Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{ val})$ with perfect residue field $\textbf{ k}_F$ and valuation ring $\mathcal{ O}_F$, there exists an integral separated smooth group scheme $\mathcal{ E}$ over $\mathcal{ O}_F$ with $\mathcal{ E}\times_{ \text{ Spec } \mathcal{ O}_F}\text{ Spec } F\cong E$. If $\text{ char}(\textbf{ k}_F)\neq 2,3$ then one can be found over $\mathcal{ O}_{ F^{ alg}}$ such that the definable group $\mathcal{ E}(\mathcal{ O})$ is the maximal generically stable subgroup of $E$. We also give some partial results on general Abelian varieties over $F$. The construction of $\mathcal{ E}$ is by means of generating a birational group law over $\mathcal{ O}_F$ by the aid of a generically stable generic type of a definable subgroup of $E$.
Categories: math.LO math.AG