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.
