On the class numbers of the fields of the p n -torsion points of elliptic curves over
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 893-915.

Let E be an elliptic curve over which has multiplicative reduction at a fixed prime p. Assume E has multiplicative reduction or potentially good reduction at any prime not equal to p. For each positive integer n we put K n :=(E[p n ]). The aim of this paper is to extend the authors’ previous results in [13] concerning with the order of the p-Sylow group of the ideal class group of K n to more general setting. We also modify the previous lower bound of the order given in terms of the Mordell–Weil rank of E() and the ramification related to E.

Soit E une courbe elliptique sur ayant réduction multiplicative en un nombre premier p. Supposons que en tout nombre premier différent de p la courbe E a une réduction multiplicative ou potentiellement bonne. Pour chaque entier positif n on pose K n :=(E[p n ]). Le but de cet article est d’étendre nos résultats précédents [13] concernant l’ordre du p-sous-groupe de Sylow du groupe des classes d’idéaux de K n à un cadre plus général. Nous modifions également la borne inférieure précédente de cet ordre donnée en termes du rang de Mordell–Weil de E() et de la ramification liée à E.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1056
Classification: 11G05,  11G07
Keywords: elliptic curves, Mordell–Weil rank, class number
Fumio Sairaiji 1; Takuya Yamauchi 2

1 Faculty of Nursing, Hiroshima International University, Hiro, Hiroshima 737-0112, Japan
2 Mathematical Institute, Tohoku University 6-3, Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, Japan
@article{JTNB_2018__30_3_893_0,
     author = {Fumio Sairaiji and Takuya Yamauchi},
     title = {On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\protect \mathbb{Q}$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {893--915},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1056},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1056/}
}
TY  - JOUR
TI  - On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\protect \mathbb{Q}$
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 893
EP  - 915
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1056/
UR  - https://doi.org/10.5802/jtnb.1056
DO  - 10.5802/jtnb.1056
LA  - en
ID  - JTNB_2018__30_3_893_0
ER  - 
%0 Journal Article
%T On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\protect \mathbb{Q}$
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 893-915
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1056
%R 10.5802/jtnb.1056
%G en
%F JTNB_2018__30_3_893_0
Fumio Sairaiji; Takuya Yamauchi. On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\protect \mathbb{Q}$. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 893-915. doi : 10.5802/jtnb.1056. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1056/

[1] M. I. Bashmakov The cohomology of abelian varieties over a number field, Usp. Mat. Nauk, Volume 27 (1972) no. 6, pp. 25-66 (translation in Russ. Math. Surv. 17 (1972), no. 1, p. 25-70) | Zbl: 0256.14016

[2] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 21, Springer, 1990 | Zbl: 0705.14001

[3] Tim Dokchitser; Vladimir Dokchitser Surjectivity of mod 2 n representations of elliptic curves, Math. Z., Volume 272 (2012) no. 3-4, pp. 961-964 | Zbl: 1315.11046

[4] Noam Elkies Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9 (2006) (preprint)

[5] Takashi Fukuda; Keiichi Komatsu; Shuji Yamagata Iwasawa λ-invariants and Mordell–Weil ranks of abelian varieties with complex multiplication, Acta Arith., Volume 127 (2007) no. 4, pp. 305-307 | Zbl: 1188.11055

[6] Ralph Greenberg Iwasawa theory past and present, Class field theory its centenary and prospect (Tokyo, 1998) (Advanced Studies in Pure Mathematics) Volume 30, Mathematical Society of Japan, 2001, pp. 335-385 | Zbl: 0998.11054

[7] Alexander Grothendieck Modèles de Néron et monodromie, Seminaire de géométrie algébrique Du Bois-Marie 1967-1969 (SGA 7) (Lecture Notes in Mathematics) Volume 288, Springer, 1972, pp. 313-523 | Zbl: 0248.14006

[8] Toshiro Hiranouchi Class numbers associated to elliptic curves over with good reduction at p (2016) (preprint)

[9] John W. Jones; David P. Roberts Database of Local Fields (available at https://math.la.asu.edu/~jj/localfields/)

[10] Serge Lang Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften, Volume 231, Springer, 1978 | Zbl: 0388.10001

[11] Tyler Lawson; Christian Wuthrich Vanishing of some Galois cohomology groups for elliptic curves, Elliptic curves, modular forms and Iwasawa theory (Springer Proceedings in Mathematics & Statistics) Volume 188, Springer, 2016, pp. 373-399 | Zbl: 06740248

[12] Jean-François Mestre; Joseph Oesterlé Courbes de Weil semi-stables de discriminant une puissance m-iéme, J. Reine Angew. Math., Volume 400 (1989), pp. 173-184 | Zbl: 0693.14004

[13] Fumio Sairaiji; Takuya Yamauchi On the class numbers of the fields of the p n -torsion points of certain elliptic curves over , J. Number Theory, Volume 156 (2015), pp. 277-289 | Zbl: 1328.11064

[14] Jean-Pierre Serre Proprietes galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972) no. 4, pp. 259-331 | Zbl: 0235.14012

[15] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, Volume 106, Springer, 1986 | Zbl: 0585.14026

[16] The LMFDB Collaboration Elliptic Curves over Number Fields (http://www.lmfdb.org/EllipticCurve/)

Cited by Sources: