On Hyperelliptic Abelian Functions of Genus 3

TL;DR

This paper proves a conjecture about the structure of the affine ring of a genus 3 hyperelliptic Jacobian as a D-module and explicitly constructs a basis using hyperelliptic functions.

Contribution

It confirms the minimal D-free resolution conjecture for genus 3 hyperelliptic Jacobians and provides an explicit basis in terms of Klein's hyperelliptic pe functions.

Findings

Proved the minimal D-free resolution conjecture for genus 3 hyperelliptic Jacobians.

Constructed an explicit linear basis of the affine ring.

Connected the basis to derivatives of Klein's hyperelliptic pe functions.

Abstract

The affine ring A of the affine Jacobian variety of a hyperelliptic curve of genus 3 is studied as a D-module. The conjecture on the minimal D-free resolution previously proposed is proved in this case. As a by-product a linear basis of A is explicitly constructed in terms of derivatives of Klein's hyperelliptic pe functions.