×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.