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Let $E$ be an elliptic curve defined over $\mathbf{Q}$. It is known that $E$ admits a modular parametrization $\phi : X_0(N) \to E$, where $N$ is the conductor of $E$.

We may normalize $\phi$ such that it maps, say, the cusp $\infty$ to the origin of $E$. By the Manin-Drinfeld theorem, it is then known that $\phi$ maps every cusp of $X_0(N)$ to a (possibly non-rational) torsion point of $E$.

I'm interested in finding elliptic curves $E$ such that the preimage of $E_{\mathrm{tors}}(\mathbf{Q})$ consists only of cusps. Since the degree of the modular parametrization goes to infinity with $N$, it seems to me reasonable to expect that there are only finitely many elliptic curves satisfying this condition, but I see no easy argument to prove it. Hence the questions :

  1. Is it known that there are only a finite number of $(E,\phi)$ as above such that $\phi^{-1}(0)$ consists only of cusps ?

  2. If the answer to Q1 is yes, is it possible to find all elliptic curves satisfying this condition ?

I might also add the following question, because I don't know the answer to it :

3: Given an explicit elliptic curve $E$ and an explicit cusp $x$ of $X_0(N)$, is there a simple way to compute the ramification index of $\phi$ at $x$ ?

Note that Q3 makes sense because $\phi$ is well-defined up to composition by a finite étale morphism $E \to E$, so the ramification index at $x$ is well-defined. I know that $\phi$ is always unramified at $\infty$ because the differential form associated to the modular form $f_E$ doesn't vanish at $\infty$, but I don't know how to compute the ramification index in general.

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    $\begingroup$ Dear Francois, Regarding your third question, isn't it just a question of computing the order of vanishing of $f_E$ at the cusp in question? Regards, Matthew $\endgroup$ – Emerton Feb 13 '12 at 19:06
  • $\begingroup$ Dear Matthew, Yes this reduces to compute the order of vanishing of $f_E$ at $x$. More generally one can ask about the Fourier expansion of $f_E | g$ for any $g \in \mathrm{SL}_2(\mathbf{Z})$ but it's not so clear to me how to compute it using modular symbols. $\endgroup$ – François Brunault Feb 13 '12 at 20:19
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Question 3 has been fully answered by Corbett and Saha in On the order of vanishing of newforms at cusps. They show in particular that the ramification index of a modular parametrization $X_0(N) \to E$ always divides 24, and depends only on the local automorphic representations $\pi_2$ and $\pi_3$ attached to $E$.

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