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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 !

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  • $\begingroup$ Genus of this curve? $\endgroup$
    – Narasimham
    Mar 25, 2015 at 13:59
  • $\begingroup$ I believe that in your result, you need to assume that $f$ has rational coefficients. $\endgroup$ Mar 25, 2015 at 21:48

1 Answer 1

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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 $H_1(X_0(N),\mathbb Z)$.

Edit: the conclusion was wrong, as Bruno noticed.

For more details about these topics, you can take a look at http://wstein.org/books/modform/modform/weight_two.html or at the nice book by W. Stein "Modular forms, a computational approach".

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  • $\begingroup$ 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. $\endgroup$
    – Mostafa
    Mar 25, 2015 at 19:21
  • $\begingroup$ 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 $H_1(X, \mathbb Z)$. $\endgroup$ Mar 25, 2015 at 21:17
  • $\begingroup$ Thanks Bruno, you're right, I was assuming in my mind that the image is discrete... $\endgroup$
    – Ferra
    Mar 26, 2015 at 7:11
  • $\begingroup$ Well, this is my fault, but I have very few notions on algebraic topology and geometry... I plan to work on it later. Tell me if this answer is correct so I can valid it ! But many thanks for your answer ! $\endgroup$
    – Stabilo
    Mar 28, 2015 at 18:43

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