Q-curves and their Manin ideals

Let C be an elliptic curve over Q and let ω denote its Néron differential (uniquely determined up to sign). According to a conjecture of Manin-Stevens, there is a modular parametrization π: X₁(N) → C defined over Q such that the so-called Manin constant c(π) is equal to 1. The constant c(π) is defined by the equality π*(ω) = c(π)f(q)dq/q, where f is a normalized newform of weight 2 on Γ₁(N). In this paper we propose a generalization of the Manin constant as a certain ideal (we call it the Manin ideal) attached to modular parametrizations of elliptic curves defined over number fields. We conjecture that its value is (1), generalizing Manin-Stevens, and show the Manin ideal to be an "integral" ideal involving only primes dividing twice the conductor. Motivated by the Manin ideal considerations, we first study several aspects of building blocks of modular abelian varieties and modular parametrizations of Q-curves. Some examples are included to provide numerical evidence of the generalized conjecture, and the paper also contains the analogous items on the Manin ideals for modular building blocks of higher dimension.