Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$.


Suppose that $C$ has good reduction at a prime $l$. Is it true that $C'$ has good reduction at $l$ if and only if $l$ is unramified in $K$? It seems to me that this should be the case (for example by looking at quadratic twists of elliptic curves) but I am not sure why it is true. Could someone provide a proof or a reference for a proof?



Good and bad reduction for twists of algebraic curves
Let $B$ be a Dedekind scheme with function field $K$. (Think of $B=\mathrm{Spec } \ \mathbb{Z}\_{p}$ for simplicity, so that $K=\mathbb{Q}\_p$.) Let $C$ be a smooth proper geometrically connected curve over $K$ of genus at least one with good reduction over $B$. Let $\mathcal{C}\to B$ be its (unique) smooth proper model.


Let $C'$ be a twist of $C$ over $K$. Let $L/K$ be a finite field extension such that $C'\_L \cong C\_L$ over $L$. One can use the theory of Neron models for curves due to Liu-Tong to prove the following (see <https://arxiv.org/abs/1312.4822>).


*Lemma.* Assume that the normalization $B'$ of $B$ in $L$ is (finite flat) etale over $B$. Then $C'$ has a smooth proper model over $B$.


**Proof.** Let $\mathcal{C}'\to B$ be the Neron model of $C'$ over $B$. Since $B'\to B$ is finite etale, the basechange $\mathcal{C}'\times\_B B'$ is the Neron model of $C'\_L$ over $B'$. Since $C'\_L\cong C\_L$, we have that $C'\_L$ has good reduction over $B'$. In particular, its Neron model is proper over $B'$. Thus, $\mathcal{C}' \times\_B B'\to B'$ is proper (as it is the Neron model of $C'\_L$ over $B'$). We conclude that $\mathcal{C}'\to B$ was already proper. In particular, $\mathcal{C}'\to B$ is a smooth proper model for $C'$ over $B$. QED


Concretely: when $K=\mathbb{Q}\_p$ and $B=\mathrm{Spec} \ \mathbb{Z}\_p$, if $C'$ is a twist of $C$ which has good reduction after passing to an unramified extension of $\mathbb{Q}\_p$, then it must have already had good reduction over $\mathbb{Q}\_p$.


