Coincidences of division fields

Daniels, Harris B.; Lozano-Robledo, Álvaro

doi:10.5802/aif.3520

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)

Annales de l'Institut Fourier, Online first, 40 p.

Abstract
Résumé

Let $E$ be an elliptic curve defined over $ℚ$ , and let $ρ_{E} : Gal (\bar{ℚ} / ℚ) \to GL (2, \hat{ℤ})$ be the adelic representation associated to the natural action of Galois on the torsion points of $E (\bar{ℚ})$ . By a theorem of Serre, the image of $ρ_{E}$ is open, but the image is always of index at least $2$ in $GL (2, \hat{ℤ})$ 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 $ℚ (E [p]) \cap ℚ (ζ_{q^{k}})$ 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 $ρ_{E} : Gal (\bar{ℚ} / ℚ) \to GL (2, \hat{ℤ})$ la représentation adélique associée à l’action naturelle de Galois sur les points de torsion de $E (\bar{ℚ})$ . Par un théorème de Serre, l’image de $ρ_{E}$ est ouverte mais toujours d’indice au moins $2$ dans $GL (2, \hat{ℤ})$ 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 $ℚ (E [p]) \cap ℚ (ζ_{q^{k}})$ 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: 2019-12-30
Revised: 2021-04-12
Accepted: 2021-06-23
Online First: 2022-07-29

DOI: 10.5802/aif.3520

Classification: 11G05, 14H52
Keywords: Elliptic Curves, Division Fields, Galois Representations.

Author's affiliations:

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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     title = {Coincidences of division fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2022},
     doi = {10.5802/aif.3520},
     language = {en},
     note = {Online first},
}

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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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