Generalised elliptic functions

Matthew England; Chris Athorne

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1655-1672
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Matthew England, and Chris Athorne. "Generalised elliptic functions." Open Mathematics 10.5 (2012): 1655-1672. <http://eudml.org/doc/269465>.

@article{MatthewEngland2012,
abstract = {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.},
author = {Matthew England, Chris Athorne},
journal = {Open Mathematics},
keywords = {Generalised elliptic functions; Sigma functions; Equivariance; generalised elliptic functions; sigma functions; equivariance},
language = {eng},
number = {5},
pages = {1655-1672},
title = {Generalised elliptic functions},
url = {http://eudml.org/doc/269465},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Matthew England
AU - Chris Athorne
TI - Generalised elliptic functions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1655
EP - 1672
AB - 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.
LA - eng
KW - Generalised elliptic functions; Sigma functions; Equivariance; generalised elliptic functions; sigma functions; equivariance
UR - http://eudml.org/doc/269465
ER -

References

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  1. [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 Zbl1149.14027
  2. [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. [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 Zbl1246.14014
  4. [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 Zbl1056.33016
  5. [5] Baker H.F., Abelian Functions, Cambridge University Press, Cambridge, 1897 Zbl0848.14012
  6. [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 Zbl34.0464.03
  7. [7] Baker H.F., An Introduction to the Theory of Multiply-Periodic Functions, Cambridge University Press, Cambridge, 1907 Zbl38.0478.05
  8. [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 Zbl1211.37082
  9. [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 Zbl1091.35065
  10. [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 Zbl0911.14019
  11. [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. [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. [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 Zbl0857.14018
  14. [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 Zbl1165.14034
  15. [15] Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C., Solitons and Nonlinear Wave Equations, Academic Press, London-New York, 1982 Zbl0496.35001
  16. [16] Eilbeck J.C., England M., Ônishi Y., Abelian functions associated with genus three algebraic curves, LMS J. Comput. Math., 2011, 14, 291–326 Zbl1304.14042
  17. [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 Zbl1003.14008
  18. [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 Zbl1210.14032
  19. [19] England M., Higher genus Abelian functions associated with cyclic trigonal curves, SIGMA Symmetry Integrability Geom. Methods Appl., 2010, 6, #025 
  20. [20] England M., Deriving bases for Abelian functions, Comput. Methods Funct. Theory, 2011, 11(2), 617–654 Zbl1256.14027
  21. [21] England M., Athorne C., Building Abelian functions with generalised Hirota operators, preprint available at http://arxiv.org/abs/1203.3409 Zbl1242.14027
  22. [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 Zbl1157.14303
  23. [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 Zbl1184.14059
  24. [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 Zbl1213.83039
  25. [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 Zbl1021.37039
  26. [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. [27] Korotkin D., Shramchenko V., On higher genus Weierstrass sigma-function, Phys. D (in press), DOI: 10.1016/j.physd.2012.01.002 Zbl1262.14033
  28. [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 Zbl0513.14024
  29. [29] McKean H., Moll V., Elliptic Curves, Cambridge University Press, Cambridge, 1997 
  30. [30] Nakayashiki A., On algebraic expressions of sigma functions for (n; s)-curves, Asian J. Math., 2010, 14(2), 175–211 Zbl1214.14028
  31. [31] Washington L.C., Elliptic Curves, 2nd ed., Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2008 Zbl1200.11043

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