2.1 Definable manifolds and their tangent bundles

Definition 2.2 For M an ${\mathcal {L}}$ -definable $C^1$ -manifold of dimension r given as above, we let the tangent space of M, $T(M)$ be the quotient of $\bigsqcup _{i=1}^n V_i\times K^r$ by the equivalence relation, denoted by $\sim _{T(M)}$ , given via the maps:

$$ \begin{align*}D\phi_{i,j}:V_{i,j}\times K^r\to V_{j,i}\times K^r\,\,;\,\, D\phi_{i,j}(c,u)=(\phi_{i,j}(c),D(\phi_{i,j})_c\cdot u).\end{align*} $$

We then write

$$ \begin{align*}T(M)=\bigsqcup_i V_i\times K^n\diagup \sim_{T(M)},\end{align*} $$

and denote (equivalence classes of) elements in $T(M)$ by $[a,u]$ , $a\in \bigsqcup _i V_i$ , $u\in K^r$ .

Note that if $M=U\subseteq K^r$ is a definable open set with the identity atlas, then ${T(M)=U\times K^r}$ .

The following are easy to verify.

Lemma 2.3 Assume that M and N are ${\mathcal {L}}$ -definable, $C^1$ -manifolds, given by atlases $(W_i,V_i,\phi _i)_{i\in I}$ and $(U_j,Z_j,\psi _j)_{j\in J}$ . If $f:M\to N$ is a $C^1$ -map (read through the charts), then there is a well-defined continuous map $Df:T(M)\to T(N)$ satisfying, whenever the elements are in the appropriate $V_i$ and $U_j$ ,

$$ \begin{align*}Dh([a,u])=[(f(a),D(\psi_j^{-1}\circ f\circ \phi_i)_a\cdot u)].\end{align*} $$

Summarizing, we have the following lemma.

3.1 The definition of $f^{\partial }$ on an open set

The following theorem of Fornasiero and Kaplan, which follows easily from their [Reference Fornasiero and Kaplan6, Lemma A.3], plays an important role here: in the Appendix, we prove the analogous result, Proposition A.1, for p-adically closed fields, and the proof could be modified to give an alternative proof in the o-minimal setting as well.

As a corollary, one obtains the following.

Definition 3.5 For $U\subseteq K^n$ open and $f:U\to K^r$ an ${\mathcal {L}}$ -definable $C^1$ -map (possibly over additional parameters), let

$$ \begin{align*}f^\partial(a)=\partial f(a)-(Df)_a\partial a.\end{align*} $$

Notice that if f is $\emptyset $ -definable, then $f^\partial (a)=0$ . For the following, see also [Reference Fornasiero and Kaplan6, Lemma 2.12].

Proof By Fact 3.4, we may write $f(x)=g(x,b)$ , for $b\in K^m$ which is ${\mathcal {L}}(\emptyset )$ -independent, and g which is a $C^1$ map, ${\mathcal {L}}(\emptyset )$ -definable. By the compatibility of $\partial $ ,

$$ \begin{align*}\partial f(a)=\partial g(a,b)=(Dg)_{(a,b)}(\partial a,\partial b)=\end{align*} $$

$$ \begin{align*}=(D_xg)_{(a,b)}\partial a+(D_yg)_{(a,b)}\partial b=(Df)_a\partial a+(D_y g)_{(a,b)}\partial b.\end{align*} $$

It follows that $f^\partial (a)=\partial f(a)-(Df)_a\partial a=(D_yg)_{(a,b)}\partial b,$ and since $b\in k$ , then so is $\partial b$ . Also, because g is a $C^1$ -function, $f^{\partial }$ is continuous.

Remark 3.7 When $p(x)=\sum _m a_m x^m$ is a polynomial over k, then $p^\partial (x)$ is a polynomial over k of the same degree:

$$ \begin{align*}p^\partial(x)=\sum_m \partial a_m x^m.\end{align*} $$

For $a\in K^n$ , we let $\nabla (a)=(a,\partial a)$ , and for $r\in \mathbb N$ , $\nabla ^r(a)=(a,\partial a,\ldots , \partial ^r a).$ We also need the following.

3.2 Prolongation of functions on open sets

Here and below, we make use of Marker’s account [Reference Marker14] of prolongations in the algebraic setting.

Definition 3.9 For $U\subseteq K^r$ open and $f:U\to K^n$ an ${\mathcal {L}}$ -definable $C^1$ -map, we let $\tau (f):U\times K^r\to K^n\times K^n$ be defined as

$$ \begin{align*}\tau(f)(a,u)=(f(a), (Df)_a\cdot u+f^{\partial}(a))=(f(a),(Df)_a\cdot (u-\partial a)+\partial f(a)).\end{align*} $$

Using Lemma 3.6 (the ${\mathcal {L}}$ -definability of $f^\partial $ ) and the ${\mathcal {L}}$ -definability of $Df$ , we have the following lemma.

Using the second equality in the definition of $\tau (f)$ and the chain rule for D, we immediately obtain the following lemma.

Lemma 3.11 If $f:U\to V$ and $h:V\to W$ are definable $C^1$ -functions on open sets, then

$$ \begin{align*}\tau(h\circ f)=\tau(h)\circ \tau(f).\end{align*} $$

3.3 The definition of $\tau (M)$ and $\tau (f)$ for definable manifolds

Definition 3.12 Assume that $M=\bigsqcup _i V_i/\sim _M$ is an ${\mathcal {L}}$ -definable $C^1$ -manifold of dimension n. Then the prolongation of M is defined as

$$ \begin{align*}\tau(M):=\bigsqcup_i V_i\times K^n/\sim_{\tau(M)},\end{align*} $$

where $(a_i,u)\sim _{\tau (M)}(a_j,v)$ if $\tau (\phi _{i,j})(a_i,u)=(a_j,v)$ .

By Lemma 3.10, $\tau (M)$ is an ${\mathcal {L}}$ -definable $C^0$ -manifold.

The following is easy to verify.

Lemma 3.13 Assume that $M=\bigsqcup _i V_i/\sim _M$ is an ${\mathcal {L}}$ -definable $C^1$ -manifold. Then

$$ \begin{align*}(a_i,u)\sim_{T(M)} (a_j,v) \Leftrightarrow (a_i,u+\partial a_i)\sim_{\tau(M)} (a_j,v+\partial a_j).\end{align*} $$

In particular, the map

$$ \begin{align*}\sigma_M:(a,u)\mapsto (a,u+\partial a)\end{align*} $$

induces a well-defined ${\mathcal {L}}_{\partial }$ -definable bijection over $M$ , between $T(M)$ and $\tau (M)$ .

Using the above lemma, we see that for $a\in M$ , the element $(a,\partial a)\in \tau (M)$ is well defined (e.g., as $\sigma _M(a,0)$ ). We thus have a well-defined map $\nabla :M\to \tau (M)$ , given in coordinates by $\nabla _M(a)=(a,\partial a)$ .

Definition 3.14 Assume that M and N are ${\mathcal {L}}$ -definable $C^1$ -manifolds, $f:M\to N$ an ${\mathcal {L}}$ -definable $C^1$ map. Then the prolongation of f, $\tau (f):\tau (M)\to \tau (N)$ , is defined by

$$ \begin{align*}\tau(f):=\sigma_N \circ Df\circ \sigma_M^{-1}.\end{align*} $$

The following is easy to verify.

Lemma 3.15 Assume that M and N as above are given via the atlases $\{(V_i,W_i,\phi _i)_{i\in I}\}$ and $\{(U_j,Z_j,\psi _j)\}$ , respectively, with $\dim M=r$ and $\dim N=n$ . If $f:M\to N$ is an ${\mathcal {L}}$ -definable $C^1$ -map, then, for $(a,u)\in V_i\times K^r$ , we have

$$ \begin{align*}\tau(f)([a,u])=[\tau(\psi_j^{-1}\circ f\circ \phi_i)(a,u)].\end{align*} $$

As a corollary, we have the following lemma.

4.1 Prolongation of ${\mathcal {L}}$ -definable groups, D-groups, and Nash D-groups

Let G be an ${\mathcal {L}}$ -definable group of dimension m. By [Reference Pillay16], it admits the structure of an ${\mathcal {L}}$ -definable $C^0$ -manifold. Since we shall be using the particular construction, in its $C^1$ -version, we repeat the details below for future use (see [Reference Pillay16, Lemmas 2.4 and Proposition 2.5]).

To be precise, [Reference Pillay16, Proposition 2.5] states the result in the $C^0$ category. However, as is commented in [Reference Pillay16, Remark 2.6], if the underlying o-minimal structure is the field $(\mathbb R,<,+,\cdot )$ , then one can obtain an analytic atlas for G, making it an analytic group. If we work, as we do here, in an o-minimal structure over an arbitrary real closed field, where definable functions are piecewise $C^1$ [Reference van den Dries23, Theorem 6.3.2], then exactly the same proof would yield the above $C^1$ -atlas, making G a $C^1$ -group. Moreover, every other definable $C^1$ -atlas on G which makes it into a $C^1$ group yields the same $C^1$ -structure, namely the identity map is a diffeomorphism of the two (this follows from the fact that definable functions are generically $C^1$ ). In addition, if G was definable over A, then since the notion of a $C^1$ -atlas is first-order, one can obtain a corresponding $C^1$ -atlas for G which is also defined over A.

From now on, one we endow every ${\mathcal {L}}$ -definable group G with its canonical $C^1$ -structure.

By purely categorical reasons, using Lemmas 2.5 and 3.17, we have (see [Reference Marker14, Section 2] for the same construction in algebraic groups) the following lemma.

Lemma 4.3 Let G be a definable group, endowed with its canonical $C^1$ -structure, and let $m:G\times G\to G$ be the group product. Then

$$ \begin{align*}\langle T(G);Dm\rangle \,\,\, \mbox{ and } \langle \tau(G); \tau(m)\rangle\end{align*} $$

are ${\mathcal {L}}$ -definable $C^0$ -groups (namely, topological groups with respect to an ${\mathcal {L}}$ -definable $C^0$ -atlas)), and the function $[a,u]\mapsto a$ is in both cases an ${\mathcal {L}}$ -definable group homomorphism from $T(G)$ and $\tau (G)$ onto G.

The map $a\mapsto [a,0]: G\to T(G)$ is an ${\mathcal {L}}$ -definable group section and $\nabla _G:G\to \tau (G)$ is an ${\mathcal {L}}_{\partial }$ -definable group section.

Our goal is to prove the following theorem.

Theorem 4.6 Let $T_\partial $ be the model companion of a complete, model complete, o-minimal theory T, with a T-compatible derivation $\partial $ . Assume that $\Gamma $ is an ${\mathcal {L}}_{\partial }$ -definable group of finite ${\mathcal {L}}_\partial $ -dimension. Then there exists an ${\mathcal {L}}$ -definable D-group $(G,s)$ and an ${\mathcal {L}}_\partial $ -definable group embedding $\Gamma \to G$ whose image is

$$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla_G(g)\}.\end{align*} $$

We first recall our result [Reference Peterzil, Pillay and Point15, Theorem 6.8]. We shall be using the following version.

Note that as a corollary of above we may obtain $X^{\prime }_1,\ldots , X^{\prime }_k\subseteq G$ pairwise disjoint, all of the same dimension as $\dim G$ , satisfying (ii), but instead of $G=\bigcup X^{\prime }_i$ we have $\dim (G\setminus \bigcup X_i')<\dim G$ . Indeed, we replace the original $X_i$ by $X_i'=X_i\setminus \bigcup _{j<i} X_j$ and remove all $X_i'$ whose dimension is smaller than $\dim G$ .

In fact, we shall prove a more precise version of Theorem 4.6.

We first prove a general fact about groups in geometric structures.

Proof We let

$$ \begin{align*}S_1=\{s\in S: \mbox{ the set } \{t\in S:s\cdot t\in S\, \& \,t\cdot s\in S\} \mbox { is large in } S\}.\end{align*} $$

By definability of dimension in geometric structures, $S_1$ is definable. By our assumptions, $S_1$ contains all generic elements of S; thus, by our assumptions, $S_0:=S_1\cap S_1^{-1}$ is also large in S. We claim that $S_0\cdot S_0$ is a subgroup of G.

We need to prove that for every $a,b,c,d\in S_0$ , we have $a b c^{-1} d^{-1}\in S_0\cdot S_0$ . We fix $g\in S$ generic over $a,b,c,d$ , and consider

$$ \begin{align*}abc^{-1}d^{-1}=(abg)(g^{-1}c^{-1}d^{-1}).\end{align*} $$

Since $b\in S_0$ , the set $\{t\in S: bt\in S\}$ is large in S, defined over b, and therefore contains g. Thus, $bg$ is in S and by our choice, it is in fact generic in S over $a,c,d$ , so in particular belongs to $S_0$ . Thus, $a(bg)\in S$ , and again generic there over $c,d$ , so belongs to $S_0$ . Similarly, $g^{-1}c^{-1}d^{-1}\in S_0$ , so $abc^{-1}d^{-1}\in S_0\cdot S_0$ . Let $H:=S_0\cdot S_0$ .

To see that $S_0$ is a large subset of H, we fix $g\in S_0$ and $h\in S$ generic over g (so $h\in S_0$ ). Then $g h, gh^{-1}\in S$ and generic there so in $S_0$ . It follows that $g\in S_0\cdot S_0$ and $h\in S_0^{-1}\cdot S_0=S_0\cdot S_0$ .

Hence, $S_0\subseteq H$ and every generic $h\in S$ over g is in H, so $S_0$ is large in H.

We are now ready to prove Theorem 4.6.

We first apply Theorem 4.7 and the subsequent corollary and deduce the existence of pairwise disjoint ${\mathcal {L}}$ -definable $X_1,\ldots , X_r\subseteq G\subseteq K^n$ , each of dimension equal to $m=\dim G$ , such that $X=\bigsqcup _j X_j$ is a large subset of G and on each j, we have an ${\mathcal {L}}$ -definable $s_j:X_j\to K^n$ , such that for g generic in $X_j$ , we have $g\in \Gamma \Leftrightarrow \partial g=s_j(g)$ . We let $s:X \to K^n$ be the union of the $s_j$ ’s.

Next, we apply Fact 4.1, and fix an ${\mathcal {L}}$ -definable large $W\subseteq G$ , $V\subseteq K^m$ open, $\sigma :V\to W$ a homeomorphism, and $g_1,\ldots , g_k\in G$ , such that the maps $\phi _i: V\to g_iW: x\mapsto g_i\sigma (x)$ endow G with a definable $C^1$ -manifold structure, and make G into a $C^1$ -group.

By intersecting W with the relative interior of X in G, we may assume that $W=X$ .

Proof Every $a\in V$ is ${\mathcal {L}}$ -interdefinable with $\sigma (a)$ , so by Lemma 3.8, $\nabla (a)$ and $\nabla (\sigma (a))$ are ${\mathcal {L}}$ -interdefinable over k. By compactness, there exists an ${\mathcal {L}}$ -definable (partial) bijection $h:W\times K^n\to V\times K^m$ , such that for each generic $a\in V$ , $h(\nabla (\sigma a))=\nabla (a)$ . Let

$$ \begin{align*}\hat s(a)=\pi_2(h(\sigma(a)),s(\sigma(a))),\end{align*} $$

where $\pi _2:V\times K^n\to K^n$ is the projection onto the second coordinate.

Now, if $s(\sigma (a))=\partial (a)$ , then $(\sigma (a),s(\sigma (a)))=\nabla (\sigma (a))$ , so

$$ \begin{align*}\hat s (a)=\pi_2( h(\nabla \sigma(a)))=\pi_2(\nabla(a))=\partial a.\end{align*} $$

The converse follows from the invertibility of h.

Going back to G, we now endow G with a finite $C^1$ -atlas $(V_i,g_iW,\phi _i)_{i\in I}$ , where $V_i=V$ for all i, and identify G with $\bigsqcup V_i/\sim _M$ . We also identify $\Gamma $ with the group $\bigsqcup \phi _i^{-1}(\Gamma \cap g_iW)/\sim _M$ . Notice that each $g_iW/\sim _M$ is large in G, and by Claim 4.9, there is an ${\mathcal {L}}$ -definable $\hat s:V\to K^m$ such that, for generic $a\in V$ , $\hat s(a) =\partial a$ if and only if $s(\sigma (a))=\partial \sigma (a)$ . Thus, by our assumption, for every generic $g\in G$ , $g\in \Gamma \Leftrightarrow \hat s(g)=\partial g$ . For simplicity, from now on, we use s instead of $\hat s$ and let $X=dom (s)$ , an ${\mathcal {L}}$ -definable large subset of G.

Consider the ${\mathcal {L}}$ -definable $C^1$ -group $\tau (G)$ as before, and the associated ${\mathcal {L}}$ -definable homomorphism $\pi :\tau (G)\to G$ , together with an ${\mathcal {L}}_\partial $ -definable group section $\nabla _G:G\to \tau (G)$ . The map s can be replaced by $x\mapsto (x,s(x))$ , so we may think of it as a function from X into $\tau (X)=X\times K^m$ with $\pi \circ s(x)=x$ .

In addition, we still have for every generic $g\in X$ , $g\in \Gamma \Leftrightarrow s(g)=\nabla _G(g)$ . By our assumptions, every generic ${\mathcal {L}}$ -type of X contains an element of $\Gamma $ ; hence, the ${\mathcal {L}}$ -definable set $X_0=\{x\in X:s(x)\in \tau (X)\}$ is large in X, so without loss of generality, $X=X_0$ . Let S be the graph of $s|X_0$ .

We claim that S satisfies the assumptions of Proposition 4.8: indeed, assume that $(a,b)$ is generic in $S^2$ . Namely, $a=(g,s(g))$ and $b=(h,s(h))$ , for $(g,h)$ generic in $X\times X$ . We need to prove that $a b\in S$ .

By [Reference Peterzil, Pillay and Point15, Lemma 6.7], applied to the function $(s,s):X\times X\to \tau (X\times X)$ , there exists $(x,y)\in X\times X$ , realizing the same ${\mathcal {L}}$ -type as $(g,h)$ such that $\nabla _{G\times G}(x,y)=(s(x),s(y))$ . But then, by our assumptions, $(x,y)\in \Gamma \times \Gamma $ , so $xy\in \Gamma $ . Because $xy$ is still ${\mathcal {L}}$ -generic in G, we have $xy\in X$ . Thus, we have

$$ \begin{align*}s(xy)=\nabla_G(xy)=\nabla_G(x) \nabla_G(y)=s(x) s(y)\end{align*} $$

(where the middle equality follows from the fact that $\nabla _G$ is a group homomorphism). Since $tp_{{\mathcal {L}}}(x,y)=tp_{{\mathcal {L}}}(g,h)$ , we also have $s(gh)=s(g)s(h)$ , hence $a b=(gh,s(gh))$ , is in S.

We similarly prove that for a generic in S, we have $a^{-1}\in S$ ; thus, S satisfies, indeed, the assumption of Proposition 4.8.

Hence, there exists an ${\mathcal {L}}$ -definable $S_0\subseteq S$ , such that $S_0$ is a large subset of the group $H=S_0\cdot S_0$ . Since $S_0$ is large in H, for every generic $(g,s(g))\in H$ , we have $\pi ^{-1}(g)\cap H$ is a singleton, which implies that $ker(\pi |H)=\{1\}$ , and hence H is the graph of a function. Also, since the group $\pi (H)$ is large in G, it necessarily equals to G.

We therefore found an ${\mathcal {L}}$ -definable group-section $\hat s:G\to \tau (G)$ , making $(G,\hat s)$ into a D-group. In addition, $x\in \Gamma \Leftrightarrow \hat s(x)=\nabla _G(x)$ , for all x generic in G.

It is left to see that

$$ \begin{align*}\Gamma=(G,\hat s)^\partial=\{x\in G:\hat s(x)=\nabla_G(x)\}.\end{align*} $$

Let $X_0=\pi (S_0)$ and $\Gamma _0=X_0\cap \Gamma $ . By the definition of S, $\Gamma =\{x\in \pi (S):S(x)=\nabla _G(x)\}$ , so $\Gamma _0=\{x\in X_0:\hat s(x)=\nabla _G(x)\}.$ We claim that $\Gamma _0\cdot \Gamma _0=\Gamma $ .

Indeed, let $\gamma \in \Gamma $ , and pick g generic in $X_0$ over $\gamma $ . By the geometric axioms, there exists $\gamma _1\equiv _{{\mathcal {L}}(\gamma )} g$ such that $\hat s(\gamma _1)=\nabla _G(\gamma _1)$ , namely $\gamma _1\in \Gamma _0$ . It follows that $\gamma \cdot \gamma _1^{-1}$ is ${\mathcal {L}}$ -generic in G over $\gamma $ and hence in $X_0$ , namely in $\Gamma _0$ . Hence, $\gamma \in \Gamma _0\cdot \Gamma _0$ .

It follows that for all $\gamma \in \Gamma $ , we have $\hat s(\gamma )=\nabla _G(\gamma )$ . To see the converse, assume that $\hat s(x)=\nabla _G(x)$ , and choose $\gamma \in \Gamma _0$ generic over x. We then have $\hat s(\gamma )=\nabla _G(\gamma )$ , and $x\cdot \gamma $ generic in $X_0$ . Because $\hat s$ is a homomorphism,

$$ \begin{align*}\hat s(x\gamma)=\hat s(x)\hat s(\gamma)=\nabla_G(x)\nabla_G(\gamma)=\nabla_G(x\gamma).\end{align*} $$

It follows that $x\gamma \in \Gamma $ and hence so is x. This ends the proof of Theorem 4.6.

4.2 The case of p-adically closed fields

Let K be a p-adically closed field, namely a field which is elementarily equivalent to a finite extension of $\mathbb Q_p$ . The field admits a definable valuation, which we may add to the field language and call this language ${\mathcal {L}}$ .

We shall use multiplicative notation for the valuation map $|\,|:K\to \{0\}\cup vK$ . Namely,

$$ \begin{align*}|0|=0<vK\,\, , |\,|:K^*\to (vK,\cdot)\, \mbox{a group homomorphism}\end{align*} $$

and

$$ \begin{align*}\forall x,y\in K \,\,|x+y|\leq \max \{|x|,|y|\}.\end{align*} $$

For $a=(a_1,\ldots ,a_n)\in K^n$ , we write $\| h\|=\max \{|a_i|:i=1,\ldots ,n\}.$ Since K is a geometric structure, we use the $acl$ -dimension below.

Definition 4.10 For $U\subseteq K^m$ open, a map $f:U\to K^n$ is called differentiable at $a\in U$ if there exists a K-linear map $T:K^m\to K^n$ such that for all $\epsilon \in vK$ there is $\delta \in vK$ , such that for all $h\in K^m$ , if $\|h\|<\delta $ , then

$$ \begin{align*}\|f(a+h)-f(a)+T(h)\|<\epsilon\|h\|.\end{align*} $$

The linear map T can be identified with $Df_a$ the $n\times m$ matrix of partial derivatives of f. We identify $M_{n\times m}(K)$ with $K^{n\cdot m}$ . The function f is called continuously differentiable on U, or $C^1$ , if it is differentiable on U and the map $x\mapsto Df_a$ is continuous.

Differentiable maps satisfy the chain rule, by the usual proof (see, for example, [Reference Schneider19, Remark 4.1] for a proof in $\mathbb Q_p$ ).

Toward our main result, we first note that p-adically closed fields satisfy the assumptions in Remark 4.2: indeed, these are geometric fields with a definable Hausdorff, non-discrete topology. Let us see that they admit $C^1$ -cell decomposition (we could not find a precise reference for that in the literature).

First, one can read-off analytic cell decomposition in finite extensions of $\mathbb Q_p$ from Scowcroft and van den Dries [Reference van den Dries and Scowcroft24, Sections 4 and 5]. More explicitly, the result is stated in [Reference Cluckers, Comte and Loeser5, Theorem 3.3] (as mentioned there, the theorem works for the Macintyre language, as well as the subanalytic one). Since being $C^1$ is a definable property, one may conclude a $C^1$ -cell decomposition for definable sets in arbitrary elementarily equivalent structures, i.e., p-adically closed fields.

Thus, as we commented in Remark 4.2, the result of Fact 4.1 holds in this setting as well and in particular, every ${\mathcal {L}}$ -definable group admits an ${\mathcal {L}}$ -definable $C^1$ -manifold which makes it into a $C^1$ -group.

We now endow K with a derivation, denoted by $\partial $ . By Tressl’s work (see [Reference Tressl21, Theorem 7.2]), the theory of p-adically closed fields with a derivation has a model companion $T_{\partial }$ . In our one derivation case, (Tressl deals with several commuting derivations), one can axiomatize $T_{\partial }$ with the following geometric axioms (see, for instance, [Reference Peterzil, Pillay and Point15, Fact 5.7(ii)]): whenever $(V,s)$ is an irreducible D-variety over K with a smooth K-point and U is a Zariski open subset of V defined over K, then there is $a\in U(K)$ such that $(a,s(a))=\nabla (a)$ . (Recall that a D-variety $(V,s)$ defined over K is a K-variety V equipped with a rational section s defined over K from V to $T(V)$ [Reference Peterzil, Pillay and Point15, Definition 2.4].)

Now, exactly as in the work of Fornasiero and Kaplan for real closed fields [Reference Fornasiero and Kaplan6, Lemmas 2.4 and 2.7 and Proposition 2.8], every nontrivial derivation is compatible with the theory of p-adically closed fields, namely compatible with every ${\mathcal {L}}(\emptyset )$ -definable $C^1$ map, as in Definition 3.1. We briefly review the details.

As in [Reference Fornasiero and Kaplan6, Lemma 2.4], it is enough to verify compatibility of $\emptyset $ -definable $C^1$ -functions in neighborhoods of $acl_{\mathcal {L}}$ -independent points in $K^n$ (we use here the fact that $acl_{\mathcal {L}}=dcl_{\mathcal {L}}$ ). By quantifier elimination, the graph of every ${\mathcal {L}}(\emptyset )$ -definable function $g{\kern-1.2pt}:{\kern-1.2pt}U{\kern-1.2pt}\to{\kern-1.2pt} K$ , for $U{\kern-1.2pt}\subseteq{\kern-1.2pt} K^n$ open, is given implicitly by $\{(x,{\kern-1pt}y){\kern-1.2pt}:{\kern-1.2pt}x{\kern-1.2pt}\in{\kern-1.2pt} U\& f(x,{\kern-1pt}y){\kern-1.2pt}={\kern-1.2pt}0\}$ , for $f(x,y)$ an irreducible polynomial over $\mathbb Z$ . Now, if $a\in U$ is ${\mathcal {L}}$ -generic over $\emptyset $ , then $(\partial f/\partial y)(a,g(a))\neq 0$ and as in [Reference Fornasiero and Kaplan6, Lemma 2.7], $\partial $ is compatible with g.

In order to develop the rest of the theory as in the o-minimal case, we prove in the Appendix (see Proposition A.1) that definable functions in p-adically closed fields satisfy the analogue of Fact 3.4.

Now, the category of K-differentiable manifolds M and their associated functors T and $\tau $ can be developed identically to Sections 1 and 2. This allows us to associate to every definable group G the definable groups $T(G)$ and $\tau (G)$ , such that the natural projections onto G are group homomorphisms. If G is a $C^1$ -group, then $T(G)$ and $\tau (G)$ are $C^0$ -groups.

By a p-adic D-group, we mean a pair $(G,s)$ where G is an ${\mathcal {L}}$ -definable $C^1$ -group and $s:G\to \tau (G)$ an ${\mathcal {L}}$ -definable homomorphic section (i.e., $\pi \circ s=id$ ).

As before, we define in models of $T_{\partial }$ , given a D-group $(G,s)$ ,

$$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla_G(g)\}.\end{align*} $$

In order to prove our main theorem in the p-adically closed field, we let T be the theory of p-adically closed fields and let $\Gamma $ be a finite-dimensional ${\mathcal {L}}_\partial $ -definable group in $K\models T_\partial $ . Since p-adically closed fields are large geometric fields, we may apply [Reference Peterzil, Pillay and Point15, Theorem 5.11] to conclude that Theorem 4.7 holds in this setting as well. Namely, $\Gamma $ embeds into an ${\mathcal {L}}$ -definable group G, with the additional ${\mathcal {L}}$ -definable $X_i$ ’s as in the theorem. Now we repeat word-for-word the proof in the o-minimal setting (see also Remark 4.2) to conclude the following theorem.

4.3 The case of pseudo-finite fields

Let ${\mathcal {L}}$ be the language of rings, and let $C=(c_{i,n})_{n\in \mathbb N, i<n}$ be an infinite countable set of new constants. Let T be the ${\mathcal {L}}(C)$ -theory of pseudo-finite fields of characteristic $0$ , namely the theory of pseudo-algebraically closed fields plus the scheme of axioms saying, for every $n\in \mathbb N$ , that there is a unique extension of degree n, and that the polynomial

$$ \begin{align*}X^n+c_{n-1,n}X^{n-1}+\cdots +c_{0,n}\end{align*} $$

is irreducible.

Since T is a model-complete theory of large fields, one can apply the Tressl machinery and so the theory of differential expansions of models of T has a model-companion [Reference Tressl21, Corollary 8.4], which has been axiomatized [Reference Tressl21, Theorem 7.2] (in case of expansions by a single derivation, one obtains a geometric axiomatization [Reference Brouette, Cousins, Pillay and Point3, Lemma 1.6]). Recall that since T has almost q.e. (see [Reference Brouette, Cousins, Pillay and Point3, Remark 1.4(2)]), the theory $T_\partial $ does too [Reference Brouette, Cousins, Pillay and Point3, Definition 1.5 and Lemma 2.3], [Reference Tressl21, Theorem 7.2(iii)].

Let ${\mathcal U}$ be our sufficiently saturated model of $T_\partial $ , a differential expansion of a pseudo-finite field, and let $\bar {\mathcal U}\supseteq \mathcal U$ be a saturated model of DCF $_0$ extending it. We work over a small submodel $(K,\partial )\models T_\partial $ .

We briefly review the construction of the algebraic prolongation $\tau (V)\subseteq \bar {\mathcal U}^n\times \bar {\mathcal U}^n$ of an irreducible algebraic variety (see [Reference Marker14] for details).

Assume that the ideal $I(V)$ is generated by polynomials $p_1,\ldots , p_m$ , over K, and let $P:\bar {\mathcal U}^n\to \bar {\mathcal U}^m$ be the corresponding polynomial map $P(x)=(p_1(x),\ldots , p_m(x))$ . The definition of $DP$ and $\tau (P)$ is defined as before using the formal derivative of polynomials (see also Remark 3.7). Then

$$ \begin{align*}T(V)=\{(x,u)\in \bar{\mathcal U}^{2n}: P(x)=0\,\& \,(DP)_x\cdot u=0\}\end{align*} $$

and

$$ \begin{align*}\tau(V)=\{(a,u)\in \bar{\mathcal U}^{2n}: a\in V\, \&\, \tau(P)(a,u)=0\}.\end{align*} $$

Both are algebraic varieties over K. For $a\in V(\bar {\mathcal U})$ , $a\mapsto \partial a$ is a section of ${\pi :\tau (V)\to V}$ , and we have

$$ \begin{align*}\tau(V)=\{(a,u)\in \bar{\mathcal U}^{2n}: a\in V\, \&\, u-\partial a\in T(V)_a\}.\end{align*} $$

So, $\tau (V)_a$ is an affine translate of the vector space $T(V)_a\subseteq \bar {\mathcal U}^n$ . In particular, $\dim (\tau (V)_a)=\dim V$ .

As described in [Reference Marker14], the above constructions of $T(V)$ and $\tau (V)$ can be extended to abstract, not necessarily affine, algebraic varieties (which are covered by finitely many affine algebraic varieties). Furthermore, if H is an algebraic group, then $T(H)$ and $\tau (H)$ are algebraic groups with the property that the map $\nabla _{H}$ is now a group morphism [Reference Marker14, Section 2].

Our goal is to prove the following theorem.

By “ $\Gamma $ and $(H,s)^\partial (K)$ are virtually isogenous,” we mean the following: there exist an ${\mathcal {L}}_{\partial }$ -definable subgroup $\Gamma _0\subseteq \Gamma $ of finite index and an ${\mathcal {L}}_{\partial }$ -definable homomorphism $\sigma :\Gamma _0\to H$ with finite kernel, whose image has finite index in the K-points of $(H,s)^\partial $ .

We first need the following lemma.

Proof This basically follows from the fact that the projection from W to $W_1,W_2$ is generically étale. However, for completeness, we include a proof. By dimension considerations, the affine space $\tau (W)_{(a,b)}$ projects onto both $\tau (W_{1})_{a}$ and $\tau (W_{2})_{b}$ .

Fix some coordinate $a_{1}$ of the tuple a. By our interalgebraicity assumption, $a_{1}$ is in the field-theoretic algebraic closure of $K(b)$ . Let $q(x)$ be the minimal monic polynomial of $a_{1}$ over $K(b)$ . $q(x) = x^{n} + f_{n-1}(b)x^{n-1} + \cdots +f_{1}(b)x + f_{0}(b)$ , where the $f_{i}(b)$ are K-rational functions of the tuple b. After getting rid of denominators, we can rewrite $q(x)$ as $q_n(b)x^{n} + q_{n-1}(b)x^{n-1} + \cdots + q_{1}(b) x + q_{0}(b)$ where the $q_{i}$ are polynomials over K.

Hence, $r(x,y)$ : $q_n(y)x^{n} + q_{n-1}(y)x^{n-1} + \cdots + q_{0}(y)$ is a polynomial (over K) in $I_{K}(W)$ .

Hence, $(\partial r/\partial x)(a_{1}, b)(u_{1}) + \sum _{j}(\partial r/\partial y_{j})(a_{1},b)(v_{j}) + r^{\partial }(a_{1},b) = 0$ for $(u_{1},...,v_{1},...,v_{j},...)\in \tau (W)_{a,b}$ .

By the minimality of $r(x,b)$ , we have $\partial r/\partial x(a_1,b)\neq 0$ , and hence

$$ \begin{align*}u_{1} = \left(\sum_{j}(\partial r/\partial y_{j})(a_{1},b)(v_{j}) + r^{\partial}(a_{1},b))/ (\partial r/\partial x)(a_{1}, b)\right).\end{align*} $$

Thus, $u_1\in dcl(K,a,b,v)$ . We similarly prove that u and v are inter-definable over $K(a,b)$ . Since $\tau (W)_{(a,b)}$ is a translate of a linear space, containing $(u,v)$ whose dimension equals $\dim W_1=\dim W_2$ , it must be the graph of a bijection.

We now return to the proof of Theorem 4.13.

By [Reference Peterzil, Pillay and Point15, Theorem 5.11], there exist an ${\mathcal {L}}$ -definable group G and a definable group embedding $:\sigma :\Gamma \to G$ . Furthermore, every generic ${\mathcal {L}}$ -type of G is realized by an element in $\sigma (\Gamma )$ and, in addition, there is a covering of G by finitely many ${\mathcal {L}}(K)$ -definable sets $X_i$ , $i=1,\ldots , m$ , and for each $X_i$ , there is a K-rational function $s_i:X_i\to \mathcal U^n$ such that for every $a\in X_i(\mathcal U)$ which is ${\mathcal {L}}$ -generic in $X_i$ over K, we have

$$ \begin{align*}a\in \sigma(\Gamma) \Leftrightarrow \partial(a)=s_i(a).\end{align*} $$

For simplicity, we assume now that $\sigma =id$ , so $\Gamma \subseteq G$ .

We may take each $X_i$ to be Zariski dense in a K-variety $V_i$ . We are only interested in those $V_i$ whose Zariski dimension is maximal, call it d, so in particular, every algebraic type in $V_i$ over K, of dimension d, is realized in $X_i$ in $\mathcal U$ , so by the axioms also realized by some $a\in X_i(\mathcal U)$ with $\partial a=s_i(a)$ , and hence, by the above, also realized in $\Gamma $ . Finally, each $X_i$ can be taken to be the $\mathcal U$ -points of $W_i=Reg(V_i):=V_i\setminus Sing(V_i)$ , namely the $\mathcal U$ -points of a smooth quasi affine K-variety.

We now apply [Reference Hrushovski and Pillay9, Theorem C] in the structure $\mathcal U$ : there exist a connected algebraic group H over K, ${\mathcal {L}}$ -definable subgroups of finite index, $G_0\subseteq G$ and $H_0\subseteq H(\mathcal U)$ , and an ${\mathcal {L}}$ -definable surjective homomorphism $f:G_0\to H_0$ whose kernel is finite, all defined over K. The group H, as an algebraic group over K, has an associated K-algebraic group $\tau (H)$ . Notice that $acl_{{\mathcal {L}}}$ equals the field $acl$ and we have $\dim H=d$ .

Let $\Gamma _0=\Gamma \cap G_0$ , a subgroup of $\Gamma $ of finite index. Since $f(G_0)$ is Zariski dense in H, so is $f(\Gamma _0)$ . As we shall now see, we can endow H with the structure of a D-group, $(H,s)$ such that

$$ \begin{align*}f(\Gamma_0)=\{h\in H_0:s(h)=\nabla_H (h)\}.\end{align*} $$

Proof We first prove the result for $g\in \Gamma _0$ such that $tr.deg(g/K)=d$ .

Since $tr.deg(g/K)=d$ , there exists a K-algebraic quasi-affine variety $W_i$ as above such that g is generic in $W_i$ over K. The ${\mathcal {L}}$ -definable function f takes values in H, and because $acl_{{\mathcal {L}}}$ is the same as the field $acl$ , there exists an algebraic correspondence $C_i\subseteq W_i\times H$ over K, such that $(g,f(g))$ is field-generic in $C_i$ . It follows that $(\nabla _{W_i}(g),\nabla _H f(g))\in \tau (C_i)\subseteq \tau (W_i)\times \tau (H)$ .

By Lemma 4.14, $\tau (C_i)_{(g,f(g))}$ induces an (algebraic) bijection over K between $\tau (W_i)_g$ and $\tau (H)_{f(g)}$ . In particular, $\nabla _{W_i}(g)$ and $\nabla _H f(g)$ are interalgebraic over K. By our construction, $\nabla _{W_i}(g)=s_i(g)$ ; hence, g and $\nabla _H f(g)$ are interalgebraic over K (notice that g is algebraic over $\nabla _{W_i}(g)$ ). Hence, $tr.deg(\nabla _H f(g)/K)= tr.deg(g/K)=d$ .

Assume now that g is an arbitrary element of $\Gamma _0$ , and let $h\in \Gamma _0$ be such that $tr.deg(h/K,g, \nabla _Hf(g))=d$ .

Since $tr.deg(hg/K)=d$ and $hg\in \Gamma _0$ , it follows from the above that

$$ \begin{align*}d=tr.deg(\nabla_{H}f(hg)/K))=tr.deg(\nabla_H f(h)\cdot \nabla_H f(g)/K),\end{align*} $$

and therefore

$$ \begin{align*}tr.deg(\nabla_Hf(h) \cdot \nabla_H f(g)/\nabla_H f(h), K)\leq d.\end{align*} $$

The elements $\nabla _Hf(h)\cdot \nabla _H f(g)$ and $\nabla _H f(g)$ are interalgebraic over K and $\nabla _Hf(h)$ , and thus $tr.deg(\nabla _H f(g)/\nabla _H f(h),K)\leq d$ .

We know that h and $\nabla _{X_i}(h)$ (and hence also $\nabla _H f(h)$ ) are interalgebraic over K (as witnessed by $s_i$ ), and as h and $\nabla _H(f(g))$ are independent over K, it follows that $\nabla _H f(h)$ and $\nabla _H f(g)$ are independent over K. Therefore,

$$ \begin{align*}tr.deg(\nabla_H f(g)/K)=tr.deg(\nabla_Hf(g)/\nabla_H f(h),K)\leq d.\\[-35pt]\end{align*} $$

We now consider the subgroup $\nabla _H f(\Gamma _0)$ of $\tau (H)$ and let $S\subseteq \tau (H)$ be its Zariski closure, an algebraic subgroup of $\tau (H)$ . By the claim above, $\dim (S)\leq \dim H$ , but since S contains $\nabla _H f(h)$ for ${\mathcal {L}}$ -generic $h\in \Gamma _0$ , we have $\dim (S)=\dim (H)$ . Consider the projection $\pi :\tau (H)\to H$ , a group homomorphism, and its restriction to S. Since H is connected, we have $\pi (S)=H$ , and hence $ker(\pi )\cap S$ is a finite subgroup of $\tau (H)_e$ . However, $\tau (H)_e=T(H)_e$ is a vector space over K, a field of characteristic $0$ , thus torsion-free. Hence, $ker(\pi )\cap S$ is trivial, so $\pi :S\to H$ is a group isomorphism. It follows that S can be viewed as a group section $s:H\to \tau (H)$ . Since S is the Zariski closure of $\nabla _H(f(\Gamma _0))$ , we have for every $g\in \Gamma _0$ , $\nabla _H(f(g))=s(f(g))$ .

Recall that $H_0=f(G_0)$ is an ${\mathcal {L}}$ -definable subgroup of finite index of $H(\mathcal U)$ .

Proof We only need to prove the $\supseteq $ inclusion.

We first prove that for every $h\in (H_0,s)^\partial $ , if $tr.deg(h/K)=d$ , then $h\in f(\Gamma _0)$ . Indeed, since $h\in H_0$ is ${\mathcal {L}}$ -generic in H over K, there exists $g\in G_0$ , necessarily ${\mathcal {L}}$ -generic in G over K, such that $h=f(g)$ . By our assumptions, there exists $g'\in \Gamma $ , such that $g'$ and g realize the same ${\mathcal {L}}$ -type over K. Thus, $g'$ is in $G_0$ so also in $\Gamma _0$ . In addition, $f(g')$ and $f(g)=h$ must realize the same ${\mathcal {L}}$ -type over K. Because S is the Zariski closure of $\nabla _H f(\Gamma _0)$ , it follows that $f(g')\in (H_0,s)^\partial $ .

Since $\nabla _H(h)=s(h)$ and $\nabla _H(f(g'))=s(f(g'))$ , it follows that for every $n\in \mathbb N$ , there is an ${\mathcal {L}}(K)$ -definable function $s_n$ such that

$$ \begin{align*}\nabla^n_H(h)=s_n(h)\, ,\,\nabla^n_H(f(g'))=s_n(f(g'))\end{align*} $$

(recall that $\nabla _H^n(g)=(g,\partial g,\ldots ,\partial ^n g)$ ). Because h and $f(g')$ realize the same ${\mathcal {L}}$ -type over K, we may conclude that for every n, $\nabla _H^n(h)$ and $\nabla _H^n(f(g')))$ realize the same ${\mathcal {L}}$ -type over K, and therefore

$$ \begin{align*}{\mathrm{tp}}_{{\mathcal{L}}(K)}(\nabla_H^{\infty}(h))={\mathrm{tp}}_{{\mathcal{L}}(K)} (\nabla_H^{\infty}(f(g'))).\end{align*} $$

( $\nabla _H^\infty (g)=(g,\partial g,\ldots , \partial ^n g,\ldots ).$ )

By [Reference Tressl21, 7.2(iii)], every ${\mathcal {L}}_{\partial }(K)$ -formula is equivalent in $T_{\partial }$ to a Boolean combination of formulas of the form $(x,\partial x,\ldots ,\partial ^n x)\in Y$ , where Y is an ${\mathcal {L}}(K)$ definable set. Thus, it follows from the above that h and $f(g')$ realize the same ${\mathcal {L}}_{\partial }$ -type over K, and therefore $h\in f(\Gamma _0)$ , as needed.

This proves that every $h\in (H_0,s)^\partial $ with $tr.deg(h/K)=d$ belongs to $f(\Gamma _0)$ . However, every $h\in (H_0,s)^\partial $ can be written as $h=h_1h_2$ , with $h_1,h_2\in (H_0,s)^\partial $ and $tr.deg(h_1/K)=tr.deg(h_2/K)=d$ . Indeed, pick $h_1\in (H_0,s)^\partial $ with $tr.deg(h_1/hK)=d$ and $h_2=h_1^{-1}h$ ). Thus, every $h\in (H_0,s)^\partial $ belongs to $f(\Gamma _0)$ . This ends the proof of Theorem 4.13.