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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.

##### MSC:
 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
##### Keywords:
elliptic curve; Q-curve; modular curve; Galois representation
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