Building Abelian Functions with Generalised Baker-Hirota Operators

TL;DR

This paper introduces a systematic method for constructing Abelian functions on Jacobian varieties of algebraic curves using a generalized Baker-Hirota operator, leading to explicit formulas and new insights into functions of genus three.

Contribution

It develops a symmetric generalization of the Baker-Hirota operator to generate Abelian functions, providing explicit formulas and new bases for genus three curves.

Findings

Explicit formulas for generalized operators

New bases of Abelian functions for genus three

Revealed similarities between functions of same genus

Abstract

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.