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

We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory T_∂, the model companion of an o-minimal L-theory T expanded by a generic derivation ∂ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007). We generalize Buium's notion of an algebraic D-group to L-definable D-groups, namely (G,s), where G is an L-definable group in a model of T, and s: G → _T(G) is an L-definable group section. Our main theorem says that every definable group of finite dimension in a model of T_∂ is definably isomorphic to a group of the form (Formula presented) for some L-definable D-group (G,s) (where ▿(g) = (g,∂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.