Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)

Abstract

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 ...

Zeta functions associated to admissible representations of compact 𝑝-adic Lie groups

“…Dirichlet generating functions have also been employed to study the distribution of finite-dimensional irreducible representations of finitely generated nilpotent groups. For such a group Γ one defines and studies the zeta function enumerating twist-isoclasses of irreducible representations of Γ; for instance, see [33,56,49,21]. Many theorems on representation zeta functions, e.g., regarding rationality, pole spectra, and functional equations, have analogues in the twist-isoclass setting; the Kirillov orbit method can be adjusted to take into account twist-isoclasses and thus yields a basic tool.…”

“…In recent years the subject of representation growth has advanced with a primary focus on zeta functions enumerating (i) irreducible representations of arithmetic lattices and compact p-adic Lie groups, (ii) twist-isoclasses of irreducible representations of finitely generated nilpotent groups; for instance, see [41,3,4,5,1] and [58,56,49,21,33,50], or the relevant surveys [38,59]. The aim of this paper is to introduce and study a new, more general zeta function that can be associated to any 'suitably tame' infinite-dimensional representation of a group.…”

Representation growth of compact linear groups

“…Such twist zeta functions have already been studied for nilpotent groups (cf. [33] and [15]). The idea is that the abscissa of ζ SL n (O) (s) is closely related to that of the twist zeta function of GL n (O).…”

The average size of the kernel of a matrix and orbits of linear groups

“…Remark (i)The author does not know if the conclusion of Theorem remains valid if is allowed to be a subgroup of . The proof of Theorem below combines basic ‐adic Lie theory and a powerful model‐theoretic result due to Cluckers [, Appendix A]. Both these ingredients are only available in characteristic 0.…”

“…Proof of Theorem Define an equivalence relation on via Our theorem will follow immediately from [, Theorem A.2] once we have established that is definable (definably in ) in the subanalytic language used in [, Appendix A]. By the preceding lemma, we may assume that .…”