# ANNALES DE L'INSTITUT FOURIER

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

Let $E$ be an elliptic curve defined over $ℚ$, and let ${\rho }_{E}:Gal\left(\overline{ℚ}/ℚ\right)\to GL\left(2,\stackrel{^}{ℤ}\right)$ be the adelic representation associated to the natural action of Galois on the torsion points of $E\left(\overline{ℚ}\right)$. By a theorem of Serre, the image of ${\rho }_{E}$ is open, but the image is always of index at least $2$ in $GL\left(2,\stackrel{^}{ℤ}\right)$ 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 $ℚ\left(E\left[p\right]\right)\cap ℚ\left({\zeta }_{{q}^{k}}\right)$ 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 $ℚ\left(E\left[n\right]\right)=ℚ\left(E\left[m\right]\right)$, when the division field is abelian.

Soit $E$ une courbe elliptique définie sur $ℚ$ et soit ${\rho }_{E}:Gal\left(\overline{ℚ}/ℚ\right)\to GL\left(2,\stackrel{^}{ℤ}\right)$ la représentation adélique associée à l’action naturelle de Galois sur les points de torsion de $E\left(\overline{ℚ}\right)$. Par un théorème de Serre, l’image de ${\rho }_{E}$ est ouverte mais toujours d’indice au moins $2$ dans $GL\left(2,\stackrel{^}{ℤ}\right)$ 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 $ℚ\left(E\left[p\right]\right)\cap ℚ\left({\zeta }_{{q}^{k}}\right)$ 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 $ℚ\left(E\left[n\right]\right)=ℚ\left(E\left[m\right]\right)$, lorsque ce corps de division est abélien.

Revised:
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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