MOTIVES FOR PERFECT PAC FIELDS WITH PRO-CYCLIC GALOIS GROUP

IMMANUEL HALUPCZOK

Abstract. Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.

In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

Contents

1. Preliminaries 1 1.1. Introduction 1 1.2. The results in detail 2 2. Galois stratifications 5 2.1. Galois covers 5 2.2. Artin symbols and colorings 5 2.3. Galois stratifications 6 2.4. Quantifier elimination to Galois stratifications 7 3. Existence of χ_{c} 7 3.1. Overview of the proof 7 3.2. Some fiber sizes 8 3.3. A version of Chebotarev’s density theorem 9 3.4. Assigning a virtual motive to a colored Galois cover 10 3.5. Assigning a virtual motive to a definable set 12 3.6. The properties of χ_{c} 13 4. Other proofs on χ_{c} 15 4.1. The uniqueness statement 15 4.2. Change of the base field k 16 5. Maps between Grothendieck rings 16 5.1. Existence of the maps θ_{ι} 16 5.2. χ_{c} is not injective 17 5.3. Compatibility of χ_{c} and θ_{ι} 18 6. Open problems 18 6.1. Uniqueness of χ_{c} 18

1991 Mathematics Subject Classification. 03C10, 03C60, 03C98, 12L12, 12E30, 12F10, 14G15, 14G27.

Key words and phrases. pseudo-finite fields, pseudo algebraically closed fields, motives. The author was supported by the Agence National de la Recherche (contract ANR-06-BLAN-

0183-01).

6.2. From motives to measure 19 6.3. Larger Galois groups for the maps θ_{ι} 19 6.4. Larger Galois groups for the maps χ_{c} 19 6.5. What exactly do we know about K_{0}(T_{Gal,k})? 19 References 20

1. Preliminaries

1.1. Introduction. To understand definable sets of a theory, it is helpful to have invariants with nice properties. For a fixed pseudo-finite field K, there are two well-known invariants of definable sets: the dimension (see [3]), and the measure (see [2]).

In a slightly different setting, Denef and Loeser constructed a much stronger invariant: they do not fix a pseudo-finite field; instead they consider definable sets in the theory of all pseudo-finite fields of characteristic zero. To each such set X they associate an element χ_{c}(X) of the Grothendieck group of Chow motives (see [4], [5]). In particular, this implies that all the usual cohomological invariants (like Euler characteristic, Hodge polynomial) are now applicable to arbitrary definable sets.

Note that the work of Denef and Loeser does not directly imply the existence of the measure of [2]: the motive associated to a set defined by a formula ϕ is invariant under equivalence in the theory of pseudo-finite fields, but not under equivalence in the theory of a fixed pseudo-finite field.

The dimension defined in [3] exists for a much larger class of fields and in [10], Hrushovski asked whether one can also generalize the measure. This question has been answered in [9]: it is indeed possible to define a measure for any perfect, pseudo algebraically closed (PAC) field with pro-cyclic Galois group. A natural question is now: Can the work of Denef-Loeser also be generalized to this setting? More precisely, fix an infinite pro-cyclic group Gal, and consider the theory of perfect PAC fields with absolute Galois group Gal. Then to any definable set X in that theory we would like to associate a virtual motive χ_{c}(X). The first goal of this article is to do this.

One reason this result seems interesting to me is the following: the map χ_{c} exists for pseudo-finite fields (by Denef-Loeser) and for algebraically closed fields (by quantifier elimination). The case of general pro-cyclic Galois groups is a common generalization of both and thus a kind of interpolation.

Comparing those maps χ_{c} for different Galois groups, one gets the feeling that they are closely related. Indeed, given an inclusion of Galois groups Gal_{2} ⊂ Gal_{1}, we will prove the existence of a map θ from the definable sets for Gal_{2} to the definable sets for Gal_{1} which is compatible with the different maps χ_{c}.

