Mathematics of Computation

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

Author(s): Martine Girard.
Journal: Math. Comp. 75 (2006), 1561-1583.
MSC (2000): Primary 11G30, 14H55, 14Q05; Secondary 14H40
Posted: May 1, 2006
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Abstract: The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space $\mathcal{M}_{3}$ of curves of genus 3.

1.: E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris.
Geometry of algebraic curves. Vol. I.
Springer-Verlag, New York, 1985. MR 0770932 (86h:14019)
2.: W. Bosma and J. Cannon, editors.
Handbook of Magma Functions, Sydney, 2002.
http://magma.maths.usyd.edu.au/.
3.: M. Girard.
Géométrie du groupe des points de Weierstrass d'une quartique lisse,
Journal of Number Theory, 94 (2002), 103-135. MR 1904965 (2003c:11069)
4.: M. Girard.
Groupe des points de Weierstrass sur une famille de quartiques lisses.
Acta Arithmetica, 105 (2002), 305-321. MR 1932565 (2003h:11067)
5.: M. Girard.
Code for computing heights of elliptic curves on number fields,
http://www.institut.math.jussieu.fr/ girard/magma/.
6.: M. Girard.
Group of Weierstrass points of a plane quartic with eight hyperflexes or more.
Technical report of the Mathematical Institute, Leiden University, June 2002.
http://www.math.leidenuniv.nl/reports/2002-14.shtml.
7.: M. Girard and P. Tzermias.
Group generated by the Weierstrass points of a plane quartic,
Proc. Amer. Math. Soc., 130 (2002), 667-672. MR 1866017 (2002h:14053)
8.: M. Hindry and J. H. Silverman.
Diophantine Geometry, An Introduction.
Springer-Verlag, New York, 2000.
Graduate Texts in Mathematics, 201. MR 1745599 (2001e:11058)
9.: J. H. Hubbard.
Sur les sections analytiques de la courbe universelle de Teichmüller.
Mem. Amer. Math. Soc., 4(166):ix+137, 1976. MR 0430321 (55:3326)
10.: M. J. Klassen and E. F. Schaefer.
Arithmetic and geometry of the curve $y\sp 3+1=x\sp 4$ .
Acta Arith., 74 (1996), 241-257. MR 1373711 (96k:11081)
11.: D. Laksov and A. Thorup.
Weierstrass points and gap sequences for families of curves.
Ark. Mat., 32 (1994), 393-422. MR 1318539 (96b:14041)
12.: D. T. Prapavessi.
On the Jacobian of the Klein curve.
Proc. Amer. Math. Soc., 122 (1994), 971-978. MR 1212286 (95b:14023)
13.: D. E. Rohrlich.
Points at infinity on the Fermat curves.
Invent. Math., 39 (1977), 95-127. MR 0441978 (56:367)
14.: J. H. Silverman.
Heights and the specialization map for families of abelian varieties.
J. Reine Angew. Math., 342 (1983), 197-211. MR 0703488 (84k:14033)
15.: A. M. Vermeulen.
Weierstrass points of weight two on curves of genus three.
Ph.D. thesis, Universiteit van Amsterdam, 1983. MR 0715084 (84j:14036)

Retrieve articles in Mathematics of Computation with MSC (2000): 11G30, 14H55, 14Q05, 14H40

Martine Girard
Affiliation: Universiteit Leiden, Mathematisch Instituut, 2300 R. A. Leiden, The Netherlands
Address at time of publication: School of Mathematics and Statistics, The University of Sydney, New South Wales, NSW 2006, Australia
Email: girard@maths.usyd.edu.au

DOI: 10.1090/S0025-5718-06-01853-9
PII: S 0025-5718(06)01853-9
Keywords: Algebraic curves, Jacobian, Weierstrass points, quartics, elliptic curves.
Received by editor(s): March 6, 2003
Received by editor(s) in revised form: April 1, 2005
Posted: May 1, 2006
Additional Notes: This research was carried out while the author was a postdoctoral fellow at Leiden University within the European Research Training Network Galois Theory and Explicit Methods in Arithmetic.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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