Selmer groups, component groups and Heegner points

We study various Selmer groups and Shafarevich-Tate groups of elliptic curves over number fields, prove new theoretical results and provide computational evidence towards the Birch and Swinnerton-Dyer conjecture. First, we exploit the Shafarevich-Tate groups of modular abelian varieties over number fields using the method of visibility with respect to embeddings into Jacobians of modular curves whose levels are multiples of the conductor of the level of the modular abelian variety. We establish criteria for the existence of visible elements and use these criteria to provide evidence for the Birch and Swinnerton-Dyer conjecture in the case when the associated Hasse-Weil L-function does not vanish at s = 1. Second, we refine and improve the Euler system method of Kolyvagin to establish in many cases the exact upper bounds on the p-primary part of the Shafarevich-Tate group of an elliptic curve over a quadratic imaginary fields as predicted by the Birch and Swinnerton-Dyer conjectural formula. This is done in the crise when the order of vanishing of the L-function 2 of the curve at s = 1 is equal to 1. When the L-function of the elliptic curve vanishes up to higher order, Kolyvagin conjectured that the Heegner point Euler system associated to the elliptic curve is nontrivial. We prove some consequences of Kolyvagin's conjecture and present the first theoretical and computational evidence for the conjecture. Finally, we prove an equidistribution result for Galois orbits of Heegner points with respect to reduction which we believe could be useful for a better understanding of Kolyvagin's conjecture.