Geodesic
Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: Abelian function; Kleinian sigma function; Jacobi inversion problem; cyclic trigonal curve. 
@article{SIGMA_2010_6_a24,
     author = {Matthew England},
     title = {Higher {Genus} {Abelian} {Functions} {Associated} with {Cyclic} {Trigonal} {Curves}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2010},
     volume = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a24/}
}
TY  - JOUR
AU  - Matthew England
TI  - Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2010
VL  - 6
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a24/
LA  - en
ID  - SIGMA_2010_6_a24
ER  - 
%0 Journal Article
%A Matthew England
%T Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves
%J Symmetry, integrability and geometry: methods and applications
%D 2010
%V 6
%U http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a24/
%G en
%F SIGMA_2010_6_a24
Matthew England. Higher Genus Abelian Functions Associated with Cyclic Trigonal Curves. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a24/

[1] Baker H. F., Abelian functions. Abel's theorem and the allied theory of theta functions, Cambridge University Press, Cambridge, 1897, (reprinted in 1995) | MR

[2] Baker H. F., “On a system of differential equations leading to periodic functions”, Acta Math., 27 (1903), 135–156 | DOI | MR | Zbl

[3] Baker H. F., Multiply periodic functions, Cambridge University Press, Cambridge, 1907, (reprinted in 2007 by Merchant Books) | Zbl

[4] Baldwin S., Eilbeck J. C., Gibbons J., Ônishi Y., “Abelian functions for cyclic trigonal curves of genus four”, J. Geom. Phys., 58 (2008), 450–467, arXiv: math.AG/0612654 | DOI | MR | Zbl

[5] Baldwin S., Gibbons J., “Genus 4 trigonal reduction of the Benny equations”, J. Phys. A: Math. Theor., 39 (2006), 3607–3639 | DOI | MR | Zbl

[6] Buchstaber V. M., Enolskii V. Z., Leykin D. V., “Kleinian functions, hyperelliptic Jacobians and applications”, Rev. Math. Math. Phys., 10 (1997), 3–120 | Zbl

[7] Buchstaber V. M., Leykin D. V., Enolskii V. Z., “Rational analogues of abelian functions”, Funct. Anal. Appl., 33 (1999), 83–94 | DOI | MR | Zbl

[8] 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), 159–171 | DOI | MR | Zbl

[9] 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 pp., arXiv: math.AG/0610019 | DOI | MR | Zbl

[10] Eilbeck J. C., Enolskii V. Z., Leykin D. V., “On the Kleinian construction of Abelian functions of canonical algebraic curves”, SIDE III “Symmetries and Integrability of Difference Equations” (Sabaudia, 1998), CRM Proc. Lecture Notes, 25, eds. D. Levi and O. Ragnisco, Amer. Math. Soc., Providence, RI, 2000, 121–138 | MR | Zbl

[11] England M., Higher genus Abelian functions associated with cyclic trigonal curves, available at http://www.maths.gla.ac.uk/~mengland/Papers/2010_HGT/

[12] 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 pp., arXiv: 0806.2377 | DOI | MR

[13] England M., Gibbons J., “A genus six cyclic tetragonal reduction of the Benney equations”, J. Phys. A: Math. Theor., 42 (2009), 375202, 27 pp., arXiv: 0903.5203 | DOI | MR

[14] Nakayashiki A., On algebraic expressions of sigma functions for $(n,s)$-curves, arXiv: 0803.2083

[15] Schreiner W., Distributed Maple, available at http://www.risc.uni-linz.ac.at/software/distmaple/