$X_{\text{sp}}(N)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of a Cartan subgroup $\begin{pmatrix} * & 0\\ 0& * \end{pmatrix}$ that is split at every prime dividing $N$. As a moduli space it parameterizes triples $(E,C,D)$ where $E$ is an elliptic curve over $k$, and $C$ and $D$ are $\Gal_k$-stable cyclic subgroups such that $E[N](\overline{k})\simeq C \oplus D$.

**Knowl status:**

- Review status: beta
- Last edited by Asimina Hamakiotes on 2024-07-12 14:02:27

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**History:**(expand/hide all)

- 2024-07-12 14:02:27 by Asimina Hamakiotes
- 2024-07-12 14:01:04 by Asimina Hamakiotes
- 2024-06-14 21:34:56 by Asimina Hamakiotes
- 2023-11-26 23:01:25 by John Voight
- 2023-11-26 23:00:53 by John Voight
- 2023-07-09 09:00:55 by Andrew Sutherland
- 2022-11-06 16:55:49 by Ciaran Schembri

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