I have a couple questions regarding symplectic matrices, specifically those with integer entries.

1) Suppose we are trying to find symplectic matrices of dimension $4$ with integer entries. We know that eigenvalues come in reciprocal pairs; we are assuming that the symplectic matrix $A$ has irreducible characteristic polynomial so the eigenvalues are distinct. Also we want the eigenvalues real. Thus, it is necessary for the characteristic polynomial of $A$ to be of the form

$$f = x^4 - nx^3 + (2+m)x^2 -nx+1$$

where $n,m \in \mathbb{Z}$. The question is, if we have an irreducible $f$ of this form, with roots $\lambda, \lambda ^{-1}, \mu, \mu^{-1}$, is it possible to construct an integer matrix which is symplectic with this characteristic polynomial?

2) Suppose the construction in 1) is possible, and we have matrix A. Are there any properties on matrices in the conjugacy class of A over $\mathbb{Q}$ which ensure that those matrices are symplectic? That is, is there a way to find matrices conjugate to a symplectic matrix A which are also symplectic?

Thanks for any help or any suggestions on literature to read.


1 Answer 1


The answer to the first question is "yes", see p 21 of


On that same page, if you look at conditions (6-8), that should answer your second question with a little work...


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