A Canonical Form for Symmetric and Skew-Symmetric Extended Symplectic Modular Matrices With Applications to Riemann Surface Theory

Abstract

The (extended) symplectic modular group $({ \wedge _n}){\Gamma _n}$ is the set of all $2n \times 2n$ integer matrices $M$ such that \[ (M{J^t}M = \pm J),M{J^t}M = J,J = \left [ {\begin {array}{*{20}{c}} 0 & I \\ { - I} & 0 \\ \end {array} } \right ],\] $I$ being the $n \times n$ identity matrix. Let ${S_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = - {}^tM\}$ and ${T_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = {}^tM\}$. We say $M \sim N$ if there exists $K \in {\Gamma _n}$ such that $M = KN{}^tK$. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.