Elliptic curves associated with simplest quartic fields

Sylvain Duquesne[1]

  • [1] Université Montpellier II Laboratoire I3M (UMR 5149) et LIRMM (UMR 5506) CC 051, Place Eugène Bataillon 34005 Montpellier Cedex, France

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 81-100
  • ISSN: 1246-7405

Abstract

top
We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information has been obtained both on the structure of the Mordell-Weil group and on integral points for an infinite family of curves of rank 2. The canonical height is the main tool we used for that study.

How to cite

top

Duquesne, Sylvain. "Elliptic curves associated with simplest quartic fields." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 81-100. <http://eudml.org/doc/249948>.

@article{Duquesne2007,
abstract = {We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information has been obtained both on the structure of the Mordell-Weil group and on integral points for an infinite family of curves of rank 2. The canonical height is the main tool we used for that study.},
affiliation = {Université Montpellier II Laboratoire I3M (UMR 5149) et LIRMM (UMR 5506) CC 051, Place Eugène Bataillon 34005 Montpellier Cedex, France},
author = {Duquesne, Sylvain},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {simplest quartic field, elliptic curve; Mordell-Weil group; rank; generator; intgral point; naive height; canonical height},
language = {eng},
number = {1},
pages = {81-100},
publisher = {Université Bordeaux 1},
title = {Elliptic curves associated with simplest quartic fields},
url = {http://eudml.org/doc/249948},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Duquesne, Sylvain
TI - Elliptic curves associated with simplest quartic fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 81
EP - 100
AB - We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information has been obtained both on the structure of the Mordell-Weil group and on integral points for an infinite family of curves of rank 2. The canonical height is the main tool we used for that study.
LA - eng
KW - simplest quartic field, elliptic curve; Mordell-Weil group; rank; generator; intgral point; naive height; canonical height
UR - http://eudml.org/doc/249948
ER -

References

top
  1. H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Math. 138, Springer-Verlag, 1993. Zbl0786.11071MR1228206
  2. J. Cremona, M. Prickett, S. Siksek, Height difference bounds for elliptic curves over number fields. Journal of Number Theory 116 (2006), 42–68. Zbl1162.11032MR2197860
  3. J. Cremona, S. Siksek, Computing a Lower Bound for the Canonical Height on Elliptic Curves over ℚ . Algorithmic Number Theory, 7th International Symposium, ANTS-VII, LNCS 4076 (2006), 275–286. Zbl1143.11333MR2282930
  4. S. Duquesne, Integral points on elliptic curves defined by simplest cubic fields. Exp. Math. 10:1 (2001), 91–102. Zbl0983.11031MR1822854
  5. M. N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de ℚ . Publ. Math. Fac. Sci. Besancon, fasc 2 (1977/1978). Zbl0471.12006MR898667
  6. E. Halberstadt, Signes locaux des courbes elliptiques en 2 et 3. C. R. Acad. Sci. Paris Sér. I Math. 326:9 (1998), 1047–1052. Zbl0933.11030MR1647190
  7. H. K. Kim, Evaluation of zeta functions at s = - 1 of the simplest quartic fields. Proceedings of the 2003 Nagoya Conference “Yokoi-Chowla Conjecture and Related Problems”, Saga Univ., Saga, 2004, 63–73. 
  8. A. J. Lazarus, Class numbers of simplest quartic fields. Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, 313–323. Zbl0712.11061MR1106670
  9. A. J. Lazarus, On the class number and unit index of simplest quartic fields. Nagoya Math. J. 121 (1991), 1–13. Zbl0719.11073MR1096465
  10. S. Louboutin, The simplest quartic fields with ideal class groups of exponents less than or equal to 2. J. Math. Soc. Japan 56:3 (2004), 717–727. Zbl1142.11365MR2071669
  11. P. Olajos, Power integral bases in the family of simplest quartic fields. Experiment. Math. 14:2 (2005), 129–132. Zbl1092.11042MR2169516
  12. O. Rizzo, Average root numbers for a nonconstant family of elliptic curves. Compositio Math. 136:1 (2003), 1–23. Zbl1021.11020MR1965738
  13. S. Siksek, Infinite descent on elliptic curves. Rocky Mountain J. Math. 25:4 (1995), 1501–1538. Zbl0852.11028MR1371352
  14. J. H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, 1986. Zbl0585.14026MR817210
  15. J. H. Silverman, Computing heights on elliptic curves. Math. Comp. 51 (1988), 339–358. Zbl0656.14016MR942161
  16. J. H. Silverman, The difference between the Weil height and the canonical height on elliptic curves. Math. Comp. 55 (1990), 723–743. Zbl0729.14026MR1035944

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.