On Algebraic Expressions of Sigma Functions for (n, s) Curves

Abstract

An expression of the multivariate sigma function associated with a (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be the Schur function determined from the gap sequence at infinity. *

Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves

“…This proof follows the ideas in [9] and [12]. First we note that the σ-function associated with the (3,7)-curve is even and then, by Theorem 3(i) in [14], we know the expansion will be a sum of monomials in u and λ with rational coefficients.…”

“…One of the key properties of the σ-function is that the part of its expansion without λ will be given by a constant multiple of the corresponding Schur-Weierstrass polynomial (see [7] or [14]) and as discussed in [12] we can set this constant to one. We can then conclude the weight of the expansion to be the weight of SW 3,7 which is +16.…”

“…For more information on these polynomials and their role in the theory see [7] and [14]. To construct the Schur-Weierstrass polynomial associated to (n, s):…”

Periods of second kind differentials of $(n,s)$-curves

“…In [Nak10], Nakayashiki presented a description of the polynomial F(Q, R) as a polynomial of (x, y; z, w) with the coefficients in homogeneous polynomials (with respect to the weights) of λ i (=coefficients of f (x, y)). Our approach is based on the explicit algebraic expression of F(Q, R), whose derivation is classically known and described as follows.…”

“…It is shown in [BL08] that models of (n, s)-curves are better adapted to the construction of multi-variable σ-functions than models suggested by Weierstrass and later models of mini-versal deformations of singularities of the form y n = x s . We also mention Klein [Kle888], [Kle890] and recent works presenting effective description of multi-variate σ-functions [Nak10] and [KSh12]. In particular, the approach of Klein (for any Riemann surface of genus 3) and [KSh12] (for an arbitrary Riemann surface of any genus) to the theory of higher genus sigma-functions is based on resolving the generalized Legendre relations in terms of theta-constants.…”