These maps θ turn out to be interesting in themselves. An open question was whether, using χ_{c}, one can get all the (additive) information about definable sets; in other words: is χ_{c} injective? We will show that it is not, by giving an example of two definable sets with the same image under χ_{c} but with different images under one of those maps θ.

We have one more result. In [5], the map χ_{c} is defined by enumerating certain properties and then existence and uniqueness of such a map is proven. We are able to weaken the conditions needed for uniqueness in the case of pseudo-finite fields. Unfortunately however, we do not get any sensible uniqueness conditions for other pro-cyclic Galois groups.

1.2. The results in detail. Let me fix some notation once and for all.

By a “group homomorphism” we will always mean a continuous group homomorphism if there are profinite groups involved.

Let Gal be a pro-cyclic group such that there do exist perfect PAC fields having Gal as absolute Galois group. This is the case if and only if Gal is torsion-free, or equivalently, if it is of the form ∏_{p∈P} ℤ_{p}, where P is any set of primes.

“definable” will always mean 0-definable. When we want to consider sets definable with parameters, we will add these parameters to the language. Indeed most of the time we will fix a field k (mostly of characteristic zero) and work in the theory T_{Gal,k} of perfect PAC (pseudo algebraically closed) fields with absolute Galois group Gal which contain k.

Models of T_{Gal,k} will be denoted by K; the algebraic closure of a field K will be denoted by K̃.

By “variety”, we mean a separated, reduced scheme of finite type. If not stated otherwise, all our varieties will be over k. The reader not so familiar with algebraic geometry can restrict herself to locally closed subsets of 𝔸^{n}, and everything will work out fine.

I will use the notion “definable set” even when there is no model around: By a “definable set (in T_{Gal,k})”, I mean a formula up to equivalence modulo T_{Gal,k}. In addition, I will permit myself to speak about “definable subsets of (arbitrary) varieties”. For affine embedded varieties, it is clear what this should mean. In general, any definable decomposition of a variety V into affine embedded ones yields the same notion of definable subsets of V. For a precise version of this, see e.g. [4]. (Denef and Loeser call this “definable subassignments”.)

For the definition of a Chow motive, see e.g. [11]. We do not repeat the definition here, as anyway the only things the reader really needs to believe about them are Theorem 3.5 and Lemma 3.6 and the existence of the map χ_{c}: K_{0}(Var_{k}) → K_{0}(Mot_{k}) mentioned below.

We will work with the following Grothendieck groups: K_{0}(T_{Gal,k}) is the group generated by the definable sets of T_{Gal,k}, modulo the relations [X_{1}] = [X_{2}] if there is a definable bijection between X_{1} and X_{2}, and [X_{1}] + [X_{2}] = [X_{1} ∪ X_{2}] if X_{1} and X_{2} are disjoint. K_{0}(Var_{k}) is the group generated by the varieties over k, modulo [V_{1}] = [V_{2}] if V_{1} and V_{2} are isomorphic and [V] = [U] + [V \ U] if U is an open subvariety of V. K_{0}(Mot_{k}) is the group generated by the Chow motives over k, modulo [m_{1}] = [m_{2}] if m_{1} and m_{2} are isomorphic and [m_{1}] + [m_{2}] = [m_{1} ⊕ m_{2}]. All three Grothendieck groups are in fact rings, where the multiplication is induced by × resp. × resp. ⊗. We will also need to tensor the Grothendieck ring of motives with ℚ; we denote this by K_{0}(Mot_{k})_{ℚ} := K_{0}(Mot_{k}) ⊗_{ℤ} ℚ.

Gillet and Soulé [7] and Guillén and Navarro Aznar [8] showed that there exists a unique morphism of rings χ_{c}: K_{0}(Var_{k}) → K_{0}(Mot_{k}) which extends the canonical map on the smooth projective varieties. Our first theorem states that this map can be extended to χ_{c}: K_{0}(T_{Gal,k}) → K_{0}(Mot_{k})_{ℚ} in such a way that it satisfies one additional property which needs some more definitions to be stated properly. Let me write down the theorem right now anyway; the reader may skip that property for the moment and come back after she has read the necessary definitions in Section 2.

