On higher genus Weierstrass sigma-function

TL;DR

This paper introduces a novel higher genus generalization of the Weierstrass sigma-function, extending classical elliptic functions to complex Riemann surfaces using theta-functions and spin structures.

Contribution

It defines odd and even higher genus sigma-functions based on theta-functions, spin line bundles, and period matrices, expanding the classical theory to higher genus surfaces.

Findings

Defined odd higher genus sigma-function using theta-functions and spin structures.

Constructed even sigma-functions as analogs of classical formulas.

Extended the sigma-function concept to moduli spaces of Riemann surfaces.

Abstract

The goal of this paper is to propose a new way to generalize the Weierstrass sigma-function to higher genus Riemann surfaces. Our definition of the odd higher genus sigma-function is based on a generalization of the classical representation of the elliptic sigma-function via Jacobi theta-function. Namely, the odd higher genus sigma-function $\sigma_{\chi}(u)$ (for $u\in \C^g$) is defined as a product of the theta-function with odd half-integer characteristic $\beta^{\chi}$, associated with a spin line bundle $\chi$, an exponent of a certain bilinear form, the determinant of a period matrix and a power of the product of all even theta-constants which are non-vanishing on a given Riemann surface. We also define an even sigma-function corresponding to an arbitrary even spin structure. Even sigma-functions are constructed as a straightforward analog of a classical formula relating even and odd sigma-functions. In higher genus the even sigma-functions are well-defined on the moduli space of Riemann surfaces outside of a subspace defined by vanishing of the corresponding even theta-constant.