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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 $X_1(n)$?

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1 Answer 1

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 $X_1(n)$.

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