no. 1
Coincidences of division fields
Daniels, Harris B.1; Lozano-Robledo, Álvaro2
1 Department of Mathematics Amherst College Amherst, MA 01002 (USA)
2 Department of Mathematics University of Connecticut Storrs, CT 06269 (USA)
Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 163-202.

Let E be an elliptic curve defined over ℚ, and let ρ E :Gal(ℚ ¯/ℚ)→GL(2,ℤ ^) be the adelic representation associated to the natural action of Galois on the torsion points of E(ℚ ¯). By a theorem of Serre, the image of ρ E is open, but the image is always of index at least 2 in GL(2,ℤ ^) 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 E/ℚ, and primes p and q such that ℚ(E[p])∩ℚ(ζ q k ) is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves E/ℚ and integers m,n such that the m-th and n-th division fields coincide, i.e., when ℚ(E[n])=ℚ(E[m]), when the division field is abelian.

Soit E une courbe elliptique définie sur ℚ et soit ρ E :Gal(ℚ ¯/ℚ)→GL(2,ℤ ^) la représentation adélique associée à l’action naturelle de Galois sur les points de torsion de E(ℚ ¯). Par un théorème de Serre, l’image de ρ E est ouverte mais toujours d’indice au moins 2 dans GL(2,ℤ ^) 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 E/ℚ et les nombres premiers p et q tels que ℚ(E[p])∩ℚ(ζ q k ) est non trivial et déterminons le degré de l’intersection. En conséquence, nous classifions toutes les courbes elliptiques E/ℚ et les entiers m,n tels que les corps de division m-ième et n-ième coïncident, c’est-à-dire lorsque ℚ(E[n])=ℚ(E[m]), lorsque ce corps de division est abélien.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3520
Classification: 11G05, 14H52
Keywords: Elliptic Curves, Division Fields, Galois Representations.
Mots-clés : Courbes elliptiques, corps de division, représentations galoisiennes.

Daniels, Harris B. 1; Lozano-Robledo, Álvaro 2

1 Department of Mathematics Amherst College Amherst, MA 01002 (USA)
2 Department of Mathematics University of Connecticut Storrs, CT 06269 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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