Let be an elliptic curve defined over , and let be the adelic representation associated to the natural action of Galois on the torsion points of . By a theorem of Serre, the image of is open, but the image is always of index at least in due to a certain quadratic entanglement amongst division fields. In this paper, we study other types of abelian entanglements. More concretely, we classify the elliptic curves , and primes and such that is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves and integers such that the -th and -th division fields coincide, i.e., when , when the division field is abelian.
Soit une courbe elliptique définie sur et soit la représentation adélique associée à l’action naturelle de Galois sur les points de torsion de . Par un théorème de Serre, l’image de est ouverte mais toujours d’indice au moins dans en raison d’un certain enchevêtrement quadratique entre les corps de division. Dans cet article, nous étudions d’autres types d’enchevêtrements abéliens. Plus concrètement, nous classifions les courbes elliptiques et les nombres premiers et tels que est non trivial et déterminons le degré de l’intersection. En conséquence, nous classifions toutes les courbes elliptiques et les entiers tels que les corps de division -ième et -ième coïncident, c’est-à-dire lorsque , lorsque ce corps de division est abélien.
Revised:
Accepted:
Published online:
Keywords: Elliptic Curves, Division Fields, Galois Representations.
Mot clés : Courbes elliptiques, corps de division, représentations galoisiennes.

@article{AIF_2023__73_1_163_0, author = {Daniels, Harris B. and Lozano-Robledo, \'Alvaro}, title = {Coincidences of division fields}, journal = {Annales de l'Institut Fourier}, pages = {163--202}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {1}, year = {2023}, doi = {10.5802/aif.3520}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3520/} }
TY - JOUR AU - Daniels, Harris B. AU - Lozano-Robledo, Álvaro TI - Coincidences of division fields JO - Annales de l'Institut Fourier PY - 2023 SP - 163 EP - 202 VL - 73 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3520/ DO - 10.5802/aif.3520 LA - en ID - AIF_2023__73_1_163_0 ER -
%0 Journal Article %A Daniels, Harris B. %A Lozano-Robledo, Álvaro %T Coincidences of division fields %J Annales de l'Institut Fourier %D 2023 %P 163-202 %V 73 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3520/ %R 10.5802/aif.3520 %G en %F AIF_2023__73_1_163_0
Daniels, Harris B.; Lozano-Robledo, Álvaro. Coincidences of division fields. Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 163-202. doi : 10.5802/aif.3520. https://aif.centre-mersenne.org/articles/10.5802/aif.3520/
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