Models of Abelian varieties over valued fields, using model theory

Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{v}\mathrm{a}\mathrm{l})$ with perfect residue field ${\mathbf{\text{k}}}_F$ 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 $\text{char}({\mathbf{\text{k}}}_F)\mathrm{\ne }2,3$ 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 $E$.