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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.

14H55 Riemann surfaces; Weierstrass points; gap sequences
14H42 Theta functions and curves; Schottky problem
Full Text: DOI
[1] Klein, F., Ueber hyperelliptische sigmafunctionen, Math. ann., 32, 3, 351-380, (1888) · JFM 20.0491.01
[2] Baker, H.F., On the hyperelliptic sigma-functions, Amer. math. J., 20, 4, 301-384, (1898) · JFM 29.0394.03
[3] Klein, F., Zur theorie der abel’schen functionen, Math. ann., 36, 1, 1-83, (1890) · JFM 22.0498.01
[4] Buchshtaber, V.M.; Leykin, D.V., Addition laws on Jacobian variety of plane algebraic curves, Proc. Steklov inst. math., 251, 1-72, (2005)
[5] Buchshtaber, V.M.; Enolskii, V.Z.; Leykin, D.V., Kleinian functions, hyperelliptic Jacobians and applications, (), 1-125, part 2 · Zbl 0911.14019
[6] Nakayashiki, A., On algebraic expressions of sigma functions for (\(n, s\)) curves, Asian J. math., 14, 2, 174-211, (2010)
[7] Eilbeck, J.C.; Enolski, V.Z.; Gibbons, J., Sigma, tau and abelian functions of algebraic curves, J. phys. A: math. theor., 43, 45, (2010), 20 pp. 455216 · Zbl 1223.14067
[8] Whittaker, E.T.; Watson, G.N., A course of modern analysis, (1927), Cambridge University Press · Zbl 0108.26903
[9] Farkas, H.M., Special divisors and analytic subloci of teichmueller space, Amer. J. math., 88, 881-901, (1996) · Zbl 0154.33101
[10] V. Enolskii, B. Hartmann, V. Kagramanova, J. Kunz, C. Lämmerzahl, P. Sirimachan, Inversion of a general hyperelliptic integral and particle motion in Horava-Lifshitz black hole space-times, J. Math. Phys., arXiv:1106.2408 (in press). · Zbl 1273.83099
[11] Fay, John D., Theta functions on Riemann surfaces, Lect. notes math., 352, (1973) · Zbl 0281.30013
[12] Fay, John D., Kernel functions, analytic torsion, and moduli spaces, Mem. AMS, 464, (1992) · Zbl 0777.32011
[13] Mumford, D., Tata lectures on theta I,II, (1984), Birkhäuser Boston
[14] Atiyah, M., The logarithm of the Dedekind-function, Math. ann., 278, 1-4, 335-380, (1987) · Zbl 0648.58035
[15] Bukhshtaber, V.M.; Leykin, D.V.; Enolskii, V.Z., Rational analogues of abelian functions, Funct. anal. appl., 33, 2, 83-94, (1999) · Zbl 1056.14049
[16] Harnad, J.; Enolskii, V.Z., Schur function expansions of KP tau-function associated to algebraic curves, Russ. math. surveys, 66, 4, 767-807, (2011) · Zbl 1231.14025
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