# Good reduction of finite etale covers of abelian varieties

Let $$R$$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $$K$$. Let $$A$$ be an abelian variety over $$K$$ with good reduction over $$R$$. Let $$X\to A$$ be a finite etale morphism with $$X$$ geometrically connected over $$K$$.

Does $$X$$ have good reduction over $$R$$? (That is, does $$X$$ have a smooth proper model over $$R$$?)

By Lang-Neron, there is finite field extension $$L/K$$ such that $$X$$ is an abelian variety over $$K$$. In this case $$X\to A$$ is an isogeny and it follows from Neron-Ogg-Shafarevich that $$X$$ has good reduction as well over $$R$$. Thus, $$X$$ has potential good reduction over $$R$$, i.e., there is a finite extension $$L/K$$ such that $$X_{R_L}$$ has a smooth proper model over $$R_L$$, where $$R_L$$ is the integral closure of $$R$$ in $$L$$.

I fear that my answer has a negative answer, but I can't think of an explicit example. Can one find an elliptic curve $$E$$ over $$\mathbb{Q}$$ with good reduction at a prime $$p$$, an $$E$$-torsor $$T$$ with bad reduction at this prime $$p$$, and a finite etale cover $$T\to E$$. But I do not see how.

• Try the cover of $y^2=x^3-x$ given by $z^2=3x$ with $p=3$. – Felipe Voloch Feb 13 at 9:20