Covers of the Sphere and Moduli of Curves

摘要对于建议#0200225，Helmut Voelklein：Hurwitz空间将给定的分支型和单调群的覆盖参数化.每个Hurwitz空间都有到亏格g曲线的模空间的自然映射(对于适当的g).这在代数几何中已经使用了很长一段时间，但只适用于单覆盖(特别是它有一个对称群作为单群)的情况。Fry和Voelklein构造了具有任意单群的覆盖的Hurwitz空间，并将其应用于Galois逆问题。所提出的研究探索了与Hurwitz空间相关的群论方法如何应用于曲线的模的研究，其中包括计算群论的算法方法，这些方法在研究有限群的生成系统上的辫子群作用时特别有用。推广用非零亏格的穿孔曲面的映射类群代替辫子群。伽罗华逆问题的应用是可以预期的。该项目部分是与G.Frey，K.Magaard和S.Shectorov合作的。群论是对(任何物体)对称图案的抽象研究。群论的许多算法都是在现代计算机代数系统中实现的，所提出的研究利用这些计算机代数系统来发现和研究高度对称的代数曲线族。代数曲线是数学、物理和密码学等应用中的基本对象，椭圆曲线密码学为互联网上的数据安全提供了加密方案。

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.