We prove that the theory of the p-adics Q p admits elimination of imaginaries provided we add a sort for GL n (Q p )/GL n (Z p ) for each n. We also prove that the elimination of imaginaries is uniform in p. Using p-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p) of these formal zeta functions that extends to the subanalytic context.As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.2010 Mathematics Subject Classification. 03C60 (03C10 11M41 20E07 20C15). Key words and phrases. Elimination of imaginaries, invariant extensions of types, cell decompositions, rational zeta functions, subgroup zeta functions, representation zeta functions 2. For all M |= T , every e ∈ M eq , every A ⊆ M, every unary Ae-definable function f e and every non-empty A-definable set D, the following holds: if A ⊇ dcl eq (e) ∩ M, e ∈ dcl eq (A, f e (c)) for any c ∈ D, and tp(e/A) implies the type of e over Ac for any c ∈ D, then f e restricted to D is A-definable.Indeed, let e be imaginary. There exist c 1 , . . . , c n ∈ dom(M) such that e ∈ dcl eq (c 1 , . . . , c n ). Let A l = dcl eq (e, c 1 , . . . , c l ) ∩ M. We know that e ∈ dcl eq (A n ) and we want to show that e ∈ dcl eq (A 0 ) = dcl eq (dcl eq (e) ∩ M); i.e., e is interdefinable with a tuple of real elements. Let us proceed by reverse induction. Suppose e ∈ dcl eq (A l+1 ), let A = A l and let c = c l+1 . Then e ∈ dcl eq (A l+1 ) = dcl eq (dcl eq (e, c 1 , . . . , c l+1 ) ∩ M) = dcl eq (dcl eq (Aec) ∩ M). So we can find d = f (e, c) and e = h(d) for some A-definable functions f, h. By unary EI and since dcl eq (Ae) ∩ M = A, any Ae-definable subset of a dominant sort is already A-definable. Thus, by hypothesis, e = h(f (e, c ′ )) for any c ′ |= tp(c/A). Let D be an A-definable set with c ∈ D and such that e = h(f (e, c ′ )) for any c ′ ∈ D. Note also that, by unary EI again, for any c ∈ D, tp(c/A) implies tp(c/Ae) and thus tp(e/A) implies tp(e/Ac). It follows from hypothesis 2 that the map f e : x → f (e, x) restricted to D is A-definable and that e ∈ dcl eq (A) = dcl eq (A l ).Definition 2.4 We will say that a theory T eliminates imaginaries up to uniform finite imaginaries (EI/UFI) if for all M |= T and e ∈ M eq , there exists a tuple d ∈ M such that e ∈ acl eq (d) and d ∈ dcl eq (e).The theory T is said to eliminate finite imaginaries (EFI) if any e ∈ acl eq (∅) is interdefinable with a tuple from M.Let us now give a criterion for elimination of imaginaries ...
We prove field quantifier elimination for valued fields endowed with both an analytic structure and an automorphism that are σ-Henselian. From this result we can deduce various Ax-Kochen-Ersov type results with respect to completeness and the NIP property. The main example we are interested in is the field of Witt vectors on the algebraic closure of F p endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first order theory and that this theory is NIP.
We show that separably closed valued fields of finite imperfection degree (either with λ‐functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets to study stably dominated types in those structures. We show that separably closed valued fields of finite imperfection degree are metastable and that the space of stably dominated types is strict pro‐definable.
Let T be an NIP L-theory andT be an enrichment. We give a sufficient condition onT for the underlying L-type of any definable (respectively invariant) type over a model ofT to be definable (respectively invariant). These results are then applied to Scanlon's model completion of valued differential fields.Let T be a theory in a language L and consider an expansion T ⊆T in a languageL. In this paper, we wish to study how invariance and definability of types in T relate to invariance and definability of types inT . More precisely, let U ⊧T be a monster model and consider some typep ∈ S(U) which is invariant over some small M ⊧T . Then the reduct p ofp to L is of course invariant under the action of theL-automorphisms of U that fix M (which we will denote asL(M )-invariant), but there is, in general, no reason for it to be L(M )-invariant. Similarly, ifp isL(M )-definable, p might not be L(M )-definable. When T is stable, and ϕ(x; y) is an L-formula, ϕ-types are definable by Boolean combinations of instances of ϕ. It follows that ifp isL(M )-invariant then p is both L(M )invariant and L(M )-definable. Nevertheless, when T is only assumed to be NIP, then this is not always the case. For example one can take T to be the theory of dense linear orders andL = {⩽, P (x)} where P (x) is a new unary predicate naming a convex non-definable subset of the universe. Then there is a definable type inT lying at some extremity of this convex set whose reduct to L = {⩽} is not definable without the predicate. In the first section of this paper, we give a sufficient condition (in the case where T is NIP) to ensure that anyL(M )-invariant (resp. definable) L-type p is also L(M )invariant (resp. definable). The condition is that there exists a model M ofT whose reduct to L is uniformly stably embedded in every elementary extension of itself. In the case where T is o-minimal for example, this happens whenever the ordering on M is complete. The main technical tool developed in this first section is the notion of external separability (Definition (1.2)). Two sets X and Y are said to be externally separable if there exists * Partially supported by ValCoMo (ANR-13-BS01-0006)
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