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 $\mathbb{Z}[i]$?

**Edit:**

Thanks for the insights.

By "modular" I meant in the sense defined in descriptive articles on FLT and from expository articles by B. Mazur, K. Ribet, and Gouvea (AMM) on the work Wiles, Frey and Ribet did on FLT in 1994-1995. Loosely put, my "assumed curve" $E/\mathbb Q[i]$ (i.e. I never did think it could be modular; I was trying to find a way to show this lack of modularity by finding a contradiction) would be modular if somehow it was to have an $L$-function that equals the corresponding complex series expansion that is a cusp form and an eigenform on the upper complex plane, so that the coefficients in the complex series expansion are the $p$-defects $a_p$ in the $L$-function.

But I was wondering how it could be possible one could have an $L$-function for an elliptic curve over $\mathbb Q[i]$, by getting reductions modulo a prime, but by doing this with a curve that is over a complex rational field extension $\mathbb Q[i]$ of the rationals (i.e. with elements $a + bi$, $a, b \in\mathbb Q$, and above the real line on $\mathbb C$). I mean how could one define an element $$ a + bi \mod p? $$ I saw nothing about this in the related literature.

I was thinking there was a theorem by B. Mazur that showed if this assumed curve $E/\mathbb Q[i]$ was not modular over $\mathbb Q[i]$, where this is a complex field extension of $\mathbb Q$, then it certainly could not be modular as $E/\mathbb Q$.

Hasse-Weil Zeta Functionof a variety defined over a number field. Knowing the name will allow you to see that there is quite a large literature here. Also Felipe is right: based on what you write, you will need to learn some basic algebraic number theory in order to make sense of all this. (If you do not have this background, you can feel free to ask very basic questions on the related site math.stackexchange.com.) $\endgroup$