The sigma function for Weierstrass semigroups <3,7,8> and <6,13,14,15,16>

TL;DR

This paper constructs algebraic sigma functions for specific Weierstrass semigroups, linking their properties to modular forms and Monstrous Moonshine, thus advancing the understanding of abelian functions on Riemann surfaces.

Contribution

It introduces a new algebraic construction of sigma functions based on Weierstrass semigroups, generalizing plane affine models and connecting to modular forms and Moonshine.

Findings

Defined sigma functions for semigroups <3,7,8> and <6,13,14,15,16>

Established properties and Jacobi inversion formulas for these sigma functions

Linked sigma functions to Norton bases and Monstrous Moonshine phenomena

Abstract

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular form have pointed to the relevance of $\tau$-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) $\sigma$-function. This paper proposes a construction of $\sigma$-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction of plane affine models of the Riemann surface. Because our definition is algebraic, we are able to consider the properties of the $\sigma$-functions including their Jacobi inversion formulae, and to give an observation of their properties to those of a Norton basis for replicable functions, in turn relevant to the Monstrous Moonshine.