Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T07:12:39.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelian functions associated with genus three algebraic curves

Part of: Infinite-dimensional Hamiltonian systems Abelian varieties and schemes Curves

Published online by Cambridge University Press:  01 November 2011

J. C. Eilbeck ,
M. England  and
Y. Ônishi
J. C. Eilbeck
Affiliation:
Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom (email: j.c.eilbeck@hw.ac.uk)
M. England
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ, United Kingdom (email: matthew.england@glasgow.ac.uk)
Y. Ônishi
Affiliation:
Faculty of Education and Human Sciences, University of Yamanashi, 4-3-11, Takeda, Kofu, 400-8511, Japan (email: yonishi@yamanashi.ac.jp)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and the advancement of efficient symbolic computation techniques have allowed for recent progress in this area. In this paper we focus on the genus three cases, comparing the two canonical classes of hyperelliptic and trigonal curves. We present new addition formulae, derive bases for the spaces of Abelian functions and discuss the differential equations such functions satisfy.

Supplementary materials are available with this article.

MSC classification

Secondary: 37K20: Relations with algebraic geometry, complex analysis, special functions 14H55: Riemann surfaces; Weierstrass points; gap sequences 14K25: Theta functions 14H45: Special curves and curves of low genus 14H40: Jacobians, Prym varieties
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 291 - 326
Copyright
Copyright © London Mathematical Society 2011

References

[1]Athorne, C., ‘Identities for hyperelliptic ℘-functions of genus one, two and three in covariant form’, J. Phys. A 41 (2008) 415202.CrossRefGoogle Scholar
[2]Athorne, C., ‘A generalization of Baker’s quadratic formulae for hyperelliptic ℘-functions’, Phys. Lett. A 375 (2011) 2689–2693.CrossRefGoogle Scholar
[3]Baker, H. F., Abelian functions: Abel’s theorem and the allied theory of theta functions (Cambridge University Press, Cambridge, 1897) Reprinted in 1995.CrossRefGoogle Scholar
[4]Baker, H. F., ‘On a system of differential equations leading to periodic functions’, Acta Math. 27 (1903) 135–156.CrossRefGoogle Scholar
[5]Baker, H. F., Multiply periodic functions (Cambridge University Press, Cambridge, 1907) Reprinted in 2007 by Merchant Books. ISBN 193399880.Google Scholar
[6]Baldwin, S., Eilbeck, J. C., Gibbons, J. and Ônishi, Y., ‘Abelian functions for cyclic trigonal curves of genus four’, J. Geom. Phys. 58 (2008) 450–467.CrossRefGoogle Scholar
[7]Baldwin, S. and Gibbons, J., ‘Genus 4 trigonal reduction of the Benny equations’, J. Phys. A 39 (2006) 3607–3639.CrossRefGoogle Scholar
[8]Buchstaber, V. M., Enolskii, V. Z. and Leykin, D. V., ‘Kleinian functions, hyperelliptic Jacobians and applications’, Rev. Math. Math. Phys. 10 (1997) 1–125.Google Scholar
[9]Buchstaber, V. M., Enolskii, V. Z. and Leykin, D. V., ‘Rational analogs of Abelian functions’, Funct. Anal. Appl. 33 (1999) 83–94.CrossRefGoogle Scholar
[10]Buchstaber, V. M., Enolskii, V. Z. and Leykin, D. V., ‘Uniformization of Jacobi varieties of trigonal curves and nonlinear equations’, Funct. Anal. Appl. 34 (2000) 159–171.CrossRefGoogle Scholar
[11]Christoffel, E. B., ‘Vollständige Theorie der Riemann’schen ϑ-function’, Math. Ann. 54 (1901) 347–399.CrossRefGoogle Scholar
[12]Eilbeck, J. C., Enolski, V. Z. and Gibbons, J., ‘Sigma, tau and abelian functions of algebraic curves’, J. Phys. A 43 (2010) 455216.CrossRefGoogle Scholar
[13]Eilbeck, J. C., Enolski, V. Z., Matsutani, S., Ônishi, Y. and Previato, E., ‘Abelian functions for trigonal curves of genus three’, Int. Math. Res. Not. IMRN 2008 (2008) 102–139.Google Scholar
[14]Eilbeck, J. C., Enolskii, V. Z. and Leykin, D. V., ‘On the Kleinian construction of Abelian functions of canonical algebraic curves’, Proceedings of the 1998 SIDE III Conference: Symmetries of Integrable Difference Equations, CRM Proceedings and Lecture Notes 25 (eds Levi, D. and Ragnisco, O.; American Mathematical Society, Providence, RI, 2000) 121–138.Google Scholar
[15]Eilbeck, J. C., Matsutani, S. and Ônishi, Y., ‘Addition formulae for abelian functions associated with specialized curves’, Philos. Trans. R. Soc. Lond. A 369 (2011) 1245–1263.Google ScholarPubMed
[16]England, M., ‘Higher genus Abelian functions associated with cyclic trigonal curves’, SIGMA 6 (2010) no. 025, 22.Google Scholar
[17]England, M., ‘Deriving bases for Abelian functions’, Comput. Methods Funct. Theory 11 (2011) no. 2.Google Scholar
[18]England, M. and Eilbeck, J. C., ‘Abelian functions associated with a cyclic tetragonal curve of genus six’, J. Phys. A 42 (2009) no. 095210, 27.CrossRefGoogle Scholar
[19]England, M. and Gibbons, J., ‘A genus six cyclic tetragonal reduction of the Benney equations’, J. Phys. A 42 (2009) no. 375202, 27.CrossRefGoogle Scholar
[20]Lang, S., Introduction to algebraic functions and Abelian functions, 2nd edn, Graduate Texts in Mathematics 89 (Springer, Berlin, 1982).CrossRefGoogle Scholar
[21]Mumford, D., Tata lectures on Theta I, Progress in Mathematics 28 (Birkhäuser, Boston, MA, 1983).CrossRefGoogle Scholar
[22]Nakayashiki, A., ‘On algebraic expressions of sigma functions for (n,s)-curves’, Asian J. Math. 14 (2010) no. 2, 175–212.CrossRefGoogle Scholar
[23]Nakayashiki, A., ‘On hyperelliptic abelian functions of genus 3’, J. Geom. Phys. 61 (2011) no. 6, 961–985.CrossRefGoogle Scholar
[24]Ônishi, Y., ‘Complex multiplication formulae for hyperelliptic curves of genus three’, Tokyo J. Math. 21 (1998) 381–431, A list of corrections is available from http://www.ccn.yamanashi.ac.jp/∼yonishi/.CrossRefGoogle Scholar
[25]Shiga, H., ‘On the representation of the Picard modular function by θ constants I, II’, Publ. Res. Inst. Math. Sci. 24 (1988) 311–360.CrossRefGoogle Scholar
Supplementary material: File

Eilbeck Supplementary Material

Eilbeck Supplementary Material

Download Eilbeck Supplementary Material(File)
File 246.9 KB