# Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$.

Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose generic fibre $\mathcal{X}_K$ surjects onto $A$.

Does $A$ have good reduction over $\mathcal{O}_K$?

If $\mathcal{O}_K$ is a dvr, the answer is positive by the theory of Neron models. But, if $\mathcal{O}_K$ is not discrete, as I've just learned from the answers to my earlier question Do abelian varieties have Neron models over arbitrary valuation rings?, the abelian variety $A$ might not have a Neron model over $\mathcal{O}_K$.

• This is already an interesting question if $\mathcal X$ is an abelian scheme and $\mathcal X_K \to A$ an isogeny. – R. van Dobben de Bruyn Aug 6 '18 at 3:07