  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.
