Hesse Pencils and 3-Torsion Structures
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three.
Keywords: Hesse pencil; Galois representation; torsion points; elliptic curves.
@article{SIGMA_2018_14_a101,
     author = {Ane S. I. Anema and Jaap Top and Anne Tuijp},
     title = {Hesse {Pencils} and {3-Torsion} {Structures}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a101/}
}
TY  - JOUR
AU  - Ane S. I. Anema
AU  - Jaap Top
AU  - Anne Tuijp
TI  - Hesse Pencils and 3-Torsion Structures
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2018
VL  - 14
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a101/
LA  - en
ID  - SIGMA_2018_14_a101
ER  - 
%0 Journal Article
%A Ane S. I. Anema
%A Jaap Top
%A Anne Tuijp
%T Hesse Pencils and 3-Torsion Structures
%J Symmetry, integrability and geometry: methods and applications
%D 2018
%V 14
%U http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a101/
%G en
%F SIGMA_2018_14_a101
Ane S. I. Anema; Jaap Top; Anne Tuijp. Hesse Pencils and 3-Torsion Structures. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a101/

[1] Anema A. S.I., The arithmetic of maximal curves, the Hesse pencil and the Mestre curve, Ph.D. Thesis, Rijksuniversiteit Groningen, 2016 http://hdl.handle.net/11370/0ef530b1-709b-4285-b68d-016a67e6e928

[2] Artebani M., Dolgachev I., “The Hesse pencil of plane cubic curves”, Enseign. Math., 55 (2009), 235–273, arXiv: math.AG/0611590 | DOI | MR | Zbl

[3] Artin M., Rodriguez-Villegas F., Tate J., “On the Jacobians of plane cubics”, Adv. Math., 198 (2005), 366–382 | DOI | MR | Zbl

[4] Bosma W., Cannon J., Playoust C., “The Magma algebra system. I. The user language”, J. Symbolic Comput., 24 (1997), 235–265 | DOI | MR | Zbl

[5] Dickson L. E., “Invariantive theory of plane cubic curves modulo 2”, Amer. J. Math., 37 (1915), 107–116 | DOI | MR | Zbl

[6] Fisher T., “The Hessian of a genus one curve”, Proc. Lond. Math. Soc., 104 (2012), 613–648, arXiv: math.NT/0610403 | DOI | MR | Zbl

[7] Fisher T., “On families of 7- and 11-congruent elliptic curves”, LMS J. Comput. Math., 17 (2014), 536–564 | DOI | MR | Zbl

[8] Fulton W., Algebraic curves. An introduction to algebraic geometry, 2008 http://www.math.lsa.umich.edu/w̃fulton/ | MR

[9] Glynn D. G., “On cubic curves in projective planes of characteristic two”, Australas. J. Combin., 17, 1–20 | MR

[10] Kuwata M., Constructing families of elliptic curves with prescribed mod 3 representation via Hessian and Cayleyan curves, arXiv: 1112.6317

[11] Repertorio di matematiche superiori (definizioni, formole, teoremi, cenni bibliografici), v. II, Geometria, Ulrico Hoepli, Milano, 1900 | Zbl

[12] Rubin K., Silverberg A., “Families of elliptic curves with constant mod $p$ representations”, Elliptic Curves, Modular Forms, {F}ermat's Last Theorem (Hong Kong, 1993), Number Theory, I, Int. Press, Cambridge, MA, 1995, 148–161 | MR | Zbl

[13] Silverman J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York, 1986 | DOI | MR | Zbl

[14] Top J., Yui N., “Explicit equations of some elliptic modular surfaces”, Rocky Mountain J. Math., 37 (2007), 663–687, arXiv: math.AG/0307230 | DOI | MR | Zbl

[15] Tuijp A., Hesse pencil in characteristic two, Bachelor's Thesis, Rijksuniversiteit Groningen, 2015 http://fse.studenttheses.ub.rug.nl/id/eprint/13074

[16] Washington L. C., Elliptic curves. Number theory and cryptography, Discrete Mathematics and its Applications, 2nd ed., Chapman Hall/CRC, Boca Raton, FL, 2008 | DOI | MR | Zbl