On definable groups and D-groups in certain fields with a generic derivation

Abstract

Abstract We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$ , the model companion of an o-minimal ${\mathcal {L}}$ -theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007). We generalize Buium’s notion of an algebraic D-group to ${\mathcal {L}}$ -definable D-groups, namely $(G,s)$ , where G is an ${\mathcal {L}}$ -definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$ -definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$ for some ${\mathcal {L}}$ -definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$ ). We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$ .