Jacobi's Inversion Problem for Genus Two Hyperelliptic Integral

TL;DR

This paper explores the inversion problem of genus two hyperelliptic integrals, aiming to uncover potential SU(2) group structures in the addition formulas of hyperelliptic theta functions, inspired by elliptic function applications in the Ising model.

Contribution

It investigates the Jacobi inversion problem for genus two hyperelliptic integrals and proposes a possible SU(2) structure in the addition formulas of hyperelliptic theta functions.

Findings

Potential SU(2) structure identified in hyperelliptic theta function addition formulas

Extension of elliptic function addition formulas to genus two hyperelliptic case

Insight into integrable structures related to hyperelliptic integrals

Abstract

Hinted by the elliptic parameterization of the Ising model, the addition formula of the elliptic function forms to give the integrable SU(2) group relation in the previous paper. We then expect that the addition formula of the Abelian function with any genus will form to give some integrable Lie group structure. In this paper, we study Jacobi's inversion problem for hyperelliptic integral with genus two and we expect some SU(2) structure for the addition formula of the hyperelliptic theta function with genus two.