$m$-symplectic matrices

Abstract

The symplectic modular group $\mathfrak {M}$ is the set of all $2n \times 2n$ matrices $M$ with rational integral entries, which satisfy $MJM’ = J,J = \left [ {\begin {array}{*{20}{c}} 0 & I \\ I & 0 \\ \end {array} } \right ]$, $I$ being the identity $n \times n$ matrix. Let $m$ be a positive integer. Then the $2n \times 2n$ matrix $N$ is said to be $m$-symplectic if it has rational integral entries and if it satisfies $NJN’ = mJ$. In this paper we consider canonical forms for $m$-symplectic matrices under left-multiplication by symplectic modular matrices (corresponding to Hermite’s normal form) and under both left- and right-multiplication by symplectic modular matrices (corresponding to Smith’s normal form). The number of canonical forms in each case is determined explicitly in terms of the prime divisors of $m$. Finally, corresponding results are stated, without proof, for $0$-symplectic matrices; these are $2n \times 2n$ matrices $M$ with rational integral entries and which satisfy $MJM’ = M’JM = 0$.