Question

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to \mathbf{P}^1$. What can we say about the branch points of $f$? Is their number bounded? How they depend on $\varphi$, $n$ and $E$?

Answer 1

"Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$". This is not generally true. We only have modularity for $K = \mathbb{Q}$.

Composing with the degree two map to the line is just confusing the issue. You get the ramification of $\phi$ and the ramification of the degree two map and the latter we know.

If we have a map $\phi: X \to E$, where $X$ is a curve of genus $g$ and $E$ is an elliptic curve, then $\phi$ is ramified at $2g-2$ points with multiplicity, by the Hurwitz formula. That answers how many branch points your map has, when you combine it with the standard formula for the genus of $X_0(n)$ which you can find in many books.

Note that your map $\phi$ is not well defined as you can compose it with isogenies and translations on $E$ and automorphisms of $X_0(n)$, so any further question about the location of the branch points has to take that into account. There are some natural choices one could make (send a cusp to the identity on $E$, ...) but I don't have anything sensible to say about the location of the branch points.

Answer 2

Delaunay's thesis (http://delaunay.perso.math.cnrs.fr/these.pdf) goes into this question with some experimental data.