The Picard Fuchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized Neumann system

TL;DR

This paper derives explicit Picard--Fuchs equations for periods of hyperelliptic integrals on genus 2 even order curves, linking them to integrable systems like the generalized Neumann system.

Contribution

It extends known Picard--Fuchs equations to even order hyperelliptic curves and connects these to action variables in integrable systems.

Findings

Derived explicit linear ODE system for hyperelliptic periods.

Connected periods to action variables of integrable systems.

Provided tools for analyzing properties of actions in these systems.

Abstract

We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the system of 5 linear ODEs for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of the parameters of the curves. The system is an extention of the well studied Picard--Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard--Fuchs equations can be used to study various properties of the actions.