Decomposability of symplectic matrices over principal ideal domain

Abstract

The characteristic polynomial of any integral symplectic matrix is palindromic. First, we say that the inverse is also true, that is for any palindromic monic polynomial f(x) of even degree, there is an integral symplectic matrix whose characteristic polynomial is f(x). Furthermore, we give a sufficient condition on decomposability of integral symplectic matrices as a symplectic direct sum of two symplectic matrices which have smaller genera and coprime characteristic polynomials. Finally, we find the number of conjugacy classes of pq-torsion in the symplectic group of genus (p+q−2)/2 over rational integers, where p and q are distinct odd primes.