On a conjecture of Agashe

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.