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

2022/03/23 by Kim, Chan-Ho
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/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 p-converse, the p-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 p Selmer groups is confirmed. 5. An application of the p-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed.

Related

• Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions
• The structure of Selmer groups of elliptic curves and modular symbols
• A higher Gross-Zagier formula and the structure of Selmer groups
• Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory
• Nonvanishing of generalised Kato classes and Iwasawa main conjectures