Covers of the Sphere and Moduli of Curves

Helmut Voelklein提案# 0200225的摘要：Hurwitz空间参数化给定分支类型和单值群的spetheris的覆盖。每个Hurwitz空间都有一个到亏格g曲线的模空间的自然映射（对于合适的g）。这在代数几何中已经应用了很长时间，但仅限于简单覆盖（特别是具有对称群作为单值群）的情况。Fried和Voelklein构造了具有任意单值群的覆盖的Hurwitz空间，并将其应用于逆Galois问题。拟议的研究探讨了如何与Hurwitz空间相关的群论方法可以应用于曲线的模的研究。这包括计算群论的算法方法，这在有限群的生成系统上的辫子群作用的研究中特别有用。推广用非零亏格的穿孔曲面的映射类群代替辫子群。本项目部分是与G. Frey，K. Magaard和S.群论是对（任何物体的）对称模式的抽象研究。群论中的许多算法都在现代计算机代数系统中实现，本研究利用这些计算机代数系统来寻找和研究高对称代数曲线族。代数曲线是数学、物理和应用的基本对象，例如，椭圆曲线密码提供了用于互联网上数据安全的加密方案。

Abstract for proposal # 0200225, Helmut Voelklein:Hurwitz spaces parametrize covers of the sphereof given ramification type and monodromy group.Each Hurwitz space has a natural map to the moduli space of genus g curves (for suitable g). Thishas been used in algebraic geometry for a long time,but only in the case of simple covers (which in particular have a symmetric group as monodromy group).Hurwitz spaces of covers with arbitrary monodromy group were constructed by Fried and Voelklein, and applied to the Inverse Galois problem. Proposed research explores how the group-theoreticmethods associated with Hurwitz spaces can be applied in the study of the moduli of curves.This includes algorithmic methods of computational group theory, which are especially useful in the study of the braid group action on generatingsystems of a finite group. Generalizations replace the braid group by the mapping class group of a punctured surface of non-zero genus. Applicationsto the Inverse Galois Problem are to be expected.The project is partially in cooperation with G. Frey, K. Magaard and S. Shpectorov.Group Theory is the abstract study of symmetry patterns (of any object). Many algorithms of Group Theoryare implemented in modern computer algebra systems.Proposed research uses these computer algebra systems to find and study families of highly symmetric algebraic curves. Algebraic curves are basic objectsin mathematics, physics and applications, e.g., cryptography.Elliptic curve cryptography provides encryption schemesused for data security on the internet.