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On higher genus Weierstrass sigma-function. (English) Zbl 1262.14033
Summary: 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 $$u\in \mathbb C^{g})$$ 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.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H42 Theta functions and curves; Schottky problem
##### Keywords:
Riemann surfaces; theta-functions; Weierstrass functions
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