
SIGMA 6 (2010), 025, 22 pages arXiv:1003.4144
http://dx.doi.org/10.3842/SIGMA.2010.025
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics
Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves
Matthew England
Department of Mathematics, University of Glasgow, UK
Received December 31, 2009, in final form March 19, 2010; Published online March 24, 2010
Abstract
We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases. We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain areas of the theory. We present solutions to the Jacobi inversion problem, sets of relations between the Abelian function, links to the Boussinesq equation and a new addition formula.
Key words:
Abelian function; Kleinian sigma function; Jacobi inversion problem; cyclic trigonal curve.
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References
 Baker H.F.,
Abelian functions. Abel's theorem and the allied theory of theta functions,
Cambridge University Press, Cambridge
1897 (reprinted in 1995).
 Baker H.F.,
On a system of differential equations leading to periodic functions,
Acta Math. 27 (1903), 135156.
 Baker H.F.,
Multiply periodic functions,
Cambridge University Press, Cambridge,
1907 (reprinted in 2007 by Merchant Books).
 Baldwin S., Eilbeck J.C., Gibbons J., Ônishi Y.,
Abelian functions for cyclic trigonal curves of genus four,
J. Geom. Phys. 58 (2008), 450467,
math.AG/0612654.
 Baldwin S., Gibbons J.,
Genus 4 trigonal reduction of the Benny equations,
J. Phys. A: Math. Theor. 39 (2006), 36073639.
 Buchstaber V.M., Enolskii V.Z., Leykin D.V.,
Kleinian functions, hyperelliptic Jacobians and applications,
Rev. Math. Math. Phys. 10 (1997), 3120.
 Buchstaber V.M., Leykin D.V., Enolskii V.Z.,
Rational analogues of abelian functions,
Funct. Anal. Appl. 33 (1999), 8394.
 Buchstaber V.M., Leykin D.V., Enolskii V.Z.,
Uniformization of Jacobi manifolds of trigonal curves, and nonlinear differential equations,
Funct. Anal. Appl. 34 (2000), 159171.
 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 (2007), Art.ID rnm140, 38 pages,
math.AG/0610019.
 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),
Editors D. Levi and O. Ragnisco,
CRM Proc. Lecture Notes, Vol. 25, Amer. Math. Soc., Providence, RI, 2000, 121138.
 England M.,
Higher genus Abelian functions associated with cyclic trigonal curves, available at
http://www.maths.gla.ac.uk/~mengland/Papers/2010_HGT/.
 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.
 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.
 Nakayashiki A.,
On algebraic expressions of sigma functions for (n,s)curves,
arXiv:0803.2083.
 Schreiner W., Distributed Maple, available at
www.risc.unilinz.ac.at/software/distmaple/.

