Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 037, 36 pages      arXiv:1203.3409      https://doi.org/10.3842/SIGMA.2012.037

Building Abelian Functions with Generalised Baker-Hirota Operators

Matthew England a and Chris Athorne b
a) Department of Computer Science, University of Bath, Bath, BA2 7AY, UK
b) School of Mathematics and Statistics, University of Glasgow, G12 8QQ, UK

Received March 16, 2012, in final form June 18, 2012; Published online June 26, 2012

Abstract
We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.

Key words: Baker-Hirota operator; $\mathcal{R}$-function; Abelian function; Kleinian function.

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References

  1. Athorne C., A novel approach to the theory of Padé approximants, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 15-27.
  2. Athorne C., Algebraic Hirota maps, in Bilinear integrable systems: from classical to quantum, continuous to discrete, NATO Sci. Ser. II Math. Phys. Chem., Vol. 201, Springer, Dordrecht, 2006, 17-33.
  3. Athorne C., Algebraic invariants and generalized Hirota derivatives, Phys. Lett. A 256 (1999), 20-24.
  4. Athorne C., Identities for hyperelliptic $\wp$-functions of genus one, two and three in covariant form, J. Phys. A: Math. Theor. 41 (2008), 415202, 20 pages, arXiv:0808.3103.
  5. Athorne C., Eilbeck J.C., Enolskii V.Z., Identities for the classical genus two $\wp$ function, J. Geom. Phys. 48 (2003), 354-368.
  6. Baker H.F., Abelian functions: Abel's theorem and the allied theory of theta functions, Cambridge University Press, Cambridge, 1995.
  7. Baker H.F., Multiply periodic functions, Cambridge University Press, Cambridge, 2007.
  8. Baker H.F., On a system of differential equations leading to periodic functions, Acta Math. 27 (1903), 135-156.
  9. Baldwin S., Eilbeck J.C., Gibbons J., Ônishi Y., Abelian functions for cyclic trigonal curves of genus 4, J. Geom. Phys. 58 (2008), 450-467, math.AG/0612654.
  10. Baldwin S., Gibbons J., Genus 4 trigonal reduction of the Benney equations, J. Phys. A: Math. Gen. 39 (2006), 3607-3639.
  11. Buchstaber V.M., Enolskii V.Z., Leykin D.V., Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys. 10 (1997), 1-125.
  12. Buchstaber V.M., Enolskii V.Z., Leykin D.V., Rational analogues of abelian functions, Funct. Anal. Appl. 33 (1999), 83-94.
  13. Cho K., Nakayashiki A., Differential structure of abelian functions, Internat. J. Math. 19 (2008), 145-171, math.AG/0604267.
  14. Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C., Solitons and nonlinear wave equations, Academic Press Inc., London, 1982.
  15. Doubilet P., On the foundations of combinatorial theory. VII. Symmetric functions through the theory of distribution and occupancy, Studies in Appl. Math. 51 (1972), 377-396.
  16. Eilbeck J.C., England M., Ônishi Y., Abelian functions associated with genus three algebraic curves, LMS J. Comput. Math. 14 (2011), 291-326, arXiv:1008.0289.
  17. Eilbeck J.C., Enolskii V.Z., Leykin D.V., On the Kleinian construction of abelian functions of canonical algebraic curves, in SIDE III - Symmetries and Integrability of Difference Equations (Sabaudia, 1998), CRM Proc. Lecture Notes, Vol. 25, Amer. Math. Soc., Providence, RI, 2000, 121-138.
  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. (2007), Art. ID rnm 140, 38 pages, math.AG/0610019.
  19. England M., Deriving bases for Abelian functions, Comput. Meth. Funct. Theor. 11 (2011), 617-654, arXiv:1103.0468.
  20. England M., Higher genus Abelian functions associated with cyclic trigonal curves, SIGMA 6 (2010), 025, 22 pages, arXiv:1003.4144.
  21. England M., Eilbeck J.C., Abelian functions associated with a cyclic tetragonal curve of genus six, J. Phys. A: Math. Theor. 42 (2009), 095210, 27 pages, arXiv:0806.2377.
  22. England M., Gibbons J., A genus six cyclic tetragonal reduction of the Benney equations, J. Phys. A: Math. Theor. 42 (2009), 375202, 27 pages, arXiv:0903.5203.
  23. Enolski 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. 61 (2011), 899-921, arXiv:1011.6459.
  24. Grammaticos B., Ramani A., Hietarinta J., Multilinear operators: the natural extension of Hirota's bilinear formalism, Phys. Lett. A 190 (1994), 65-70.
  25. Hietarinta J., Gauge symmetry and the generalization of Hirota's bilinear method, J. Nonlinear Math. Phys. 3 (1996), 260-265.
  26. Hietarinta J., Grammaticos B., Ramani A., Integrable trilinear PDE's, in Nonlinear Evolution Equations & Dynamical Systems: NEEDS '94 (Los Alamos, NM), World Sci. Publ., River Edge, NJ, 1995, 54-63, solv-int/9411003.
  27. Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.
  28. Korotkin D., Shramchenko V., On higher genus Weierstrass sigma-function, Phys. D, to appear, arXiv:1201.3961.
  29. Lang S., Introduction to algebraic and abelian functions, Graduate Texts in Mathematics, Vol. 89, 2nd ed., Springer-Verlag, New York, 1982.
  30. Lascoux A., Schützenberger M.P., Formulaire raisonne de fonctions symetriques, Universite Paris 7, 1985.
  31. Matsumura H., Commutative algebra, W.A. Benjamin, Inc., New York, 1970.
  32. McKean H., Moll V., Elliptic curves: function theory, geometry, arithmetic, Cambridge University Press, Cambridge, 1997.
  33. Nakayashiki A., On algebraic expressions of sigma functions for $(n,s)$ curves, Asian J. Math. 14 (2010), 175-211, arXiv:0803.2083.
  34. Olver P.J., Sanders J.A., Transvectants, modular forms, and the Heisenberg algebra, Adv. in Appl. Math. 25 (2000), 252-283.
  35. Rota G.C., On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340-368.
  36. Washington L.C., Elliptic curves: number theory and cryptography, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008.

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