Preimage of torsion points under modular parametrizations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:57:00Z http://mathoverflow.net/feeds/question/88355 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88355/preimage-of-torsion-points-under-modular-parametrizations Preimage of torsion points under modular parametrizations François Brunault 2012-02-13T16:41:25Z 2012-02-13T16:41:25Z <p>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 $E$.</p> <p>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 $E$.</p> <p>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 :</p> <blockquote> <ol> <li><p>Is it known that there are only a finite number of $(E,\phi)$ as above such that $\phi^{-1}(0)$ consists only of cusps ?</p></li> <li><p>If the answer to Q1 is yes, is it possible to find all elliptic curves satisfying this condition ?</p></li> </ol> </blockquote> <p>I might also add the following question, because I don't know the answer to it :</p> <blockquote> <p>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 $x$ ?</p> </blockquote> <p>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.</p>