On class numbers of division fields of abelian varieties

Jędrzej Garnek

doi:10.5802/jtnb.1077

Jędrzej Garnek

Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 227-242.

Résumé
Abstract

Soit $A$ une variété abélienne définie sur un corps de nombres $K$ . On fixe un nombre premier $p$ et pour tout nombre naturel $n,$ on note $K_{n}$ le corps engendré sur $K$ par les coordonnées des points de $p^{n}$ -torsion de $A$ . Nous donnons une minoration de l’ordre de la $p$ -partie du groupe de classes de $K_{n}$ pour $n ≫ 0$ , en construisant une extension non ramifiée suffisamment grande de $K_{n} .$ Cette minoration dépend du rang du groupe de Mordell–Weil de $A$ et de la réduction des points de $p$ -torsion en nombres premiers au-dessus de $p$ .

Let $A$ be an abelian variety defined over a number field $K$ . Fix a prime $p$ and a natural number $n$ and consider the field $K_{n}$ , obtained by adjoining to $K$ all the coordinates of the $p^{n}$ -torsion points of $A$ . We give a lower bound on the $p$ -part of the class group of $K_{n}$ for large $n$ , by finding a large unramified extension of $K_{n}$ . This lower bound depends on the Mordell–Weil rank of $A$ and the reduction of $p$ -torsion points modulo primes above $p$ .

Reçu le : 2018-05-27
Révisé le : 2018-09-23
Accepté le : 2018-11-15
Publié le : 2019-07-28

MR 3994728

DOI : https://doi.org/10.5802/jtnb.1077
Classification : 11R29, 11G10
Mots clés : division fields, class number, abelian varieties

@article{JTNB_2019__31_1_227_0,
     author = {J\k{e}drzej Garnek},
     title = {On class numbers of division fields of abelian varieties},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {227--242},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     doi = {10.5802/jtnb.1077},
     mrnumber = {3994728},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/}
}

Jędrzej Garnek. On class numbers of division fields of abelian varieties. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 227-242. doi : 10.5802/jtnb.1077. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/

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