THE SIGMA FUNCTION FOR WEIERSTRASS SEMIGROUPS 〈3, 7, 8〉 AND 〈6, 13, 14, 15, 16〉

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular forms have pointed to the relevance of τ-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) σ-function. Klein originally generalized Weierstrass' σ-function to hyperelliptic curves in a geometric way, then from a modular point of view to trigonal curves. Recently a modular generalization for all curves was given, as well as the geometric one for certain affine planar curves, known as (n, s) curves, and their generalizations known as "telescopic" curves. This paper proposes a construction of σ-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction for plane affine models of the Riemann surface. Our examples are not telescopic. Because our definition is algebraic, in the sense of being related to the field of meromorphic functions on the curve, we are able to consider properties of the σ-functions such as Jacobi inversion formulae, and to observe relationships between their properties and those of a Norton basis for replicable functions, in turn relevant to the Monstrous Moonshine.

Abelian functions associated with genus three algebraic curves

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and the advancement of efficient symbolic computation techniques have allowed for recent progress in this area. In this paper we focus on the genus three cases, comparing the two canonical classes of hyperelliptic and trigonal curves. We present new addition formulae, derive bases for the spaces of Abelian functions and discuss the differential equations such functions satisfy.Supplementary materials are available with this article.

Addition formulae for Abelian functions associated with specialized curves

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of curves.