Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$.
Let $K$ be a number field and let $X$ be a curve over $K$.
Define $f_K(X)$ to be the value of $f$ after base change to $\overline{\mathbf{Q}}$.
Now, a priori, this function $f_K$ is not well-defined. In fact, one has to choose an embedding $K\subset \overline{\mathbf{Q}}$.
I do think that this is independent of the choice of embedding under some mild (or maybe none?) hypthesis on $f$. But why exactly?
Is $f_K$ well-defined? Do isomorphism $K$-curves take the same value?
Let me say that one can ask the same question for any function on some class of varieties over $\overline{k}$, where $k$ is a field.