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The paramodular group $K(N)$ of level $N$ is defined as $$K(N)=\left\{ \left(\begin{matrix} * & N* & * &*\\ * & * & * & */N\\ * & N* &*&*\\ N*&N*&N*&* \end{matrix}\right) : *\in{\Bbb Z}\right\}\cap \mathrm{Sp}(4, {\Bbb Q})$$ where $\mathrm{Sp}(4, {\Bbb Q})$ is the symplectic group with rational entries.

The paramodular group is defined more intrinsically as the stabilizer in $\Sp(4,\Q)$ of $\Z\oplus\Z\oplus\Z\oplus N\Z$ (column vectors). Thus it comprises the $\Sp(4,\Q)$ matrices $$\left(\begin{matrix} * & * & * &*/N\\ * & * & * & */N\\ * & * &*&*/N\\ N*&N*&N*&*\end{matrix}\right)$$ but the paramodular group conditions imply that such matrices satisfy the stronger entrywise conditions shown above.

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• Review status: beta
• Last edited by Jerry Shurman on 2016-03-31 13:57:03
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