On higher genus Weierstrass sigma-function

PHYSICA D-NONLINEAR PHENOMENA

卷:241

Article

Korotkin, D.²  Shramchenko, V.¹ 

[1] Univ Sherbrooke, Dept Math, Sherbrooke, PQ J1K 2R1, Canada

[2] Concordia Univ, Dept Math & Stat, Montreal, PQ H3G 1M8, Canada

关键词: Riemann surfaces;    Theta-functions;    Weierstrass functions;   

DOI  :  10.1016/j.physd.2012.01.002

来源: Elsevier

【 摘 要 】

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 the Jacobi theta-function. Namely, the odd higher genus sigma-function sigma(chi) (u) (for u is an element of 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. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free