Galois actions on Q - curves and

We prove two “large images” results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p > 13 not dividing d, provided that d is divisible by q (but d 6= q) with q = 2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing dD, where D = disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p ∤ 6 and that d is square-free. 1 Semistable Q-curves over real quadratic fields Let K be a quadratic field, and let E be a degree d Q-curve defined over K. Let D = disc(K). Assume that E is semistable, i.e., that E has good or semistable reduction at every finite place β ofK. Recall that we can attach to E a compatible family of Galois representations {σλ} of the absolute Galois group of Q: these representations can be seen as those attached to the Weil restriction A of E to Q, which is an abelian surface with real multiplication by F := Q( √ ±d) (cf. [E]). Let us call U the set of primes dividing D. For primes not in U , it is clear that A is also semistable, so in particular for supported by BFM2003-06092 supported by a MECD postdoctoral grant at the Centre de Recerca Matemática from Ministerio de Educación y Cultura