Theorem 1.1. Suppose Gal = ∏_{p∈P} ℤ_{p} (where P is any set of primes) is a procyclic group and k is a field of characteristic zero. Then there exists a ring homomorphism χ_{c}: K_{0}(T_{Gal,k}) → K_{0}(Mot_{k})_{ℚ} extending the homomorphism χ_{c}: K_{0}(Var_{k}) → K_{0}(Mot_{k}) with the following property: if V ↠^{G} W is a Galois cover such that all prime factors of |G| lie in P, then

(∗) χ_{c}(X(V ↠^{G} W, \{1\})) = \frac{1}{|G|}χ_{c}(V) .

If Gal = ℤ̂, then a ring homomorphism with these properties is unique.

In the case Gal = ℤ̂, this result is almost the same as the one of Denef-Loeser (Theorem 6.4.1 of [5]). However, our condition (∗) needed to get uniqueness is a weakening of the one in [5].

As the reader will guess, even if Gal ≠ ℤ̂ we have a precise map χ_{c} in mind (which we will construct during the proof). Unfortunately we can not prove that condition (∗) is strong enough to define χ_{c} uniquely in the general case, and we do not have any good replacement for (∗). The best uniqueness statement we can offer is this: there is a unique map χ_{c} as in Theorem 1.1 compatible with Definition 3.7 (which in reality is spread over Section 3.4). Below we will give another (equivalent) tentative definition of χ_{c}.

Our map χ_{c} will not really depend on the base field k: if we have a second field k′ containing k, then there are canonical ring homomorphisms K_{0}(T_{Gal,k}) → K_{0}(T_{Gal,k′}) and K_{0}(Mot_{k})_{ℚ} → K_{0}(Mot_{k′})_{ℚ}, which we will both denote by ⊗_{k}k′. The map χ_{c} is compatible with these homomorphisms:

Proposition 1.2. In the setting just described (with χ_{c} satisfying Definition 3.7) we have, for any definable set X of T_{Gal,k}, χ_{c}(X ⊗_{k} k′) = χ_{c}(X) ⊗_{k} k′.

In [9], the idea behind the proof of the existence of a measure for definable sets of perfect PAC fields with pro-cyclic Galois group was to reduce the general case to the pseudo-finite case. However, this was just an idea, not a real reduction. For the present problem, it turns out that we really can define maps θ_{ι} between the different Grothendieck groups which enable us to reduce the existence of a map χ_{c}: K_{0}(T_{Gal,k}) → K_{0}(Mot_{k})_{ℚ} to the case Gal = ℤ̂. More generally, we will prove the following theorem.

Theorem 1.3. Suppose Gal_{1} and Gal_{2} are two torsion-free pro-cyclic groups, ι: Gal_{2} ↪ Gal_{1} is an injective map, and k is any field (not necessarily of characteristic zero). Denote the theories T_{Gal_{i},k} by T_{i} for i = 1, 2. Then the following defines a ring homomorphism θ_{ι}: K_{0}(T_{2}) → K_{0}(T_{1}): Suppose K_{1} is a model of T_{1}. Then the fixed field K_{2} := K̃_{1}^{ι(Gal_{2})} is a model of T_{2} containing K_{1}. For any X_{2} ⊂ 𝔸^{n} definable in T_{2}, we define θ_{ι}(X_{2})(K_{1}) := X_{2}(K_{2}) ∩ K_{1}^{n}.

To reduce the existence of χ_{c} to the case ℤ̂, apply this theorem to ι: Gal ↪ ℤ̂, where ι maps Gal to the appropriate factor ∏_{p∈P} ℤ_{p} of ℤ̂ (such that ℤ̂/Gal is torsion-free). Then define χ_{c} as the composition χ̂_{c} ∘ θ_{ι}, where χ̂_{c}: K_{0}(T_{ℤ̂,k}) → K_{0}(Mot_{k})_{ℚ} is the known map in the pseudo-finite case. Verification of the properties of χ_{c} is not very difficult, as soon as one knows the meaning of (∗), and using explicit computations done in the proof of Theorem 1.3.

