abstract: We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences: itemize0pt Kato's Kolyvagin system is non-trivial. It is the cyclotomic analogue of Kolyvagin's conjecture. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture. The rank zero p-converse, the p-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. The conjecture of Kurihara on the semi-local description of mod p Selmer groups is confirmed. An application of the p-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.
Chan-Ho Kim (Fri,) studied this question.