The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves
arXiv:2203.12159
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. 1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture. 2. 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. 3. The rank zero -converse, the -parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. 4. The conjecture of Kurihara on the semi-local description of mod Selmer groups is confirmed. 5. An application of the -adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.
Final version. To appear in American Journal of Mathematics