Let $E$ be an elliptic curve, $\mathbb{Q}$ the field of rational numbers, $\mathbb{Z}[i]$ the ring of Gaussian integers and let $K$ be a number field that is some field extension of $\mathbb{Q}$.
Is it possible for an elliptic curve $E/K$ to be modular when $K$ is some field $\not =$ $\mathbb{Q}$ but is instead some field extension of $\mathbb{Q}$? If yes, then what are the conditions for which this would hold?
Is it possible for some elliptic curve $E/\mathbb{Q}[i]$ to be modular while the corresponding curve $E/\mathbb{Q}$ would not be modular? If yes, then what are the conditions for which this would hold?
If the answer to the first question in (2) is yes, then I would like to know how does one define such a modular elliptic curve $E/\mathbb{Q}[i]$ as being reduced modulo a prime? Is it possible for $E/\mathbb{Q}[i]$ to have good reduction or multiplicative reduction modulo a prime? Or does one have to use p-adic integers in some way, when dealing with field extensions of $\mathbb{Z}[i]$?
