Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves

Toshiro Hiranouchi; Tatsuya Ohshita

doi:10.5802/jtnb.1258

Toshiro Hiranouchi ¹ ; Tatsuya Ohshita ²

¹ Department of Basic Sciences Graduate School of Engineering Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi Fukuoka 804-8550, Japan
² Department of Mathematics Cooperative Faculty of Education Gunma University, Maebashi Gunma 371-8510, Japan

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 591-657.

Résumé
Abstract

Dans cet article, nous étudions une relation entre certains quotients de groupes des classes d’idéaux et le module d’Iwasawa cyclotomique $X_{\infty}$ du dual de Pontrjagin du groupe de Selmer fin d’une courbe elliptique $E$ sur $ℚ$ . Nous considérons l’extension galoisienne $K_{n}^{E}$ de $ℚ$ engendrée par les coordonnées des points de $p^{n}$ -torsion de $E$ et introduisons le quotient $A_{n}^{E}$ du $p$ -Sylow du groupe des classes d’idéaux de $K_{n}^{E}$ découpé par la représentation galoisienne modulo $p^{n}$ sur le groupe $E [p^{n}]$ . Nous décrivons le comportement asymptotique des $A_{n}^{E}$ en utilisant le module d’Iwasawa $X_{\infty}$ . En particulier, sous certaines conditions, nous obtenons une formule asymptotique à la Iwasawa pour l’ordre de $A_{n}^{E}$ en utilisant les invariants d’Iwasawa de $X_{\infty}$ .

In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_{\infty}$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $ℚ$ . We consider the Galois extension field $K_{n}^{E}$ of $ℚ$ generated by coordinates of all $p^{n}$ -torsion points of $E$ , and introduce a quotient $A_{n}^{E}$ of the $p$ -Sylow subgroup of the ideal class group of $K_{n}^{E}$ cut out by the modulo $p^{n}$ Galois representation $E [p^{n}]$ . We describe the asymptotic behavior of $A_{n}^{E}$ by using the Iwasawa module $X_{\infty}$ . In particular, under certain conditions, we obtain an asymptotic formula as Iwasawa’s class number formula on the order of $A_{n}^{E}$ by using Iwasawa’s invariants of $X_{\infty}$ .

Reçu le : 2022-05-13
Révisé le : 2022-10-20
Accepté le : 2022-11-14
Publié le : 2023-10-10

DOI : 10.5802/jtnb.1258

Classification : 11R29, 11G05, 11R23
Mots clés : class number, elliptic curve, Iwasawa theory

Affiliations des auteurs :

Toshiro Hiranouchi ¹ ; Tatsuya Ohshita ²

¹ Department of Basic Sciences Graduate School of Engineering Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi Fukuoka 804-8550, Japan
² Department of Mathematics Cooperative Faculty of Education Gunma University, Maebashi Gunma 371-8510, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Asymptotic behavior of class groups and cyclotomic {Iwasawa} theory of elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {591--657},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Toshiro Hiranouchi; Tatsuya Ohshita. Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 591-657. doi : 10.5802/jtnb.1258. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1258/

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