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On the modularity of \(\mathbb Q\)-curves. (English) Zbl 1009.11038
An elliptic curve \(E\) over a number field \(K\) is called a \(\mathbb Q\)-curve, if it is isogenous over \(K\) to each of its Galois conjugates. Further, a \(\mathbb Q\)-curve that is a quotient of the Jacobian variety \(J_1(N)/ \overline{\mathbb Q}\) is called modular.
Let \(E/K\) be a \(\mathbb Q\)-curve, and, for each \(\sigma\in \text{Gal} (\overline{\mathbb Q}/\mathbb Q)\), let \(\mu_\sigma: E^\sigma\to E\) be a nonzero isogeny. Then we define \(b_E\in H^2 (\text{Gal} (\overline{\mathbb Q}/\mathbb Q), \pm 1)\) by \[ b_E(\sigma,\tau)= \text{sgn} (\mu_\sigma \mu_\tau^\sigma \mu_{\sigma\tau}^{-1}). \] Denote by \((b_E)_3\) the restriction of \(b_E\) to \(H^2 (\text{Gal} (\overline{\mathbb Q}_3/\mathbb Q_3), \pm 1)\). In this paper the authors prove that if \(E/K\) has potentially ordinary or multiplicative reduction at a prime of \(K\) over 3 and \((b_E)_3\) is trivial, then \(E\) is modular. Furthermore, they associate to \(E\) an \(\ell\)-adic Galois representation \(\rho_{E,\ell}\) of \(\text{Gal} (\overline{\mathbb Q}/\mathbb Q)\) and prove that if for some (hence every) prime \(\ell>3\), the projective representation associated to \(\rho_{e,\ell}\) is unramified to 3, then \(E\) is modular. As a consequence of these results, it follows that the \(\mathbb Q\)-curve \[ E_{A,B,C}: y^2= x^3+ 2(1+i)Ax^2+ (B+iA^2)x, \] discussed by H. Darmon in [“Serre’s conjectures”, Seminar on Fermat’s Last Theorem (Toronto 1993/94), V. K. Murty (ed.), CMS Conf. Proc. 17, 135–153 (1995; Zbl 0848.11019)], is modular.

11G05 Elliptic curves over global fields
11F80 Galois representations
11G18 Arithmetic aspects of modular and Shimura varieties
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
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