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Let $N \ge 1$. The modular curve $X_1(N)$ is an algebraic curve defined over $\Q$ (or even $\Z[1/N]$) whose points over any algebraically closed field $k$ of characteristic not dividing $N$ correspond to isomorphism classes of pairs $(E,P)$, where $E$ is an elliptic curve over $k$ and $P \in E(k)$ is a torsion point of order exactly $N$, with the exception of particular points called cusps which correspond to degenerate such pairs. If $N \ge 4$, the same description of points is valid even if $k$ is not algebraically closed.

The complex manifold $X_1(N)(\C)$ is biholomorphic to the quotient $\Gamma_1(N) \backslash \mathcal{H}^*$ of the completed upper half plane.

There is an action of the group $(\Z/N\Z)^*$ on $X_1(N)$, in which $c \bmod N$ maps $(E,P)$ to $(E,cP)$. The subgroup $\{\pm 1\}$ acts trivially, because $(E,-P)$ is isomorphic to $(E,P)$ via $[-1]$. The quotient of $X_1(N)$ by $(\Z/N\Z)^* / \{\pm1\}$ is the modular curve $X_0(N)$.

Knowl status:
  • Review status: beta
  • Last edited by Bjorn Poonen on 2022-03-24 16:44:37
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