Conjugacy classes of p-torsion in symplectic groups over S-integers

Abstract

For any odd prime $p$ we consider representations of a group of order $p$ in the symplectic group $Sp(p-1,Z[1/n])$ of $(p-1)\times(p-1)$-matrices over the ring $Z[1/n]$, $0\neq n\in N$. We construct a relation between the conjugacy classes of subgroups $P$ of order $p$ in the symplectic group and the ideal class group in the ring $Z[1/n]$. This is used for the study of these classes. In particular we determine the centralizer $C(P)$ and $N(P)/C(P)$ where $N(P)$ denotes the normalizer.