$p$-Selmer group and Modular symbols

TL;DR

This paper establishes a link between the size of the $p$-Selmer group of an elliptic curve and analytic quantities derived from modular symbols, confirming a conjecture by Kurihara.

Contribution

It proves that the $p$-Selmer group's dimension is governed by modular symbol-related analytic quantities, advancing understanding of elliptic curves and Selmer groups.

Findings

Dimension of $p$-Selmer group controlled by modular symbols

Confirmation of Kurihara's conjecture

Analytic quantities predict Selmer group size

Abstract

In this paper, we prove that the dimension of the $p$-Selmer group for an elliptic curve is controlled by certain analytic quantities associated with modular symbols, which is conjectured by Kurihara.