$\mathbb{Q}$-curves and the Lebesgue–Nagell equation

Bennett, Michael A.; Michaud-Jacobs, Philippe; Siksek, Samir

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 495-510

In this paper, we consider the equation

$$x^2-q^{2k+1}=y^n,q\nmid x,2\mid y,$$

for integers $x,q,k,y$ and $n$, with $k\ge 0$ and $n\ge 3$. 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 $\mathbb{Q}$-curve defined over the real quadratic field $K=\mathbb{Q}\left(\sqrt{q}\right)$, and using the modular method with multi-Frey techniques.

Dans cet article, nous considérons l’équation

$$x^2-q^{2k+1}=y^n,q\nmid x,2\mid y,$$

pour des entiers $x,q,k,y$ et $n$, avec $k\ge 0$ et $n\ge 3$. 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 $\mathbb{Q}$-courbe de Frey–Hellegouarch définie sur le corps quadratique réel $K=\mathbb{Q}\left(\sqrt{q}\right)$, et en combinant la méthode modulaire avec des techniques multi-Frey.

Classification : 11D41, 11D61, 11F80, 11G05
Keywords: Lebesgue–Nagell, Elliptic curves, Frey curve, multi-Frey, $\mathbb{Q}$-curves, modularity, level-lowering, Galois representations, newforms.

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