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I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of elliptic curves at the level of The Arithmetic of Elliptic Curves by Silverman.

I get lost in the linked paper around the third paragraph ("There is a canonical construction of algebraic points...") when they start talking about imaginary quadratic fields, Hilbert class fields, and ideal class groups.

I want to know if there is a book, or set of books, that can teach me enough to understand the gross-zagier formula and all of the terms within it. I'm not expecting to understand it any time soon, but I'm not really sure where to look from here, or even what fields to explore. Perhaps Hartshorne?

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    $\begingroup$ How much class field theory do you know? $\endgroup$ – Bombyx mori Apr 17 '18 at 1:35
  • $\begingroup$ @Bombyxmori Not much at all. $\endgroup$ – TeaFor2 Apr 17 '18 at 1:51
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    $\begingroup$ This is more or less required to understand the topic in depth. Since you are still an undergraduate, may be you can learn the subject first. Hartshorne will be necessary in future as well. One question, though - why pick Gross-Zagier in particular? To me this is the same as picking up some field medalist's paper and studying all subjects (perhaps 40+ other papers, books) related to it. Not sure if this is a good strategy. $\endgroup$ – Bombyx mori Apr 17 '18 at 1:53
  • $\begingroup$ @Bombyx mori I’m currently researching in a seemingly completely unrelated field (a family of functions encoding the eigenvalues of the Laplacian of a compact Riemannian manifold), and I started seeing this formula pop up in the literature. I’m really motivated to understand the weird connection there; and it’s not a school project so ive got plenty of time. $\endgroup$ – TeaFor2 Apr 17 '18 at 2:14
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    $\begingroup$ Not wanting to quibble, but how can you understand elliptic curves at that level without knowing basic algebraic number theory, which would include at least number fields, rings of integers, and ideal class groups. For example, how can you prove the Mordell-Weil theorem without knowing that the ideal class group is finite? In any case, to read Gross-Zagier, you'll certainly need to learn algebraic number theory and basic class field theory, plus the theory of modular forms and modular curves, which are required to construct the Heegner points that lie at the heart of the G-Z formula. $\endgroup$ – Joe Silverman Apr 17 '18 at 2:19
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Section 6 of Zagier's chapter in 1-2-3 of modular forms might be of use for you

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I highly recommend "Primes of the form $x^2 + ny^2$" by David Cox. You can probably start with chap 2, which gives a very readable summary of basic algebra theory and the statements of class field theory. He then applies these materials to study ring class fields and elliptic curves with complex multiplication, Weber functions etc. It does not contain a proof of Gross-Zagier, but it certainly will help you fill in a lot of the background material. And it's beautifully written! Good luck!

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