的解决方案，以前没有受益于模块化函数的类比。该奖项还支持博士后Y.Kopeliovich的一项调查。Kopeliovich将继续使用具有特征的theta函数，以给出从模曲线到球面的映射的详细次数界限。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。这一特定的建议通过分类和产生多项式函数，称为例外，特别适合于对数据进行编码，从而应用于密码学。

Modular curves are moduli spaces for structures on elliptic curves. Modular functions come from the homogeneous space structure-through SL(2,R) action- on an affine subset of the modular curve. This award provides funds for an investigation that applies modular curve thinking to moduli spaces that have no homogeneous space structure. The key definition is a Modular Tower attached to the data coming from any finite group G and a prime p dividing the order of G. The investigation will cover applications to the following topics. (a) Invariants attached to theta functions of specific types of curves. (b) Curves over finite fields with special relations among their zeta function zeros. (c) Structured conclusions on the inverse Galois problem. (d) A program for analyzing properties of the Grothendieck- Teichmuller group using projective systems of real points on a modular tower. The objective is to show that modular towers produce solutions to classical problems not previously benefiting from analogs of modular functions. This award also supports an investigation by postdoctoral student Y. Kopeliovich. Kopeliovich will continue using theta functions with characteristic for giving detailed degree bounds on maps from modular curves to the sphere. 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. This particular proposal has applications to cryptography, through the classification and production of polynomial functions, called exceptional, specifically suited for encoding data.