Building Abelian Functions with Generalised Baker-Hirota Operators

Abstract

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.

On Degenerate Sigma-Functions in Genus 2

“…It was shown that the Hirota derivative plays the role of a partial intertwining operator in the representation theory of sl(2, C) and so can serve for generation of algebraic invariants [16]. A generalization of bilinear Hirota operators possessing an equivariance property is applied for constructing a basis in the space of Abelian functions with poles of at most a given order [17].…”

Some new addition formulae for Weierstrass elliptic functions

“…More specifically, the right hand side will be a sum of terms, each a product of three functions, one in each of the variables and with all functions taken from the set {1, ℘, ℘ ′ , ℘ ′′ , ℘ ′′′ }. Such an expression is clear from the linear algebra when considering the space of elliptic functions graded by pole order (for more details on such spaces see for example [6,24]). This also clarifies why r 7 = 0: since there is no elliptic function of weight 1 to include in the right hand side.…”