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

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.