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Definable Equivalence Relations and Zeta Functions of Groups

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Definable Equivalence Relations and Zeta Functions of Groups Definable equivalence relations and zeta functions of groups Ehud Hrushovski, Ben Martin and Silvain Rideau, with an appendix by Raf Cluckers September 1, 2017 In memory of Fritz Grunewald. Abstract We prove that the theory of the p-adics Qp admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp) 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 func- tions obtained by counting twist isomorphism classes of irreducible representa- tions of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup arXiv:math/0701011v5 [math.LO] 31 Dec 2017 zeta functions of finitely generated nilpotent groups. 1 Introduction 2010 Mathematics Subject Classification. 03C60 (03C10 11M41 20E07 20C15). Key words and phrases. Elimination of imaginaries, invariant extensions of types, cell decompo- sitions, rational zeta functions, subgroup zeta functions, representation zeta functions 1 This paper concerns the model theory of the p-adic numbers Qp and applications to certain counting problems arising in group theory. Recall that a theory (in the model-theoretic sense of the word) is said to have elimination of imaginaries (EI) if the following holds: for every model M of the theory, for every -definable subset D of some M n and for every -definable equivalence relation R on∅ D, there exists an ∅ -definable function f : D M m, for some m, such that the fibers of f over f(D) are∅ precisely the equivalence→ classes of R. In other words, elimination of imaginaries states that every pair (D′, E′) (consisting of an -definable set D′ and an -definable equivalence relation E′ on it) reduces to a pair (D,∅ E) where E is equality—here,∅ as in descriptive set theory, we say that (D′, E′) reduces to (D, E) if there exists a -definable ∅ map f : D′ D with xE′y f(x)Ef(y). The theory→ of Q (in the language⇐⇒ of rings with a predicate for val(x) val(y)) does p ≥ not admit EI [67]: for example, no such f exists for the definable equivalence relation R on Qp given by xRy if val(x y) 1, because Qp/R is countably infinite but any m − ≥ definable subset of Qp is either finite or uncountable. Our first main theorem gives a n p-adic EI result when we add for each n a sort Sn for the family of Zp-lattices in Qp . These new sorts are called the geometric imaginaries. The language − consists of the LG valued field sort and the sorts Sn (with some more structure described in Section 2). Theorem 1.1 The theory of Q eliminates imaginaries in the language −. p LG To be precise, we prove a version of this that holds for any finite extension of Qp (Theorem 2.6). Suppose we are given not just a single definable equivalence relation for some fixed Qp, but one for every Qp. For our applications to zeta functions below, we want to control the behavior of the elimination of imaginaries as we vary the prime p. Our second main result is that the theory of ultraproducts of Qp also eliminates imaginaries if we add similar sorts. Theorem 1.2 The theory of non-principal ultraproducts Q / eliminates imagi- p p U naries in the language − provided we add some constants.Q LG See Theorem 2.7 for a more precise statement of what constants are needed to eliminate imaginaries. This last result implies that the elimination of imaginaries in Qp is uniform in p; see Corollary 2.9 for a precise statement of this uniformity. In fact, we prove a more general result (Corollary 2.17), which yields EI both for Qp and for ultraproducts: given two theories T , T satisfying certain hypotheses, T has EI if T does. In our application, T is the theory of algebraically closed valued e e e 2 fields of mixed characteristic (ACVF0,p) or equicharacteristic zero (ACVF0,0) and T is either the theory of a finite extension of Qp or the theory of an ultraproduct of Qp where p varies, with appropriate extra constants in each case (in fact in the latter case Corollary 2.17 does not apply immediately but a variant does). The notion of an invariant extension of a type plays a key part in our proof. If T is a theory, M = T , A M and p is a type over A then an invariant extension of p is a type q over| M such⊆ that q A = p and q is Aut(M/A)-invariant. The theory | ACVF is not stable; in [40, 41], Haskell, the first author and Macpherson used invariant extensions of types to study the stability properties of ACVF and to define notions of forking and independence. They proved that ACVF plus some extra sorts admits EI. As an important consequence of Theorems 2.6 and 2.7, we prove the following rationality and uniformity result for zeta functions Sp(t) counting the number of equivalence classes in some uniformly definable family of equivalence relations. (Here t = (t1,...,tr) is a tuple of indeterminates and Sp(t) is a power series in the ti; we −si obtain a zeta function in the more usual sense by setting ti = p , where the si are complex variables.) Theorem 1.3 The zeta functions (Sp(t))p prime are uniformly rational. We also give a version of Theorem 1.3 for a uniformly definable family of equivalence relations over a local field of positive characteristic (Corollary 6.8). See Section 6 for definitions and a precise statement (Theorem 6.1). Roughly speaking, uniform rationality means that each Sp(t) can be expressed as a rational function with coefficients in Q, where the denominator is a product of functions of the form (1 patb) or pn with a, b, n independent of p, and the numerator is a polynomial in − t such that each coefficient comes from counting the Fp-points of a fixed variety over Z. In particular, we obtain that Sp(t) is rational not just for all sufficiently large primes, but for every prime; this is crucial for our applications to representation growth below, as well as to the following result, which deals with the abscissa of convergence. Theorem 1.4 Let Sp(t) be as above and suppose we are in the one-variable case (r =1, −s t = t1). Define ζp(s) = Sp(p ). Assume that the constant term of ζp(s) is 1 for all but finitely many primes and set ζ(s)= p ζp(s). Then the abscissa of convergence of ζ(s) is rational (or ). Q −∞ In fact, Theorem 6.1 yields a kind of “double uniformity”: the ultraproduct for- malism allows us to vary not just the prime p, but also the choice of an extension Lp of Qp. For an application of this double uniformity, see the end of Section 8. 3 To describe the proof of Theorem 1.3, let us now come back to the meaning of our elimination of imaginaries result. It shows that any (D′, E′) can be reduced to a (D, E) of a special kind—namely, the equivalence relation on GLN (Qp) for some N whose equivalence classes are the left GLN (Zp)-cosets. The quotients D/E have a specific geometric meaning—but can one explain abstractly in what way they are special? One useful observation is that we have reduced an arbitrary equivalence relation to a quotient by a definable group action. Another concerns volumes: the E-classes have volumes that are motivically invertible (in fact, each class is equivalent to a polydisk of an appropriate dimension and size). Indeed, it is only the latter property of the geometric imaginaries that is actually used in the proof of Theorem 6.1. This proof relies on representing the number of classes of some definable equivalence relation E on some definable set D as an integral. The idea, going back to Denef and Igusa, is simple: the number of classes of E on D equals the volume of D, for any measure such that each E-class has measure one. The question is how to come up (definably) with such a measure. The setting is that we already have the Haar measure µ on Qp (normalized so that µ(Zp) = 1), and for simplicity—one can easily reduce to this case—let us assume each E-class [x]E D/E has finite, nonzero measure. The problem then is to show that there exists a definable∈ function f : D Q such that the measure of each E-class [x] is of the form → p E µ([x] )= f(x) , (1.1) E | | where f(x) denotes the p-adic norm. Then we can replace µ with f −1µ. In practice, f is usually| | given explicitly (cf. [39, Section 2]). For more complicated| | equivalence relations, however, such as the one for representation zeta functions in Section 8, it is not clear a priori that such an f can be found, even in principle. This point is beautifully brought out in work by Raf Cluckers; we are very pleased to have his permission to include it here as an Appendix.
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