Abstract: The symplectic modular group is the set of all matrices with rational integral entries, which satisfy , being the identity matrix. Let be a positive integer. Then the matrix is said to be -symplectic if it has rational integral entries and if it satisfies . In this paper we consider canonical forms for -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 . Finally, corresponding results are stated, without proof, for 0-symplectic matrices; these are matrices with rational integral entries and which satisfy .
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- C. C. MacDuffee, The theory of matrices, Chelsea, New York, 1946.
- H. Maass, Die Primzahlen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 124 (1951), 87-122. MR 13, 823. MR 0047075 (13:823g)
- M. Sugawara, On the transformation theory of Siegel's modular group of the -th degree, Proc. Imp. Acad. Japan 13 (1937), 335-338. MR 1568477
- E. Spence, Matrix divisors of mI, Acta Arith. 20 (1972). MR 0320032 (47:8573)
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Keywords: Symplectic modular group, unimodular matrices, canonical forms, elementary divisor theory, multiplicative function
Article copyright: © Copyright 1972 American Mathematical Society