HYPERELLIPTIC MODULAR CURVES X0(n) AND ISOGENIES OF ELLIPTIC CURVES OVER QUADRATIC FIELDS

Let n be an integer such that the modular curve X0(n) is hyperelliptic of genus ≥ 2 and such that the Jacobian of X0(n) has rank 0 over Q. We determine all points of X0(n) defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to Q-isomorphism, all but finitely many elliptic curves with n-isogenies over quadratic fields are in fact Q-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to Q-isomorphism, every Q-curve E over a quadratic field K admitting an n-isogeny is d-isogenous, for some d | n, to the twist of its Galois conjugate by some quadratic extension L of K; we determine d and L explicitly.