Some remarks on the symplectic group $Sp(2g, \mathbb{Z})$

TL;DR

This paper investigates the possible finite orders of elements in the symplectic group over integers, providing bounds, classifications, and explicit computations for specific cases like Sp(4, Z).

Contribution

It determines which orders of elements can occur in Sp(2g, Z), establishes an upper bound on element orders, and explicitly computes element orders in Sp(4, Z).

Findings

Identifies possible element orders in Sp(2g, Z).

Establishes an upper bound on maximal element order.

Computes explicit element orders in Sp(4, Z).

Abstract

Let $G=\Sp(2g,\mathbb{Z})$ be the symplectic group over the integers. Given $m\in \mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper, we determine the possible values of $m\in \mathbb{N}$ for which the above problem has a solution. We also show that there is an upper bound on the maximal order of an element in $G$. As an illustration, we apply our results to the group $\Sp(4,\mathbb{Z})$ and determine the possible orders of elements in it. Finally, we use a presentation of $\Sp(4,\mathbb{Z})$ to identify some finite order elements and do explicit computations using the presentation to verify their orders.