Let $X$ be a curve over $\overline{\mathbf{Q}}$.
I can prove that $X$ can be defined over some number field. (Take two equations defining $X$ in $\mathbf{P}^3$ and consider the number field containing the coefficients of this equation.)
Suppose that $X$ can be defined over a number field $K$.
It seems to be a basic fact that there are infinitely many non-isomorphic curves $Y$ over $K$, called twists I believe, such that $Y$ is isomorphic to $X$ over $\overline{\mathbf{Q}}$. Why is this?
Now, this actually bothers me a bit. Because I was hoping to have only a finite number of such "twists".
How can I guarantee that there are only a finite number of twists of a given curve $X/K$?
Let me be more precise.
For example, if I also want all the twists of $X$ over $K$ to have semi-stable reduction over $K$, is the number then finite?
Another example, if I also want all the twists of $X$ over $K$ to have good reduction over $K$, is the number then finite?
I might be asking for something that's impossible.