Gross–Zagier conjecture on the index of a Heegner point

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, let $L/\mathbb{Q}$ be an imaginary quadratic field satisfying the Heegner hypothesis, and let $P_L\in E_L(L)$ be the associated Heegner point. Let $m$ be the Manin constant of $E/\mathbb{Q}$, let $c(E/\mathbb{Q})$ be the product of the Tamagawa numbers, let $2u_L$ be the number of roots of unity in $L$, and let $\Sha(E_L/L)$ denote the Tate–Shafarevich group.

Gross–Zagier conjecture. If $P_L$ has infinite order in $E_L(L)$, then $P_L$ generates a subgroup of finite index in $E_L(L)$, and this index equals

c(E/\mathbb{Q)\cdot m\cdot u_L\cdot\sqrt{|\Sha(E_L/L)|}.

This conjecture predicts an explicit formula for the Mordell–Weil index of a Heegner point in terms of local Tamagawa factors, the Manin constant, and the Tate–Shafarevich group. The supplied text gives no resolution status.