Geodesic
Bring's curve: its period matrix and the vector of Riemann constants
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Bring's curve is the genus 4 Riemann surface with automorphism group of maximal size, $S_5$. Riera and Rodríguez have provided the most detailed study of the curve thus far via a hyperbolic model. We will recover and extend their results via an algebraic model based on a sextic curve given by both Hulek and Craig and implicit in work of Ramanujan. In particular we recover their period matrix; further, the vector of Riemann constants will be identified.
Keywords: Bring's curve; vector of Riemann constants. 
@article{SIGMA_2012_8_a64,
     author = {Harry W. Braden and Timothy P. Northover},
     title = {Bring's curve: its period matrix and the vector of {Riemann} constants},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2012},
     volume = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a64/}
}
TY  - JOUR
AU  - Harry W. Braden
AU  - Timothy P. Northover
TI  - Bring's curve: its period matrix and the vector of Riemann constants
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2012
VL  - 8
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a64/
LA  - en
ID  - SIGMA_2012_8_a64
ER  - 
%0 Journal Article
%A Harry W. Braden
%A Timothy P. Northover
%T Bring's curve: its period matrix and the vector of Riemann constants
%J Symmetry, integrability and geometry: methods and applications
%D 2012
%V 8
%U http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a64/
%G en
%F SIGMA_2012_8_a64
Harry W. Braden; Timothy P. Northover. Bring's curve: its period matrix and the vector of Riemann constants. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a64/

[1] Berndt B.C., Ramanujan's notebooks. Part III, Springer-Verlag, New York, 1991 | DOI | MR | Zbl

[2] Braden H.W., Northover T.P., “Klein's curve”, J. Phys. A: Math. Theor., 43 (2010), 434009, 17 pp. ; arXiv: 0905.4202 | DOI | MR | Zbl

[3] Breuer T., Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series, 280, Cambridge University Press, Cambridge, 2000 | MR | Zbl

[4] Bujalance E., Etayo J.J., Gamboa J.M., Gromadzki G., Automorphism groups of compact bordered Klein surfaces. A combinatorial approach, Lecture Notes in Mathematics, 1439, Springer-Verlag, Berlin, 1990 | MR | Zbl

[5] Craig M., “A sextic Diophantine equation”, Austral. Math. Soc. Gaz., 29 (2002), 27–29 | MR | Zbl

[6] Craig M., “On Klein's quartic curve”, Austral. Math. Soc. Gaz., 31 (2004), 115–120 | MR | Zbl

[7] Deconinck B., van Hoeij M., “Computing Riemann matrices of algebraic curves”, Phys. D, 152/153 (2001), 28–46 | DOI | MR | Zbl

[8] Dye R.H., “A plane sextic curve of genus $4$ with $A_5$ for collineation group”, J. London Math. Soc., 52 (1995), 97–110 | DOI | MR | Zbl

[9] Edge W.L., “Bring's curve”, J. London Math. Soc., 18 (1978), 539–545 | DOI | MR | Zbl

[10] Edge W.L., “Tritangent planes of Bring's curve”, J. London Math. Soc., 23 (1981), 215–222 | DOI | MR | Zbl

[11] Farkas H.M., Kra I., Riemann surfaces, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1980 | DOI | MR | Zbl

[12] Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin, 1973 | MR | Zbl

[13] Hulek K., “Geometry of the Horrocks–Mumford bundle”, Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Amer. Math. Soc., Providence, RI, 1987, 69–85 | MR

[14] Kallel S., Sjerve D., “Invariant spin structures on Riemann surfaces”, Ann. Fac. Sci. Toulouse Math. (6), 19 (2010), 457–477 ; arXiv: math.GT/0610568 | DOI | MR | Zbl

[15] Northover T.P., Riemann surfaces with symmetry: algorithms and applications, Ph.D. thesis, Edinburgh University, 2011

[16] Rauch H.E., Lewittes J., “The Riemann surface of Klein with 168 automorphisms”, Problems in Analysis, Papers Dedicated to Salomon Bochner, 1969, Princeton Univ. Press, Princeton, NJ, 1970, 297–308 | MR | Zbl

[17] Riera G., Rodríguez R.E., “The period matrix of Bring's curve”, Pacific J. Math., 154 (1992), 179–200 | MR | Zbl

[18] Vinnikov V., “Selfadjoint determinantal representations of real plane curves”, Math. Ann., 296 (1993), 453–479 | DOI | MR | Zbl

[19] Weber M., “Kepler's small stellated dodecahedron as a Riemann surface”, Pacific J. Math., 220 (2005), 167–182 | DOI | MR | Zbl