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.
An elliptic curve E over the rational numbers is a genus one curve defined over the rationals with a point defined over the rationals. The Weil-Chatelet group of E over rationals consists of genus one curves defined over the rationals with E as their Jacobian. The Weil-Chatelet group contains the Tate-Shafarevich group which consists of genus one curves with E as their Jacobian which have local points but no point defined over the rationals (i.e. these curves should have a rational point but they don't). While the Weil-Chatelet group is certainly infinite the Tate-Shafaravich group of E over the rationals is conjectured to be finite. The main object of this project was the Tate-Shafarevich group. One question that one may ask is whether a version of the finiteness of the Tate-Shafarevich group is preserved when the rationals are replaced by an anticyclotomic Z_p extension of some imaginary quadratic field. One of the outcomes of this project is a positive answer to the above question for primes of supersingular reduction and imaginary quadratic fields satisfying the Heegner hypothesis. Another question that one may pose (and Cassels did so in 1961) is how the Tate-Shafarevich group sits inside the Weil-Chatelet group. More precisely, Cassels asks whether the elements of the Tate-Shafarevich group are infinitely divisible in the Weil-Chatelet group. In joint work with Jakob Stix, we have shown that elements of the Tate-Shafarevich group of E over the rationals are divisible in the Weil-Chatelet group by any integer which is not divisible by 2, 3, 5, and 7. We have also studied the above question in the case when the rationals are replaced by a number field. We have shown that similar results hold in that case and moreover the set of excluded primes depends only on the degree of the number field.
