On class numbers of division fields of abelian varieties

Jędrzej Garnek

doi:10.5802/jtnb.1077

Jędrzej Garnek ¹

¹ Graduate School, Adam Mickiewicz University Faculty of Mathematics and Computer Science Umultowska 87, 61-614 Poznan, Poland

Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 227-242.

Abstract
Résumé

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 $p$ .

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 $p$ .

Received: 2018-05-27
Revised: 2018-09-23
Accepted: 2018-11-15
Published online: 2019-07-28

MR: 3994728

DOI: 10.5802/jtnb.1077

Classification: 11R29, 11G10
Keywords: division fields, class number, abelian varieties

Author's affiliations:

Jędrzej Garnek ¹

¹ Graduate School, Adam Mickiewicz University Faculty of Mathematics and Computer Science Umultowska 87, 61-614 Poznan, Poland

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On class numbers of division fields of abelian varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Jędrzej Garnek. On class numbers of division fields of abelian varieties. Journal de théorie des nombres de Bordeaux, Volume 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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