The structure of Selmer groups of elliptic curves and modular symbols

TL;DR

This paper investigates the structure of Selmer groups of elliptic curves over the rationals at a prime p, using modular symbols and Kolyvagin systems without relying on the main conjecture or height pairing non-degeneracy.

Contribution

It extends previous work by analyzing Selmer groups without assuming the main conjecture or height pairing non-degeneracy, employing modular symbols and Kolyvagin systems.

Findings

Selmer group structure determined by modular symbols and Kolyvagin systems.

No reliance on the main conjecture or p-adic height pairing non-degeneracy.

Provides new insights into Selmer groups of elliptic curves.

Abstract

For an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \cite{Ku6}, assuming the main conjecture and the non-degeneracy of the $p$-adic height pairing, we proved that the structure of the Selmer group with respect to $p$-power torsion points is determined by some analytic elements $\tilde{\delta}_{m}$ defined from modular symbols. In this paper, we do not assume the main conjecture nor the non-degeneracy of the $p$-adic height pairing, and study the structure of Selmer groups, using these analytic elements and Kolyvagin systems of Gauss sum type.