  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.
