Binomial squares in pure cubic number fields

Franz Lemmermeyer[1]

  • [1] Mörikeweg 1 73489 Jagstzell Germany

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 3, page 691-704
  • ISSN: 1246-7405

Abstract

top
Let K = ℚ ( ω ) , with ω 3 = m a positive integer, be a pure cubic number field. We show that the elements α ∈ K × whose squares have the form a - ω for rational numbers a form a group isomorphic to the group of rational points on the elliptic curve E m : y 2 = x 3 - m . This result will allow us to construct unramified quadratic extensions of pure cubic number fields K .

How to cite

top

Lemmermeyer, Franz. "Binomial squares in pure cubic number fields." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 691-704. <http://eudml.org/doc/251103>.

@article{Lemmermeyer2012,
abstract = {Let $K = \mathbb\{Q\}(\omega )$, with $\omega ^3 = m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^\times $ whose squares have the form $a - \omega $ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m: y^2 = x^3 - m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.},
affiliation = {Mörikeweg 1 73489 Jagstzell Germany},
author = {Lemmermeyer, Franz},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {binomial squares; pure cubic number fields; elliptic curves; unramified quadratic extension of pure cubic number fields},
language = {eng},
month = {11},
number = {3},
pages = {691-704},
publisher = {Société Arithmétique de Bordeaux},
title = {Binomial squares in pure cubic number fields},
url = {http://eudml.org/doc/251103},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Lemmermeyer, Franz
TI - Binomial squares in pure cubic number fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/11//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 3
SP - 691
EP - 704
AB - Let $K = \mathbb{Q}(\omega )$, with $\omega ^3 = m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^\times $ whose squares have the form $a - \omega $ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m: y^2 = x^3 - m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.
LA - eng
KW - binomial squares; pure cubic number fields; elliptic curves; unramified quadratic extension of pure cubic number fields
UR - http://eudml.org/doc/251103
ER -

References

top
  1. P. Barrucand, H. Cohn, A rational genus, class number divisibility, and unit theory for pure cubic fields. J. Number Theory 2 (1970), 7–21. Zbl0192.40001MR249398
  2. M. Bhargava, A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. ArXiv:1006.1002v2. Zbl1307.11071
  3. G. Billing, Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlecht Eins. Nova Acta Reg. Soc. Ups. (IV) 11 (1938). Zbl0018.05401
  4. H. Cohen, J. Martinet, Heuristics on class groups: some good primes are not too good. Math. Comp. 63 (1994), no. 207, 329–334. Zbl0827.11067MR1226813
  5. H. Cohn, A classical invitation to algebraic numbers and class fields. Springer-Verlag, 1978. Zbl0395.12001MR506156
  6. H. Eisenbeis, G. Frey, B. Ommerborn, Computation of the 2-rank of pure cubic fields. Math. Comp. 32 (1978), 559–569. Zbl0385.12001MR480416
  7. L. Euler, Vollständige Anleitung zur Algebra (E387, E388). St. Petersburg, 1770. 
  8. T. Honda, Pure cubic fields whose class numbers are multiples of three. J. Number Theory 3 (1971), 7–12. Zbl0222.12004MR292795
  9. D. Husemöller, Elliptic Curves. 2nd ed., Springer-Verlag, 2004. Zbl0605.14032MR2024529
  10. J.-L. Lagrange, Sur la solution des problèmes indéterminés du second degré. Mem. Acad. Sci. Berlin, 1769. 
  11. J.-L. Lagrange, Additions à l’analyse indéterminée. Lyon, 1774. 
  12. F. Lemmermeyer, A note on Pépin’s counter examples to the Hasse principle for curves of genus 1 . Abh. Math. Sem. Hamburg 69 (1999), 335–345. Zbl0949.11019MR1722943
  13. F. Lemmermeyer, Why is the class number of ℚ ( 11 3 ) even? Math. Bohemica, to appear. Zbl1274.11162
  14. T. Nagell, Solution complète de quelques équations cubiques à deux indéterminées. J. Math. Pures Appl. 4 (1925), 209–270. Zbl51.0135.02
  15. pari, available from http://pari.math.u-bordeaux.fr 
  16. sage, available from http://sagemath.org Zbl1312.68206
  17. P. Satgé, Un analogue du calcul de Heegner. Invent. Math. 87 (1987), 425–439. Zbl0616.14023MR870738
  18. J. Silverman, J. Tate, Rational Points on Elliptic Curves. Springer-Verlag, 1992. Zbl0752.14034MR1171452

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.