Generalised elliptic functions

Abstract

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study.The first approach discussed sees the functions defined as logarithmic derivatives of the σ-function, a modified Riemann θ-function. We can make use of known properties of the sigma function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given.The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future.

On the equivariant algebraic Jacobian for curves of genus two

“…A related question is why the tensor products of the coordinate modules have the pole structures they do, related as they are to the dimensions of the irreducible components in a not quite linear way. This kind of structure is also apparent in analytic treatments of the Jacobian via generalised ℘-functions [5] where the Hirota derivative plays an important rôle.…”

Modular Forms

“…Following the classical treatments by Klein and Baker [25,4] and the more modern discussions, e.g. in [14,21], we construct Abelian functions via the Riemann theta function.…”

“…(1) The definition of the Kleinian σ-function above differs from that in [14,21] by a constant depending on the curve V .…”

“…(2) In [14,21], the characteristic is fixed as the Riemann characteristic of the base point ∞. In particular, this implies that α, β are half-integral.…”