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.

Received : 2018-05-28
Revised : 2018-09-24
Accepted : 2018-11-16
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1077
Classification:  11R29,  11G10
Keywords: division fields, class number, abelian varieties
@article{JTNB_2019__31_1_227_0,
     author = {J\k edrzej Garnek},
     title = {On class numbers of division fields of abelian varieties},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {227-242},
     doi = {10.5802/jtnb.1077},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2019__31_1_227_0/}
}
Garnek, Jędrzej. 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/item/JTNB_2019__31_1_227_0/

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