Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some remarks on the symplectic group $Sp(2g, \mathbb{Z})$   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.

10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga