Models of Abelian varieties over valued fields, using model theory

Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field, there exists an integral separated smooth group scheme $\mathcal{E}$ over $\mathcal{O}_F$ with $\mathcal{E}\times_{\text{Spec } \mathcal{O}_F}\text{Spec } F\cong E$. 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 $E$

Contracting Endomorphisms of Valued Fields

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in $q$. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed). The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma$ which is locally infinitely contracting and fails to be onto. Namely we prove the existence of a model complete theory $\widetilde{\mathrm{VFE}}$ amalgamating the theories $\mathrm{SCFE}$ and $\widetilde{\mathrm{VFA}}$ introduced in [4] and [9], respectively. In characteristic zero, we also prove that $\widetilde{\mathrm{VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the valuation group

The dp-rank of abelian groups

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik–Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A/pA is infinite and for every prime p, there are only finitely many natural numbers n such that (p^n A)[p]/(p^{n + 1} A)[p] is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu