zbMATH — the first resource for mathematics

Abelian functions for cyclic trigonal curves of genus 4. (English) Zbl 1211.37082
Summary: We discuss the theory of generalized Weierstrass \(\sigma\) and \(\wp\) functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case [Tokyo J. Math. 27, No. 2, 299–312 (2004; Zbl 1160.11332)]. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve \(y^3=x^5+ \lambda_4x^4+ \lambda_3x^3+ \lambda_2x^2+ \lambda_1x+ \lambda_0\) is discussed in detail, including a list of some of the associated partial differential equations satisfied by the \(\wp\) functions, and the derivation of addition formulae.

14H45 Special algebraic curves and curves of low genus
14H55 Riemann surfaces; Weierstrass points; gap sequences
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
Full Text: DOI arXiv
[1] Abramowitz, M.; Stegun, I., Handbook of mathematical functions, (1965), Dover, p. 635 eq. 18.5.6
[2] Accola, R., On cyclic trigonal Riemann surfaces, I, Trans. amer. math. soc., 283, 2, 423-449, (1984) · Zbl 0584.14016
[3] Baker, H.F., Abelian functions, (1897), Cambridge Univ. Press Cambridge · Zbl 0848.14012
[4] Baker, H.F., On the hyperelliptic sigma functions, Amer. J. math., 20, 301-384, (1898) · JFM 29.0394.03
[5] Baker, H.F., Multiply periodic functions, (1907), Cambridge Univ. Press Cambridge · JFM 38.0478.05
[6] Buchstaber, V.M.; Enolskii, V.Z.; Leykin, D.V., Kleinian functions, hyperelliptic Jacobians and applications, Rev. math. math. phys., 10, 1-125, (1997) · Zbl 0911.14019
[7] Buchstaber, V.M.; Enolskii, V.Z.; Leykin, D.V., Rational analogs of abelian functions, Funct. anal. appl., 33, 83-94, (1999)
[8] Buchstaber, V.M.; Enolskii, V.Z.; Leykin, D.V., Uniformization of Jacobi varieties of trigonal curves and nonlinear equations, Funct. anal. appl., 34, 159-171, (2000) · Zbl 0978.58012
[9] Baldwin, S.; Gibbons, J., Genus 4 trigonal reduction of the benney equations, J. phys. A, 39, 3607-3639, (2006) · Zbl 1091.35065
[10] Buchstaber, V.M.; Leykin, D.V., Polynomial Lie algebras, Funct. anal. appl., 36, 4, 267-280, (2002) · Zbl 1027.17020
[11] Buchstaber, V.M.; Leykin, D.V., Addition laws on Jacobian varieties of plane algebraic curves, Proc. Steklov inst. math., 251, 54-126, (2005)
[12] Eilbeck, J.C.; Enolskii, V.Z., Bilinear operators and the power series for the Weierstrass \(\sigma\) function, J. phys. A, 33, 791-794, (2000) · Zbl 0955.33015
[13] Eilbeck, J.C.; Enolskii, V.Z.; Leykin, D.V., On the Kleinian construction of abelian functions of canonical algebraic curves, (), 121-138 · Zbl 1003.14008
[14] Eilbeck, J.C.; Enolskii, V.Z.; Matsutani, S.; Ônishi, Y.; Previato, E., Abelian functions for purely trigonal curves of genus three, Int. math. res. notices, (2007), 38 pages. Art. ID rnm140 · Zbl 1210.14032
[15] Lang, S., Introduction to algebraic functions and abelian functions, Number 89 in Grad. text in math., (1982), Springer-Verlag
[16] Mumford, D., Abelian varieties, (1985), Oxford Univ. Press · Zbl 0199.24601
[17] Ônishi, Y., Complex multiplication formulae for hyperelliptic curves of genus three, Tokyo J. math., 21, 381-431, (1998), A list of correction is available from: · Zbl 1016.11019
[18] Y. Ônishi, Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case. http://arxiv.org/abs/math.NT/0503696, 2005 · Zbl 1222.14066
[19] Schreiner, Wolfgang; Mittermaier, Christian; Bosa, Karoly, Distributed Maple: parallel computer algebra in networked environments, J. symbolic comput., 35, 305-347, (2003) · Zbl 1045.68167
[20] A. Nakayashiki, On algebraic expressions of sigma functions for \((n, s)\)-curves, preprint · Zbl 1214.14028
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.