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Deriving Bases for Abelian Functions Matthew England

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Abstract

We present a new method to explicitly define Abelian functions associated with algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the procedure with the functions associated with a trigonal curve of genus four. The main motivation for the construction of such bases is that it allows systematic methods for the derivation of the addition formulae and differential equations satisfied by the functions. We present a new 3-term 2-variable addition formulae and a complete set of differential equations to generalise the classic Weierstrass identities for the case of the trigonal curve of genus four.

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References

  1. C. Athorne, Identities for hyperelliptic ℘-functions of genus one, two and three in covariant form. J. Phys. A 41 (2008), 415202.

    Article  MathSciNet  Google Scholar 

  2. H. F. Baker Abelian Functions: Abel’s Theorem and the Allied Theory of theta Functions, Cambridge University Press, 1897 (reprinted in 1995).

  3. H. F. Baker, On a system of differential equations leading to periodic functions. Acta Math. 27 (1903), 135–156.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. F. Baker, Multiply Periodic Functions, Cambridge University Press, Cambridge, 1907 (reprinted in 2007 by Merchant Books. ISBN 193399880).

    MATH  Google Scholar 

  5. S. Baldwin, J. C. Eilbeck, J. Gibbons and Y. Onishi, Abelian functions for cyclic trigonal curves of genus four. J Geom. Phys. 58 (2008), 450–467.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Baldwin and J. Gibbons, Genus 4 trigonal reduction of the Benny equations, J Phys. A 39 (2006), 3607–3639.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. M. Buchstaber, V. Z. Enolskii and D. V. Leykin. Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys. 10 (1997), 1–125.

    Google Scholar 

  8. V. M. Buchstaber, V. Z. Enolskii and D. V. Leykin, Rational analogs of Abelian functions, Funct. Anal. Appl. 33 (1999), 83–94.

    Article  MATH  Google Scholar 

  9. V. M. Buchstaber, V. Z. Enolskii and D. V. Leykin, Uniformization of Jacobi varieties of trigonal curves and nonlinear equations, Funct. Anal. Appl. 34 (2000), 159–171.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Cho and A. Nakayashiki. Differential structure of Abelian functions, Int. J Math. 19 (2008), 145–171.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. C. Eilbeck, V. Z. Enolski and J. Gibbons, Sigma, tau and abelian functions of algebraic curves, J. Phys. A 43 (2010), 455216.

    Article  MathSciNet  Google Scholar 

  12. J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Onishi and E. Previato, Abelian functions for trigonal curves of genus three, Int. Math. Res. Not., page Art.ID: rnm140 (38 pages), 2007.

  13. J. C. Eilbeck, V. Z. Enolskii and D. V. Leykin. On the Kleinian construction of Abelian functions of canonical algebraic curves, in: D. Levi and O. Ragnisco (eds.), Proceedings of the 1998 SIDE III Conference, 1998: Symmetries of Integrable Differences Equations, volume CRMP/25 of CRM Proceedings and Lecture Notes, 2000, pp. 121–138.

  14. J. C. Eilbeck, S. Matsutani and Y. Onishi, Addition formulae for abelian functions associated with specialized curves, Phil. Trans. R. Soc. A 369 (2011), 1245–12638.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. C. Eilbeck, M. England and Y. Ônishi. Abelian functions associated with genus three algebraic curves, to appear in LMS J. Comput. Math., preprint available at arXiv:1008.0289v2, 2011.

  16. M. England, http://www.maths.gla.ac.uk/∼mengland/Papers/2011_DBAF/.

  17. —, Higher genus Abelian functions associated with cyclic trigonal curves, SIGMA, 6:025 (2010), 22 pages

  18. M. England and J. C. Eilbeck, Abelian functions associated with a cyclic tetragonal curve of genus six, J. Phys. A 42, (2009), 095210 (27pp).

    Article  MathSciNet  Google Scholar 

  19. M. England and J. Gibbons, A genus six cyclic tetragonal reduction of the Benney equations, J. Phys. A 42 (2009), 375202 (27pp).

    Article  MathSciNet  Google Scholar 

  20. S. Lang, Introduction to Algebraic Functions and Abelian Functions, Graduate Texts in Mathematics 89, Springer-Verlag, 2nd. edition, 1982.

  21. A. Nakayashiki, On algebraic expressions of sigma functions for (n, s)-curves, Asian J. Math. 14 no.2 (2010), 175–212.

    MathSciNet  MATH  Google Scholar 

  22. A. Nakayashiki, On hyperelliptic abelian functions of genus 3, J. Geom. Phys. 61 no.6 (2011), 961–985.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Nakayashiki and F. Smirnov, Cohomologies of affine hyperelliptic Jacobi varieties and integrable systems, Comm. Math. Phys. 217 (2001), 623–652.

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Ônishi. Complex multiplication formulae for hyperelliptic curves of genus three, Tokyo J. Math. 21 (1998), 381–431; corrections availabe from http://web.cc.iwate-u.ac.jp/∼onishi/.

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematics and Statistics, University of Glasgow, G12 8QQ, UK

    Matthew England

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  1. Matthew England
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England, M. Deriving Bases for Abelian Functions Matthew England. Comput. Methods Funct. Theory 11, 617–654 (2012). https://doi.org/10.1007/BF03321878

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  • DOI: https://doi.org/10.1007/BF03321878

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