Global divisibility conjecture for Heegner points and Tamagawa numbers

Let $E_{/\mathbb{Q}}$ be an elliptic curve of conductor $N$, let $K$ be the quadratic imaginary field associated with a Heegner discriminant, and let $y_K\in E(K)$ be the corresponding Heegner point. For an odd prime $p$, assume

$$p\nmid N$$

and that the mod-$p$ Galois representation

$$\rho_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\operatorname{GL}(E[p])$$

is surjective, equivalently $\operatorname{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})\cong\operatorname{GL}_2(\mathbb{F}_p)$. Let $m_\infty$ be the non-negative integer measuring the global $p$-divisibility of the Heegner points used in Kolyvagin's construction, and let $c_q$ be the Tamagawa number of $E$ at $q$. Global divisibility conjecture. If $p$ satisfies this hypothesis, then

$$m_\infty=\operatorname{ord}_p\left(\prod_{q\mid N}c_q\right).$$

This is the reformulation of the Birch and Swinnerton-Dyer formula obtained by comparing Kolyvagin's formula for the $p$-primary part of ${\cyr X}(E/K)$ with the conjectural Tamagawa correction factor. The conjecture concerns the relationship between global divisibility of Heegner points and local Tamagawa numbers, and remains open in the source.