In my reading about modular curves, the author states that a certain group $G$ acts on the function field $K$ of the modular curve $X_1(N)$, hence it acts on $X_1(N)$ itself. What is this induced action? I cannot think of anything particularly natural. Is there always a natural action of a group $G$ on a curve $X$ if $G$ acts on $K(X)$?
2026-04-02 23:38:02.1775173082
Induced Galois Action on Modular Curve
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