So in principle, the existence of χ_{c} is reduced to Theorem 1.3. On the other hand, one has the feeling that it should also be possible to construct this map directly when Gal ≠ ℤ̂, so we will do this. The proof in [4] in the pseudo-finite case uses finite fields a lot, so to generalize it, we first had to “clean it up”. in our proof, the only place where finite fields appear is in a proof of Lemma 3.3, which is a kind of qualitative Chebotarev density theorem for T_{Gal,k} (and which might be interesting in itself).

By the way, we now have a second not-so-beautiful definition of χ_{c} in the general case at hand: in the above reduction, we chose a map ι: Gal → ℤ̂, but it can be seen from Theorem 1.3 that θ_{ι} is independent of this choice (as we fixed the image). So one could also define χ_{c} := χ̂_{c} ∘θ. We will check that the two different definitions of χ_{c} are equivalent.

Another interesting application of Theorem 1.3 is the case Gal_{1} = Gal_{2} = Gal, but with a non-trivial injection ι: Gal ↪ Gal. One thus gets endomorphisms of the ring K_{0}(T_{Gal,k}), which might reveal a lot of information about its structure. Indeed this really gives new information: we will construct a whole family of pairs of definable sets X_{1} and X_{2} such that χ_{c}(X_{1}) = χ_{c}(X_{2}) but χ_{c}(θ(X_{1})) ≠ χ_{c}(θ(X_{2})), thereby proving:

Proposition 1.4. Let k be a field of characteristic zero and let Gal be a non-trivial torsion-free pro-cyclic group. Then the map χ_{c}: K_{0}(T_{Gal,k}) → K_{0}(Mot_{k})_{ℚ} is not injective.

The remainder of the article is organized as follows:

• In Section 2, we define Galois covers, the Artin symbol (in a version suited for this article) and Galois stratifications. After that, the reader will understand condition (∗) of Theorem 1.1. At the end of the section, we will state a version of “almost quantifier elimination” from [6] which we will be need several times.

• The construction of the map χ_{c} of Theorem 1.1 will be done in Section 3, after first proving some lemmas, in particular the version of Chebotarev’s density theorem for T_{Gal,k}. We also check that the constructed map has the required properties.

• In Section 4, we prove the remaining statements on χ_{c}, i.e. the uniqueness in the ℤ̂ case and compatibility with change of the base field k (Proposition 1.2).

• Section 5 is devoted to the maps θ_{ι}. We prove Theorem 1.3 and, as an example application, Proposition 1.4. In addition, we will check that the maps χ_{c} for different Galois groups (as constructed in Section 3.4) are compatible with the maps θ_{ι} for suitable ι.

• Finally Section 6 lists some open problems.

2. Galois stratifications

A standard technique to get hold of definable sets of perfect PAC fields with not-too-large Galois group is the quantifier elimination to Galois formulas. In this section, we define the necessary objects and then, in Section 2.4, state this quantifier elimination result in the version of Fried-Jarden [6].

In most of the article, we will work with parameters in a field k. However, at some point we will need Galois covers defined over rings, so for the definitions in this section assume that all varieties are defined over some fixed ring R.

We also fix a pro-cyclic group Gal = ∏_{p∈P} ℤ_{p} for some set of primes P, and we let T := T_{Gal,R} be the theory of perfect PAC fields with absolute Galois group Gal containing R.

2.1. Galois covers.

Definition 2.1. (1) A Galois cover consists of two integral and normal varieties V and W (over our ring R) and a finite étale map f: V → W such that for G := Aut_{W} (V)^{opp}, we have canonically W ≅ V/G (where G acts from the right on V). We denote a Galois cover by f: V ↠^{G} W and call G the group of that cover. The action of G on V will be denoted by v.g (for v ∈ V, g ∈ G).

