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I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:

(1.) Finding a uitable imaginary-quadratic extension $K$ of $\mathbb{Q}$ with no rank growth and calculating the Néron-Tate height pairings $\left<x,\sigma(x)\right>$ of Heegner points $x$ of conductor $1$ with their conjugates.

(2.) Showing that the function $\sum_{m\geq 1}\left<x,\sum_{\sigma\in\operatorname{Cl}_{K}}T_{m}\sigma(x)\right>$ pairs with $f$ under the Petersson scalar product to $L'(f,1).$

When you read the paper for the first time it feels like you are being hit in the face with a bombastic load of calculations the sense of which you understand only when you have reached the final chapter.

In order to get myself a little more "familiar" with the methodology of the paper, in particular with step one, I want to ask how it can be generalized.

Remember that Heegner points of an elliptic curve $E$ of conductor $N$ over an imaginary quadratic field $K$ can be parametrized by equivalence classes of maps $$\mathbb{C}/\mathcal{O}_{c} → \mathbb{C}/\mathcal{N}_{c}$$ where $\mathcal{N}$ is a ray class of $K$ dividing $N,$ $c$ is a natural number prime to the discriminant of $K,$ $$\mathcal{O}_{c}=\mathbb{Z}+c\cdot\mathcal{O}_{K},$$ and $\mathcal{N}_{c}:=\mathcal{O}_{c}\cap\mathcal{N}.$

My question is related to step (1.) mentioned above: Can the method of Gross and Zagier be adapted or has it been adapted to calculate the height pairings of Heegner points and their conjugates of conductor greater than $1$? I am not asking about $p$-adic heights (there seems to be tons of literature about that) but about Néron-Tate heights. A reference would also be nice.

Remark: This is not about "How do you solve the BSD conjecture", I am rather trying to get a grasp of the methodology that is used.

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  • $\begingroup$ I don't feel that I am the right person to answer this. Certainly the Gross-Zagier formula has been generealised. However the particular situation is not completely clear (to me and others). For instance Bradshaw and Stein wstein.org/papers/bs-heegner/bs-heegner.pdf lamented that there are wrong and contradictory formulae published. It is said that "The Gross-Zagier Formula on Shimura Curves" by Xinyi Yuan, Shou-wu Zhang & Wei Zhang should give you the correct formula. $\endgroup$ – Chris Wuthrich Sep 5 '16 at 19:37

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