We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed sees the functions defined as logarithmic derivatives of the σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given. The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.
@article{bwmeta1.element.doi-10_2478_s11533-012-0083-x,
author = {Matthew England and Chris Athorne},
title = {Generalised elliptic functions},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1655-1672},
zbl = {1258.14038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0083-x}
}
Matthew England; Chris Athorne. Generalised elliptic functions. Open Mathematics, Tome 10 (2012) pp. 1655-1672. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0083-x/
[1] Athorne C., Identities for hyperelliptic ℘-functions of genus one, two and three in covariant form, J. Phys. A, 2008, 41(41), #415202 http://dx.doi.org/10.1088/1751-8113/41/41/415202 | Zbl 1149.14027
[2] Athorne C., A generalization of Baker’s quadratic formulae for hyperelliptic ℘-functions, Phys. Lett. A, 2011, 375(28–29), 2689–2693 http://dx.doi.org/10.1016/j.physleta.2011.05.056
[3] Athorne C., On the equivariant algebraic Jacobian for curves of genus two, J. Geom. Phys., 2012, 62(4), 724–730 http://dx.doi.org/10.1016/j.geomphys.2011.12.016 | Zbl 1246.14014
[4] Athorne C., Eilbeck J.C., Enolskii V.Z., Identities for the classical genus two ℘-function, J. Geom. Phys., 2003, 48(2–3), 354–368 http://dx.doi.org/10.1016/S0393-0440(03)00048-2 | Zbl 1056.33016
[5] Baker H.F., Abelian Functions, Cambridge University Press, Cambridge, 1897 | Zbl 0848.14012
[6] Baker H.F., On a system of differential equations leading to periodic functions, Acta Math., 1903, 27, 135–156 http://dx.doi.org/10.1007/BF02421301 | Zbl 34.0464.03
[7] Baker H.F., An Introduction to the Theory of Multiply-Periodic Functions, Cambridge University Press, Cambridge, 1907 | Zbl 38.0478.05
[8] Baldwin S., Eilbeck J.C., Gibbons J., Ônishi Y., Abelian functions for cyclic trigonal curves of genus 4, J. Geom. Phys., 2008, 58(4), 450–467 http://dx.doi.org/10.1016/j.geomphys.2007.12.001 | Zbl 1211.37082
[9] Baldwin S., Gibbons J., Genus 4 trigonal reduction of the Benney equations, J. Phys. A, 2006, 39(14), 3607–3639 http://dx.doi.org/10.1088/0305-4470/39/14/008 | Zbl 1091.35065
[10] Buchstaber V.M., Enolskii V.Z., Leykin D.V., Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys., 1997, 10(2), 3–120 | Zbl 0911.14019
[11] Buchstaber V.M., Enolskii V.Z., Leykin D.V., Rational analogs of Abelian functions, Funct. Anal. Appl., 1999, 33(2), 83–94 http://dx.doi.org/10.1007/BF02465189
[12] Buchstaber V.M., Enolskii V.Z., Leykin D.V., Uniformization of Jacobi varieties of trigonal curves and nonlinear equations, Funct. Anal. Appl., 2000, 34(3), 159–171 http://dx.doi.org/10.1007/BF02482405
[13] Cassels J.W.S., Flynn E.V., Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, London Math. Soc. Lecture Note Ser., 230, Cambridge University Press, Cambridge, 1996 http://dx.doi.org/10.1017/CBO9780511526084 | Zbl 0857.14018
[14] Cho K., Nakayashiki A., Differential structure of Abelian functions, Internat. J. Math., 2008, 19(2), 145–171 http://dx.doi.org/10.1142/S0129167X08004595 | Zbl 1165.14034
[15] Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C., Solitons and Nonlinear Wave Equations, Academic Press, London-New York, 1982 | Zbl 0496.35001
[16] Eilbeck J.C., England M., Ônishi Y., Abelian functions associated with genus three algebraic curves, LMS J. Comput. Math., 2011, 14, 291–326 | Zbl 1304.14042
[17] Eilbeck J.C., Enolskii V.Z., Leykin D.V., On the Kleinian construction of Abelian functions of canonical algebraic curves, In: Symmetries and Integrability of Difference Equations, Sabaudia, May 16–22, 1998, CRM Proc. Lecture Notes, 25, American Mathematical Society, Providence, 2000, 121–138 | Zbl 1003.14008
[18] Eilbeck J.C., Enolski V.Z., Matsutani S., Ônishi Y., Previato E., Abelian functions for trigonal curves of genus three, Int. Math. Res. Not. IMRN, 2007, #140 | Zbl 1210.14032
[19] England M., Higher genus Abelian functions associated with cyclic trigonal curves, SIGMA Symmetry Integrability Geom. Methods Appl., 2010, 6, #025
[20] England M., Deriving bases for Abelian functions, Comput. Methods Funct. Theory, 2011, 11(2), 617–654 | Zbl 1256.14027
[21] England M., Athorne C., Building Abelian functions with generalised Hirota operators, preprint available at http://arxiv.org/abs/1203.3409 | Zbl 1242.14027
[22] England M., Eilbeck J.C., Abelian functions associated with a cyclic tetragonal curve of genus six, J. Phys. A, 2009, 42(9), #095210 http://dx.doi.org/10.1088/1751-8113/42/9/095210 | Zbl 1157.14303
[23] England M., Gibbons J., A genus six cyclic tetragonal reduction of the Benney equations, J. Phys. A, 2009, 42(37), #375202 http://dx.doi.org/10.1088/1751-8113/42/37/375202 | Zbl 1184.14059
[24] Enolskii V.Z., Hackmann E., Kagramanova V., Kunz J., Lämmerzahl C., Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in general relativity, J. Geom. Phys., 2011, 61(5), 899–921 http://dx.doi.org/10.1016/j.geomphys.2011.01.001 | Zbl 1213.83039
[25] Enolskii V.Z., Pronine M., Richter P.H., Double pendulum and θ-divisor, J. Nonlinear Sci., 2003, 13(2), 157–174 http://dx.doi.org/10.1007/s00332-002-0514-0 | Zbl 1021.37039
[26] Farkas H.M., Kra I., Riemann Surfaces, 2nd ed., Grad. Texts in Math., 71, Springer, New York, 1992 http://dx.doi.org/10.1007/978-1-4612-2034-3
[27] Korotkin D., Shramchenko V., On higher genus Weierstrass sigma-function, Phys. D (in press), DOI: 10.1016/j.physd.2012.01.002 | Zbl 1262.14033
[28] Lang S., Introduction to Algebraic and Abelian Functions, 2nd ed., Grad. Texts in Math., 89, Springer, NewYork-Berlin, 1982 http://dx.doi.org/10.1007/978-1-4612-5740-0 | Zbl 0513.14024
[29] McKean H., Moll V., Elliptic Curves, Cambridge University Press, Cambridge, 1997
[30] Nakayashiki A., On algebraic expressions of sigma functions for (n; s)-curves, Asian J. Math., 2010, 14(2), 175–211 | Zbl 1214.14028
[31] Washington L.C., Elliptic Curves, 2nd ed., Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2008 | Zbl 1200.11043