Capitulation Discriminants of Genus One Curves
In this thesis we study genus one curves of degree n embedded in projective space P n−1 . We are speci cally interested in the curves de ned over Q that are counterexamples to the Hasse principle, that is the curves that have points everywhere locally but no Q-points. These curves correspond to non-trivial elements of the Tate-Shafarevich group of the Jacobian curve E of C. Such a curve C will always admit a point over a degree n eld extension F of Q, and we say C capitulates over F, and that the discriminant of F is a capitulation discriminant for C. Our aim will be to obtain a bound for the capitulation discriminant in terms of the invariants of the minimal model of the elliptic curve E. To do so, we study the minimal free resolution of the ideal de ning curve C, and extend the classical theory of genus one models of degree n, where n = 1, 2, 3, 4 or 5, to all odd values of n. In particular we prove a formula giving a equation for the Jacobian curve E in terms of the invariants of a genus model associated to C, and we prove a minimization result, stating that if C is everywhere locally soluble, then it can be de ned by an integral genus one model with invariants as small as possible. In parallel, we obtain a formula for the discriminant of a rank n ring, realized as a ring of functions of n points in the projective space Pn−2, for all n. Our methods extend the classical results that give parametrizations of rank n rings for n ≤ 5. We then study rings obtained by intersecting the curve C with a hyperplane, and use geometry of numbers together with a compactness argument to prove a bound for the capitulation discriminant. For small values of n we then explain how to make this bound e ective. In the second part of the thesis we study Kolyvagin classes in the p-Selmer group of an elliptic curve E, for p an odd prime. Our aim will be to obtain explicit geometric representation of these classes, which frequently correspond to non-trivial elements of the Tate-Shafarevich group of E, as degree p curves embedded in the projective space P p−1 . These classes are de ned using Heegner points, which are points of E de ned over a degree 2p dihedral extension L of Q. We explain how to compute Heegner points explicitly, and then give a method, based on Galois descent, to obtain a geometric representation of the Kolyvagin class as an element of H^1(Q, E[p]), from its definition in terms of the Heegner point as an element of H^1(L, E[p]). We remark that our method will naturally result in nice integral models for these curves, i.e. minimal in the sense of the first part of the thesis. At the end we give examples, including an example of a genus one curve in P 6 that represents a non-trivial element of the 7-torsion part of the Tate-Shafarevich group of the Jacobian E.