Number theory
The truth about torsion in the CM case
[La vérité sur la torsion dans le cas CM]

Présenté par : Jean-Pierre Serre

Clark, Pete L.1 ; Pollack, Paul1
1 Department of Mathematics, University of Georgia, Athens, GA, 30602, USA
Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 683-688.

Soit TCM(d) la taille maximale du sous-groupe de torsion d'une courbe elliptique à multiplications complexes, définie sur un corps de nombres de degré d. Nous montrons qu'il existe C une constante absolue, effective, telle que TCM(d)≤Cdlog⁡log⁡(d) pour tout d≥3.

Let TCM(d) be the maximum size of the torsion subgroup of an elliptic curve with complex multiplication over a degree d number field. We show there is an absolute, effective constant C such that TCM(d)≤Cdlog⁡log⁡d for all d≥3.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.05.004
Clark, Pete L. 1 ; Pollack, Paul 1

1 Department of Mathematics, University of Georgia, Athens, GA, 30602, USA 
@article{CRMATH_2015__353_8_683_0,
     author = {Clark, Pete L. and Pollack, Paul},
     title = {The truth about torsion in the {CM} case},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {683--688},
     publisher = {Elsevier},
     volume = {353},
     number = {8},
     year = {2015},
     doi = {10.1016/j.crma.2015.05.004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.05.004/}
}
TY  - JOUR
AU  - Clark, Pete L.
AU  - Pollack, Paul
TI  - The truth about torsion in the CM case
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 683
EP  - 688
VL  - 353
IS  - 8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.05.004/
DO  - 10.1016/j.crma.2015.05.004
LA  - en
ID  - CRMATH_2015__353_8_683_0
ER  - 
%0 Journal Article
%A Clark, Pete L.
%A Pollack, Paul
%T The truth about torsion in the CM case
%J Comptes Rendus. Mathématique
%D 2015
%P 683-688
%V 353
%N 8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.05.004/
%R 10.1016/j.crma.2015.05.004
%G en
%F CRMATH_2015__353_8_683_0
Clark, Pete L.; Pollack, Paul. The truth about torsion in the CM case. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 683-688. doi : 10.1016/j.crma.2015.05.004. http://www.numdam.org/articles/10.1016/j.crma.2015.05.004/

[1] Aoki, N. Torsion points on abelian varieties with complex multiplication, Kitasakado, 1994, World Sci. Publ., River Edge, NJ, USA (1995), pp. 1-22

[2] Bateman, P.T.; Chowla, S.; Erdős, P. Remarks on the size of L(1,χ), Publ. Math. (Debr.), Volume 1 (1950), pp. 165-182

[3] Bourdon, A.; Clark, P.L.; Stankewicz, J. Torsion points on CM elliptic curves over real number fields (submitted for publication) | arXiv

[4] Breuer, F. Torsion bounds for elliptic curves and Drinfeld modules, J. Number Theory, Volume 130 (2010), pp. 1241-1250

[5] Clark, P.L.; Cook, B.; Stankewicz, J. Torsion points on elliptic curves with complex multiplication (with an appendix by Alex Rice), Int. J. Number Theory, Volume 9 (2013), pp. 447-479

[6] Cohen, H. Advanced Topics in Computational Number Theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, 2000

[7] Davenport, H. Multiplicative Number Theory, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000

[8] Haneke, W.; Haneke, W. Über die reellen Nullstellen der Dirichletschen L-Reihen, Corrigendum, Acta Arith., Volume 22 (1973), pp. 391-421

[9] Hindry, M.; Silverman, J. Sur le nombre de points de torsion rationnels sur une courbe elliptique, C. R. Acad. Sci. Paris, Ser. I, Volume 329 (1999) no. 2, pp. 97-100

[10] Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers, Oxford University Press, Oxford, UK, 2008

[11] Goldfeld, D.M.; Schinzel, A. On Siegel's zero, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 2 (1975), pp. 571-583

[12] Pintz, J. Elementary methods in the theory of L-functions. II. On the greatest real zero of a real L-function, Acta Arith., Volume 31 (1976), pp. 273-289

[13] Rosen, M. A generalization of Mertens' theorem, J. Ramanujan Math. Soc., Volume 14 (1999), pp. 1-19

[14] Silverberg, A. Points of finite order on abelian varieties, p-Adic Methods in Number Theory and Algebraic Geometry, Contemp. Math., vol. 133, Amer. Math. Soc., Providence, RI, USA, 1992, pp. 175-193

[15] Silverman, J. Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, 1994

Cité par Sources :