Binomial squares in pure cubic number fields
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 691-704.

Let K=(ω), with ω 3 =m a positive integer, be a pure cubic number field. We show that the elements αK × 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.

Soit K=(ω), avec ω 3 =m>1 un nombre entier, un corps de nombres cubique. Nous montrons que les éléments αK × avec α 2 =a-ω (où a est un nombre rationnel) forment un groupe qui est isomorphe au groupe des points rationnels de la courbe elliptique E m :y 2 =x 3 -m. Nous démontrons aussi comment utiliser cette observation pour construire des extensions quadratiques non ramifiées de K.

DOI: 10.5802/jtnb.817
Lemmermeyer, Franz 1

1 Mörikeweg 1 73489 Jagstzell Germany
@article{JTNB_2012__24_3_691_0,
     author = {Lemmermeyer, Franz},
     title = {Binomial squares in pure cubic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {691--704},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     doi = {10.5802/jtnb.817},
     zbl = {1269.11108},
     mrnumber = {3010635},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.817/}
}
TY  - JOUR
AU  - Lemmermeyer, Franz
TI  - Binomial squares in pure cubic number fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2012
SP  - 691
EP  - 704
VL  - 24
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.817/
DO  - 10.5802/jtnb.817
LA  - en
ID  - JTNB_2012__24_3_691_0
ER  - 
%0 Journal Article
%A Lemmermeyer, Franz
%T Binomial squares in pure cubic number fields
%J Journal de théorie des nombres de Bordeaux
%D 2012
%P 691-704
%V 24
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.817/
%R 10.5802/jtnb.817
%G en
%F JTNB_2012__24_3_691_0
Lemmermeyer, Franz. Binomial squares in pure cubic number fields. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 691-704. doi : 10.5802/jtnb.817. http://www.numdam.org/articles/10.5802/jtnb.817/

[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. | MR | Zbl

[2] M. Bhargava, A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. ArXiv:1006.1002v2.

[3] G. Billing, Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlecht Eins. Nova Acta Reg. Soc. Ups. (IV) 11 (1938). | Zbl

[4] H. Cohen, J. Martinet, Heuristics on class groups: some good primes are not too good. Math. Comp. 63 (1994), no. 207, 329–334. | MR | Zbl

[5] H. Cohn, A classical invitation to algebraic numbers and class fields. Springer-Verlag, 1978. | MR

[6] H. Eisenbeis, G. Frey, B. Ommerborn, Computation of the 2-rank of pure cubic fields. Math. Comp. 32 (1978), 559–569. | MR | Zbl

[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. | MR | Zbl

[9] D. Husemöller, Elliptic Curves. 2nd ed., Springer-Verlag, 2004. | MR | Zbl

[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. | MR | Zbl

[13] F. Lemmermeyer, Why is the class number of (11 3) even? Math. Bohemica, to appear. | EuDML | Zbl

[14] T. Nagell, Solution complète de quelques équations cubiques à deux indéterminées. J. Math. Pures Appl. 4 (1925), 209–270. | EuDML | JFM

[15] pari, available from http://pari.math.u-bordeaux.fr

[16] sage, available from http://sagemath.org

[17] P. Satgé, Un analogue du calcul de Heegner. Invent. Math. 87 (1987), 425–439. | EuDML | MR | Zbl

[18] J. Silverman, J. Tate, Rational Points on Elliptic Curves. Springer-Verlag, 1992. | MR | Zbl

Cited by Sources: