Genus one curves, Selmer groups and Tate-Shafarevich groups

这项建议由两部分组成。第一部分旨在完成Ciperiani与A.Wiles合作已经开始的一类曲线的分析。他们希望证明定义在Q上的每条亏格1曲线都有一个定义在Q上的可解扩张上的点。第二部分的主题是关于素数p的素数p的虚二次扩张的Z_p-扩张上的Selmer群和Tate-Shafarevic群的p-初等部分的结构。Ciperiani的研究是在算术代数几何领域。这门学科结合了代数几何和数论的技巧。一方面，代数几何从分析可以在平面上用多项式定义的图形开始。另一方面，数论的历史根源在于对自然数的研究。抛开这些差异不谈，这两个主题始终相互影响。

This proposal consist of two parts. The first part aims to complete the analysis of genus one curves that Ciperiani has already started in collaboration with A. Wiles. They hope to show that every genus one curve defined over Q has a point defined over some solvable extension of Q. The subject of the second part of this proposal concerns the structure of the p-primary part of Selmer groups and Tate-Shafarevic groups over Z_p-extensions of an imaginary quadratic extension of Q for primes p of good reduction.Ciperiani's research is in the field of arithmetic algebraic geometry.This subject combines techniques of algebraic geometry and number theory. On the one hand, algebraic geometry started by analyzing figures that could be defined in the plane by polynomials. On the other hand, number theory has its historical roots in the study of natural numbers. Independently of these differences, these two subjects have always influenced each other.