Sigma, tau and Abelian functions of algebraic curves

TL;DR

This paper compares three methods for deriving differential relations of Abelian functions linked to algebraic curves, emphasizing the role of sigma and tau functions in their construction.

Contribution

It introduces and contrasts three approaches for constructing differential relations of Abelian functions, highlighting the use of sigma and tau functions.

Findings

Three methods for Abelian functions construction are compared.

Tau function plays a central role in two of the methods.

The paper clarifies the relationships between different approaches.

Abstract

We compare and contrast three different methods for the construction of the differential relations satisfied by the fundamental Abelian functions associated with an algebraic curve. We realize these Abelian functions as logarithmic derivatives of the associated sigma function. In two of the methods, the use of the tau function, expressed in terms of the sigma function, is central to the construction of differential relations between the Abelian functions.