Genus one curves, Selmer groups and Tate-Shafarevich groups

本建议包括两个部分。第一部分旨在完成Ciperiani与A.怀尔斯他们希望证明定义在Q上的每一个亏格一条曲线都有一个定义在Q的某个可解扩张上的点。本建议的第二部分的主题涉及结构的p-主要部分的塞尔默群和泰特-Shafarevic群在Z_p-扩展的虚二次扩展Q为素数p良好的减少。Ciperiani的研究是在算术代数几何领域。这一主题结合了技术的代数几何和数论。一方面，代数几何开始于分析可以在平面上用多项式定义的图形。另一方面，数论在自然数的研究中有其历史根源。独立于这些差异，这两个主题总是相互影响的。

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.