Models of Abelian varieties over valued fields, using model theory

Abstract

Given an elliptic curve E over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field ${\text{k}}_F$ and valuation ring $O_F$, there exists an integral separated smooth group scheme $E$ over $O_F$ with $E{\times }_{\text{Spec }{O}_F}\text{Spec }F\cong E$. If $\mathrm{char}\left({\text{k}}_F\right)\ne 2,3$ then one can be found over $O_{F^{alg}}$ such that the definable group $E\left(O\right)$ is the maximal generically stable subgroup of E. We also give some partial results on general Abelian varieties over F.

The construction of $E$ is by means of generating a birational group law over $O_F$ by the aid of a generically stable generic type of a definable subgroup of E.