  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 $E$.
