On a conjecture of Agashe

TL;DR

This paper investigates Agashe's conjecture relating torsion, Shafarevich-Tate groups, and component groups of twisted elliptic curves, providing a proof that extends previous results without assuming optimality.

Contribution

We prove a more general version of Agashe's conjecture for elliptic curve twists, removing the optimality condition and supporting the Birch and Swinnerton-Dyer conjecture.

Findings

Confirmed the divisibility relation predicted by Agashe's conjecture.

Extended the proof to non-optimal elliptic curves.

Supported the BSD conjecture for rank zero cases.

Abstract

Let $E/\mathbb{Q}$ be an optimal elliptic curve, $-D$ be a negative fundamental discriminant coprime to the conductor $N$ of $E/\mathbb{Q}$ and let $E^{-D}/\mathbb{Q}$ be the twist of $E/\mathbb{Q}$ by $-D$. A conjecture of Agashe predicts that if $E^{-D}/\mathbb{Q}$ has analytic rank $0$, then the square of the order of the torsion subgroup of $E^{-D}/\mathbb{Q}$ divides the product of the order of the Shafarevich-Tate group of $E^{-D}/\mathbb{Q}$ and the orders of the arithmetic component groups of $E^{-D}/\mathbb{Q}$, up to a power of $2$. This conjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more general statement without using the optimality hypothesis.