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Rational points on twists of X0(63)
Bruin, Nils; Fernández González, Julio; González i Rovira, Josep; Lario Loyo, Joan Carles
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II; Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. TN - Teoria de Nombres
Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois representation with cyclotomic determinant, and let $N>1$ be an integer that is square mod $p$. There exist two twisted modular curves $X^+(N,p)_\varrho$ and $X^+(N,p)'_\varrho$\, defined over~$\mathbb{Q}$ whose rational points classify the quadratic $\mathbb{Q}$-curves of degree $N$ realizing $\varrho$. The paper focuses on the only genus-three instance: the case $N\!=7,\,p=3$. From an explicit description of the automorphism group of the modular curve $X_0(63)$, it follows that the twisted curves are isomorphic over $\mathbb{Q}$ in this case. We also obtain a plane quartic equation for the twists and then produce the desired $\mathbb{Q}$-curves, provided that the set of rational points on this quartic can be determined. The existence of elliptic quotients and of an unramified double cover $X(7,3)_\varrho$ having a genus-two quotient permits a variety of combinations of covers and Prym-Chabauty methods to determine these rational points. We include two examples where these methods apply.
Number theory
Galois representations
Elliptic curves
Genus-three curves
Prym varieties
Chabauty methods
Quadratic Q-curves
Galois, Teoria de
Corbes algèbriques
Nombres, Teoria dels
Classificació AMS::11 Number theory
Attribution-NonCommercial-NoDerivs 2.5 Generic

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