# Preimage of torsion points under modular parametrizations

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.

• 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 – Emerton Feb 13 '12 at 19:06
• 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. – François Brunault Feb 13 '12 at 20:19

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 $$E$$.