On class numbers of division fields of abelian varieties
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 227-242.

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 n0, 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.

Published online:
DOI: 10.5802/jtnb.1077
Classification: 11R29,  11G10
Keywords: division fields, class number, abelian varieties
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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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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