In this paper, we consider the equation
for integers and , with and . We extend work of the first and third-named authors by finding all solutions in the cases and . We do this by constructing a Frey–Hellegouarch -curve defined over the real quadratic field , and using the modular method with multi-Frey techniques.
Dans cet article, nous considérons l’équation
pour des entiers et , avec et . Nous prolongeons le travail des premier et troisième auteurs en trouvant toutes les solutions dans les cas et . Nous faisons ceci en construisant une -courbe de Frey–Hellegouarch définie sur le corps quadratique réel , et en combinant la méthode modulaire avec des techniques multi-Frey.
Revised:
Accepted:
Published online:
Mots-clés : Lebesgue–Nagell, Elliptic curves, Frey curve, multi-Frey, $\mathbb{Q}$-curves, modularity, level-lowering, Galois representations, newforms.
Michael A. Bennett 1; Philippe Michaud-Jacobs 2; Samir Siksek 2

@article{JTNB_2023__35_2_495_0, author = {Michael A. Bennett and Philippe Michaud-Jacobs and Samir Siksek}, title = {$\protect \mathbb{Q}$-curves and the {Lebesgue{\textendash}Nagell} equation}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {495--510}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {35}, number = {2}, year = {2023}, doi = {10.5802/jtnb.1254}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1254/} }
TY - JOUR AU - Michael A. Bennett AU - Philippe Michaud-Jacobs AU - Samir Siksek TI - $\protect \mathbb{Q}$-curves and the Lebesgue–Nagell equation JO - Journal de théorie des nombres de Bordeaux PY - 2023 SP - 495 EP - 510 VL - 35 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1254/ DO - 10.5802/jtnb.1254 LA - en ID - JTNB_2023__35_2_495_0 ER -
%0 Journal Article %A Michael A. Bennett %A Philippe Michaud-Jacobs %A Samir Siksek %T $\protect \mathbb{Q}$-curves and the Lebesgue–Nagell equation %J Journal de théorie des nombres de Bordeaux %D 2023 %P 495-510 %V 35 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1254/ %R 10.5802/jtnb.1254 %G en %F JTNB_2023__35_2_495_0
Michael A. Bennett; Philippe Michaud-Jacobs; Samir Siksek. $\protect \mathbb{Q}$-curves and the Lebesgue–Nagell equation. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 2, pp. 495-510. doi : 10.5802/jtnb.1254. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1254/
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