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Let $N \ge 1$. The modular curve $X_0(N)$ is an algebraic curve defined over $\Q$ (or even $\Z[1/N]$) whose points over any algebraically closed field of characteristic not dividing $N$ correspond to isomorphism classes of pairs $(E,\phi)$, where $E$ is an elliptic curve and $\phi$ is an isogeny with domain $E$ and kernel a cyclic group of order $N$, with the exception of particular points called cusps which correspond to degenerate such pairs.

The complex manifold $X_0(N)(\C)$ is biholomorphic to the quotient $\Gamma_0(N) \backslash \mathcal{H}^*$ of the completed upper half plane.

Knowl status:
  • Review status: beta
  • Last edited by Bjorn Poonen on 2022-03-24 16:36:51
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