Coincidences of division fields
Annales de l'Institut Fourier, Online first, 40 p.

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

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

Received:
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Accepted:
Online First:
DOI: 10.5802/aif.3520
Classification: 11G05,  14H52
Keywords: Elliptic Curves, Division Fields, Galois Representations.
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)
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Daniels, Harris B.; Lozano-Robledo, Álvaro. Coincidences of division fields. Annales de l'Institut Fourier, Online first, 40 p.

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