Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be a compact connected Riemann surface of genus $1$ and suppose that there is a finite morphism $X\to \mathbf{P}^1$ of degree $n$ which ramifies totally over $0$ and $\infty$. Let $f^{-1}(0)$ be the "origin".

Q. Is the point $f^{-1}(\infty)$ of order $n$?

If yes, do I understand correctly that the elliptic curve $E=(X,f^{-1}(0))$ with the point $f^{-1}(\infty)$ defines a point on the modular curve $X_1(n)$?

share|cite|improve this question
up vote 1 down vote accepted

Let $O$ be the preimage of $0$, and let $P$ be the preimage of $\infty$. Then your hypothesis is that $\text{div}(f) = n(P - O)$, and hence that, indeed, $P$ is a point of order $n$ on $X$ when you take $O$ to be the origin.

Thus $(X,P)$ does define a point on $X_1(n)$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.