Let be an abelian variety defined over a number field . Fix a prime and a natural number and consider the field , obtained by adjoining to all the coordinates of the -torsion points of . We give a lower bound on the -part of the class group of for large , by finding a large unramified extension of . This lower bound depends on the Mordell–Weil rank of and the reduction of -torsion points modulo primes above .
Soit une variété abélienne définie sur un corps de nombres . On fixe un nombre premier et pour tout nombre naturel on note le corps engendré sur par les coordonnées des points de -torsion de . Nous donnons une minoration de l’ordre de la -partie du groupe de classes de pour , en construisant une extension non ramifiée suffisamment grande de Cette minoration dépend du rang du groupe de Mordell–Weil de et de la réduction des points de -torsion en nombres premiers au-dessus de .
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1077
Classification: 11R29, 11G10
Keywords: division fields, class number, abelian varieties
Author's affiliations:
@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/} }
TY - JOUR TI - On class numbers of division fields of abelian varieties JO - Journal de Théorie des Nombres de Bordeaux PY - 2019 DA - 2019/// SP - 227 EP - 242 VL - 31 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1077/ UR - https://www.ams.org/mathscinet-getitem?mr=3994728 UR - https://doi.org/10.5802/jtnb.1077 DO - 10.5802/jtnb.1077 LA - en ID - JTNB_2019__31_1_227_0 ER -
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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