Conceptual understanding of the Gross-Zagier theorem.

Question

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.

Answer 1

Some comments, too extensive to fit into the comment box:

(1) There is a fairly recent reworking of at least some parts of the proof in the book "Heegner points and Rankin $L$-series", MSRI Publ. 49. (Brian Conrad in particular has a paper in there reworking the deformation theory arguments.)

(2) The theorem is a computation: one computes the height of the Heegner point, using Neron-Tate local heights, and relates the answer (a sum of contributions from each place) to a corresponding expression for the derivative.

(3) It is Kolyvagin's work which shows that if the Heegner point is non-zero, then it generates the Mordell-Weil group (up to finite index); so if you want motivation for the truth of Gross--Zagier, you can think of it as being a consequence of BSD + Kolyvagin. (This may be ahistorical, though.)

(4) Historically, Birch was the one who computed Heegner points on elliptic curves, and found that they were generators of the Mordell--Weil group (up to finite index) precisely when the rank was one. This was a big source of encouragement for Gross (as he explained at one point when I was in grad school), because it meant that there should be a relation between the derivative at 1 and the height of the Heegner point, and one just had to find it.

(5) The arithmetico-geometric parts of Gross--Zagier are wonderful; I wouldn't at all think of it as futile to study them. I've not studied the analytic parts, but no doubt they're equally wonderful.

(6) You might start with the Crelle paper of Gross--Zagier, which essentially treats the case of level one. Since the modular curve of level one has genus 0, the height is necessarily zero, and so one gets a very nice formula relating the sum of the finite local heights to the archimedean local height. And one can prove the same formula another way, using a special case of the analytic arguments that in the general setting compute the derivative. The fact that the same formula is obtained these two different ways is a special case of the general Gross--Zagier formula; but it may be simpler to understand the two sides and the comparison between them in this level one setting.

(7) As far as I understand, Kato says nothing in the analytic rank one case. For BSD in this case, one needs Gross--Zagier plus Kolyvagin.

Answer 2

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.

Answer 3

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.