TITLE: Conceptual understanding of the Gross-Zagier theorem.
QUESTION [22 upvotes]: The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more conceptual understanding.
The more conceptual attempts I know are the following:
$1$. The work of Kolyvagin on Birch-Swinnerton-Dyer conjecture, in which he re-proves part of Gross-Zagier using Euler systems. The problem with this is that some of the original Gross-Zagier is still needed for getting the results on BSD conjecture(if I understand things correctly. Please point out if I am wrong).
$2$. The volume of Darmon and Zhang published by MSRI, in which they attempt a $p$-adic theory. Again this is going away from the original complex analytic case. Again please correct me if I am wrong.
So I am wondering whether anybody published a more conceptual approach to the complex analytic Gross-Zagier theorem. I would be grateful for any references.

REPLY [4 votes]: Indeed, there is a conceptual understanding of this via "incoherent Siegel-Weil Formula",cft S.Kudla `s papers.See also the last section of recent preprint of Gan-Gross-Prasad.

REPLY [4 votes]: In my current (no very deep) understanding, there are two possible ways to make the proof of the Gross-Zagier more conceptual.
The first is to recognize in each terms  of the equation products of local terms which are local linear functionals. Now, a famous theorem of Saito and Tunnell states that such linear functionals live in a dimension 1 vector space. So there are proportional and the Gross-Zagier amounts to specifying the factor of proportionality. This requires a large amount of representation theory, but I think now this program has been completed. Using Gross-Prasad conjecture in place of Saito-Tunnell, GZ can apparently be extended widely.
The second is to observe that the $p$-adic variant of GZ is in fact easier to prove (this is because $p$-adic heights naturally factor through the first Bloch-Kato cohomology group). Conceptually, this is maybe not so surprising because Heegner points verify the distribution relations of an Euler system, so they are naturally linked to the $p$-adic $L$-function. Hence, to prove the Gross-Zagier theorem, all there is to do is to relate the special value of the derivative of the $p$-adic $L$-function to the value of the derivative of the complex $L$-function. But here is the rub: as far as I know, proving that the derivative of the $p$-adic $L$-function interpolates $p$-adically the derivative of the $L$-function is more or less equivalent to showing that the $p$-adic height pairing is not degenerate. So this looks hopeless.