# Modular parametrization of elliptic curve

Let $$f$$ be a cusp form of weight $$2$$ on $$\Gamma_0(N)$$ and assume that $$f$$ is a Hecke form and a newform. Then, we easily see that $$C(\gamma)=2i\pi \int_{\tau}^{\gamma \tau}{f(\tau')d\tau'} \quad (\gamma \in \Gamma_0(N))$$ is independant of $$\tau$$ (assumed in $$\mathcal{H}$$) and defines a group morphism from $$\Gamma_0(N)$$ to $$(\textbf{C},+)$$.

I can't find any references for the following statement (I think that it must be well-known) :

## Theorem :

The image of $$C$$ in $$(\textbf{C},+)$$ is a lattice unless $$f$$ is trivial (i.e. $$f=0$$).

I am looking for any references or ideas of a proof. Many thanks !

• Genus of this curve? – Narasimham Mar 25 '15 at 13:59
• I believe that in your result, you need to assume that $f$ has rational coefficients. – Bruno Joyal Mar 25 '15 at 21:48

I think this should follow from the fact that there exists a perfect pairing $$\langle\cdot ,\cdot\rangle \colon H_1(X_0(N),\mathbb R)\times H^0(X_0(N),\Omega^1_{\mathbb C})\to \mathbb C$$ $$\langle\{\alpha,\beta\},f\rangle=\int_{\alpha}^{\beta}f$$ Here $\{\alpha,\beta\}$ is the real homology class on $X_0(N)$ of any path from $\alpha$ to $\beta$ in $\mathcal H^*$ and $f$ is a cuspform of weight $2$ and level $\Gamma_0(N)$ (under the canonical identification of $H^0(X_0(N),\Omega^1_{\mathbb C})$ with $S_2(\Gamma_0(N))$).
Now there is a theorem which tells you that for any congruence subgroup $G\leq \text{SL}_2(\mathbb Z)$ and any fixed $\alpha\in \mathcal H^*$ the map $$G\to H_1(X_G,\mathbb Z)$$ $$g\mapsto \{\alpha,g\alpha\}$$ is a surjective group homomorphism which does not depend on $\alpha$. (For a proof of this, you can look at the paper "Parabolic points and zeta-functions of modular curves" by Y.Manin). This implies that the image of your homomorphism coincides with the image of the homomorphism $$\langle\cdot,f\rangle\colon H_1(X_0(N),\mathbb Z)\to \mathbb C$$ $$\gamma\mapsto \int_{\gamma}f$$ obtained by restricting the bilinear pairing I defined above to $H_1(X_0(N),\mathbb Z)$.
• Why global holomorphic forms on $X_0(N)$ corresponds to weight 2 cusp forms? I think all weight 2 modular forms induce a holomorphic form on the quotient. – Mostafa Mar 25 '15 at 19:21
• Hi Ferra, there is a problem in the above proof in the sense that $\mathbb C$ contains free $\mathbb Z$-modules of arbitrary rank, thus there is no reason a priori why the rank should be $\leq 2$. In fact, the fact that $f$ is an eigenform, which you have not used, is crucial; for a generic $f$, I would even think that the period mapping is injective on $H_1(X, \mathbb Z)$. – Bruno Joyal Mar 25 '15 at 21:17