Galois Groups and Matrices

9970357本文主要讨论正则逆Galois问题与两个问题的联系：第一，N. Katz在1996年出版的“Rigid Local Systems”一书中提出的一些问题; Frey和Kani提出的关于椭圆曲线的亏格2覆盖和椭圆曲线的相关算术性质的一些问题。 理论和应用性质的最新发展--即有限简单群的分类以及强大的计算机代数系统的出现--重新点燃了伽罗瓦理论的许多研究活动，围绕着伽罗瓦理论的逆问题，数学的主要开放问题之一：哪种类型的对称性可以发生的问题（代数方程的解）。

9970357This proposal is about connections between the Regular Inverse Galois Problem and two other topics: Firstly, the theory of the middle convolution developed by N. Katz in his book " Rigid Local Systems", published in 1996; and secondly, certain questions raised by Frey and Kani about genus 2 covers of elliptic curves and related arithmetic properties of elliptic curves.Galois theory, the study of symmetries between solutions of high degree algebraic equations, has been a centerpiece of algebra for hundreds of years. Recent developments of both theoretical and applied nature --- namely, the classification of finite simple groups as well as the advent of powerful computer algebra systems ---have rekindled a lot of research activity in Galois theory, centering around the Inverse Problem of Galois Theory, one of the major open problems of mathematics: The question which type of symmetries can occur (among the solutions of algebraic equations).