Agashe–Stein divisibility conjecture for optimal modular abelian varieties

Let $A/\mathbb{Q}$ be an abelian variety with $L(A,1)\neq 0$ that is an optimal quotient of $J_0(N)/\mathbb{Q}$ attached to a newform. Let $c_{\infty}(A)$ be the number of connected components of $A(\mathbb{R})$, let $m_A$ be the Manin constant, let $\Sha(A/\mathbb{Q})$ be the Shafarevich–Tate group, and let $c_p(A)$ denote the Tamagawa number at $p$. Agashe–Stein divisibility conjecture. The order of the torsion subgroup of the dual abelian variety satisfies

$$|A^{\vee}(\mathbb{Q})_{\textrm{tors}}|\mid c_{\infty}(A)\cdot m_A\cdot|\Sha(A/\mathbb{Q})|\cdot\prod_p c_p(A).$$

This divisibility is presented as a consequence of the analytic-rank-zero Birch and Swinnerton-Dyer conjecture together with the known integrality result of Agashe and Stein. The paper proves it unconditionally in many cases, including semistable elliptic curves, but the stated conjecture is not established in full generality.