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The principal congruence subgroup of level $N$, $\Gamma(N)$, of the integral symplectic group $\Sp(2g,\Z)$ is the subgroup of $\Sp(2g,\Z)$ that consists of matrices congruent to the identity matrix modulo $N$: \[ \Gamma(N)=\left\{ \begin{pmatrix}a&b\\c&d\end{pmatrix}\in \Sp(2g,{\Z}) : \begin{pmatrix}a&b\\c&d\end{pmatrix} \equiv \begin{pmatrix}1_g&0\\0&1_g\end{pmatrix} \bmod N \right\} \] where $1_g$ stands for the identity matrix of size $g$. For $g=1$, it is the subgroup $\Gamma(N)$ of $\SL(2,\Z)$.

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  • Review status: beta
  • Last edited by Fabien Cléry on 2021-05-06 12:37:14
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