# Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:

Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered pairs $(E,E^{'})$ of elliptic curves together with cyclic isogeny $E\rightarrow E^{'}$ of degree $N$. Gross uses on some steps the cyclic isogeny between two elliptic curves over $\mathbb{C}$. One of the books that I have read to understand the theory of modular curves is "A first course in Modular forms, written by Fred Diamond and Jerry Shurman".

Theorem 1.5.1.(page 38)

Let $N$ be a positive integer.

(a) The moduli space for $\Gamma_{0}(N)$ is $$S_{0}(N)=\{[E_{\tau},\langle1/N+\Lambda_{\tau}\rangle]:\tau\in H\},$$ with $H$ is the upper half plan. Two points $[E_{\tau},\langle1/N+\Lambda_{\tau}\rangle]$ and $[E_{\tau^{'}},\langle1/N+\Lambda_{\tau^{'}}\rangle]$ are equal if and only if $\Gamma_{0}(N)\tau=\Gamma_{0}(N)\tau^{'}$. Thus there is a bijection $$\psi:S_{0}(N)\rightarrow Y_{0}(N), [\mathbb{C}/\Lambda_{\tau},\langle1/N+\Lambda_{\tau}\rangle]\mapsto \Gamma_{0}(N)\tau.$$

So, how can we make the link between the equivalence classes of the enhanced elliptic curves for $\Gamma_{0}(N)$ (= the equivalence classes of $(E,C)$ where $E$ is a complex elliptic curve and $C$ is a cyclic subgroup of $E$ of order $N$) defined in Diamond/Shurman's book and the cyclic isogenies $E\rightarrow E^{'}$ of degree $N$ used by Gross.

I also ask if there is any other paper which explains the theory of Heegner points explicitly?

I have look at Darmon's note and Gross-Zagier paper "Heegner points and derivatives of L-series" and it seems that the both were influenced by Gross's paper! Is there any other paper which explains Heegner points explecitely and independently of Gross's paper?

(I keep this post open for any further question about Gross's paper and I apologise for any mistakes in my English.)

Thank you.

• Given $(E,C)$ you form $E \rightarrow E/C$, and given $f:E \rightarrow E'$ you form $(E, \ker f)$. But if you're trying to do something arithmetic (such as analyze fields of definition of specific analytically-defined points relative to a specific $\mathbf{Q}$-structure) then you need a definition of $Y_0(N)$ that goes beyond the upper half-plane description. Commented Nov 28, 2013 at 4:08

## 2 Answers

I am not sure whether this is anything like what you are looking for, but Brian Birch has a very nice historical survey of Heegner points avant Gross.

Coates has a Bourbaki talk on that subject, which should help you (the Birch reference on the beginnings is nice too).