Affine parts of abelian surfaces as complete intersection of three quartics

TL;DR

This paper demonstrates that a specific integrable system with three quartic invariants forms an abelian surface, which can be described as a complete intersection of three quartic hypersurfaces, and is integrable via genus two hyperelliptic functions.

Contribution

It establishes a connection between an algebraic integrable system and abelian surfaces, showing the variety completes into a Jacobian of a genus two hyperelliptic curve.

Findings

The affine variety completes into an abelian surface.

The system is algebraically completely integrable.

Integration can be performed using genus two hyperelliptic functions.

Abstract

We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a genus two hyperelliptic curve. This system is algebraic completely integrable and it can be integrated in genus two hyperelliptic functions.