The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves

abstract: We reveal a new and refined application of (a \emph{weaker} statement than) the Iwasawa main conjecture for elliptic curves to the \emph{structure} of Selmer groups of elliptic curves of \emph{arbitrary} rank. For a large class of elliptic curves, we obtain the following arithmetic consequences: \begin{itemize}\labelwidth0pt \item[$\bullet$] Kato's Kolyvagin system is non-trivial. It is the cyclotomic analogue of Kolyvagin's conjecture. \item[$\bullet$] 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. \item[$\bullet$] The rank zero $p$-converse, the $p$-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. \item[$\bullet$] The conjecture of Kurihara on the semi-local description of mod $p$ Selmer groups is confirmed. \item[$\bullet$] An application of the $p$-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.