On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$

Sairaiji, Fumio; Yamauchi, Takuya

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 893-915.

Soit $E$ une courbe elliptique sur $\mathbb{Q}$ 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 $K_n:=\mathbb{Q}\left(E\left[p^n\right]\right)$. 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 $E\left(\mathbb{Q}\right)$ et de la ramification liée à $E$.

Let $E$ be an elliptic curve over $\mathbb{Q}$ 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 $K_n:=\mathbb{Q}\left(E\left[p^n\right]\right)$. 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 $E\left(\mathbb{Q}\right)$ and the ramification related to $E$.

Classification : 11G05, 11G07
Mots-clés : elliptic curves, Mordell–Weil rank, class number

[1] Bashmakov, M. I. 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 | MR | Zbl

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

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

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

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

[6] Greenberg, Ralph 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 | DOI | MR | Zbl

[7] Grothendieck, Alexander 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

[8] Hiranouchi, Toshiro Class numbers associated to elliptic curves over $\mathbb{Q}$ with good reduction at $p$ (2016) (preprint)

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

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

[11] Lawson, Tyler; Wuthrich, Christian 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 | DOI | MR | Zbl

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

[13] Sairaiji, Fumio; Yamauchi, Takuya On the class numbers of the fields of the $p^n$-torsion points of certain elliptic curves over $\mathbb{Q}$, J. Number Theory, Volume 156 (2015), pp. 277-289 | DOI | MR | Zbl

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

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

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