ℚ-curves and the Lebesgue–Nagell equation
Michael A. Bennett1 ; Philippe Michaud-Jacobs2 ; Samir Siksek2
1 Department of Mathematics University of British Columbia Vancouver, B.C., V6T 1Z2, Canada
2 Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 495-510.

Dans cet article, nous considérons l’équation

x 2 -q 2k+1 =y n ,q∤x,2∣y,

pour des entiers x,q,k,y et n, avec k≥0 et n≥3. Nous prolongeons le travail des premier et troisième auteurs en trouvant toutes les solutions dans les cas q=41 et q=97. Nous faisons ceci en construisant une ℚ-courbe de Frey–Hellegouarch définie sur le corps quadratique réel K=ℚ(q), et en combinant la méthode modulaire avec des techniques multi-Frey.

In this paper, we consider the equation

x 2 -q 2k+1 =y n ,q∤x,2∣y,

for integers x,q,k,y and n, with k≥0 and n≥3. We extend work of the first and third-named authors by finding all solutions in the cases q=41 and q=97. We do this by constructing a Frey–Hellegouarch ℚ-curve defined over the real quadratic field K=ℚ(q), and using the modular method with multi-Frey techniques.

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DOI : 10.5802/jtnb.1254
Classification : 11D41, 11D61, 11F80, 11G05
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

1 Department of Mathematics University of British Columbia Vancouver, B.C., V6T 1Z2, Canada
2 Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {$\protect \mathbb{Q}$-curves and the {Lebesgue{\textendash}Nagell} equation},
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Michael A. Bennett; Philippe Michaud-Jacobs; Samir Siksek. $\protect \mathbb{Q}$-curves and the Lebesgue–Nagell equation. Journal de théorie des nombres de Bordeaux, Tome 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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