A Canonical Form for a Symplectic Involution

TL;DR

This paper introduces an algorithmic canonical form for symplectic involutions in integer symplectic groups, with applications to Riemann surfaces, simplifying their classification and analysis.

Contribution

It provides a new canonical form for symplectic involutions in Sp(2g, Z), enhancing understanding and computational handling of these transformations.

Findings

Developed an explicit algorithm for the canonical form.

Applied the canonical form to Riemann surface analysis.

Simplified classification of symplectic involutions.

Abstract

We present a canonical form for a symplectic involution $S\in Sp(2g,\mathbb{Z})$, $S^2=1$; the construction is algorithmic. Application is made in the Riemann surface setting.