Let $N \in \mathbb{N}$. The *modular curve* $X_1(N)$ is an algebraic curve defined over $\Q$ whose point correspond to pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a torsion point on $E$ of order exactly $N$, with the exception of particular points called *cusps* which correspond to degenerate such pairs.

The set of its complex points $X_1(N)(\C)$ is naturally isomorphic to the quotient of the completed upper half plane $\Gamma_1(N) \backslash \mathcal{H}^*$ as a Riemann surface.

If $c$ is an integer such that $\gcd(c,N) = 1$, and $(E,P)$ is a pair as above, then so is $(E,cP)$; if $c' \equiv c \bmod N$ then $(E,c'P) = (E,cP)$. This yields an action of the group $(\Z/N\Z)^*$ on $X_1(N)$, with kernel $\{\pm 1\}$ (because $(E,-P)$ is always equivalent with $(E,P)$). The quotient of $X_1(N)$ by $(\Z/N\Z)^* / \{\pm1\}$ is the modular curve $X_0(N)$.

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- Review status: beta
- Last edited by Noam D. Elkies on 2019-03-19 15:02:22

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