(2) We say that a Galois cover f′: V′ ↠^{G′} W is a refinement of f: V ↠^{G} W, if there is a finite étale map g: V′ ↠ V such that f′ = f ∘ g.

(3) If W′′ is a locally closed subset of W and V′′ is a connected component of

f^{−1}(W′′), then we call V′′ ↠^{G′′} W′′ the restriction of V ↠^{G} W to W′′, where G′′ := Aut_{W′′}(V′′)^{opp}.

Remark 2.2. (1) If f′: V′ ↠^{G′} W is a refinement of f: V ↠^{G} W, then we have

a canonical surjection π: G′ ↠ G.

(2) If V′′ ↠^{G′′} W′′ is a restriction of V ↠^{G} W, then we have a canonical injection G′′ ↪ G. Different choices of the connected component of f^{−1}(W′′) yield isomorphic restricted Galois covers.

2.2. Artin symbols and colorings. Using a Galois cover V ↠^{G} W, we would like to decompose W into subsets according to the Artin symbol of the elements. However, the usual definition of Artin symbol needs a canonical generator of the Galois group Gal (usually the Frobenius of a finite field); the Artin symbol is then the image of the generator under a certain map ρ: Gal → G (which is unique only up to conjugation by G). If one does not have such a canonical generator, then one still can consider the image of ρ. This is what one uses as Artin symbol in our case.

Definition 2.3 (and Lemma). Suppose f: V ↠^{G} W is a Galois cover and K ⊧ T is a model.

(1) Suppose v ∈ V (K̃) such that f(v) ∈ W (K). Then there is a unique group homomorphism ρ: Gal(K̃/K) → G satisfying σ(v) = v.ρ(σ) for any σ ∈ Gal(K̃/K). The decomposition group Dec(v) := im ρ ⊂ G of v is the image of that homomorphism.

(2) For w ∈ W (K), let the Artin Symbol Ar(w) of w be the set {Dec(v) | v ∈ V (K̃), f(v) = w} of decomposition groups of all preimages of w.

Ar(w) consists exactly of one conjugacy class of subgroups of G, and these subgroups are isomorphic to a quotient of Gal. The quotients of Gal are just the cyclic groups Q such that all prime factors of |Q| lie in P (where P was the set of primes such that Gal = ∏_{p∈P} ℤ_{p}). We introduce some notation for this:

Definition 2.4. Given a finite group G, we will call those subgroups of G which are isomorphic to a subgroup of Gal the permitted subgroups. We denote the set of all permitted subgroups of G by Psub(G). If Q is a finite cyclic group, then we denote by Ppart(Q) the “permitted part of Q”, i.e. the biggest permitted subgroup of Q.

(The “P” in “Psub” and “Ppart” stands for permitted and/or the set P of primes.) The interest of Ppart(Q) is the following. We will sometimes identify Gal = ∏_{p∈P} ℤ_{p} with the corresponding factor of ℤ̂ and consider homomorphisms ρ: ℤ̂ → G. Then the image of Gal in G is just ρ(Gal) = Ppart(im ρ).

Given a Galois cover V ↠^{G} W, we now define subsets of W using the Artin symbol:

Definition 2.5. (1) A coloring of a Galois cover V ↠^{G} W is a subset C of the permitted subgroups of G which is closed under conjugation. A Galois cover together with a coloring is called a colored Galois cover.

(2) Given a colored Galois cover (V ↠^{G} W, C) and a model K ⊧ T, we define the set X(V ↠^{G} W, C)(K) := {w ∈ W (K) | Ar(w) ⊂ C}.

Note that X(V ↠^{G} W, C) is definable, i.e. there is a formula ϕ such that for any model K ⊧ T we have ϕ(K) = X(V ↠^{G} W, C)(K).

Remark 2.6. (1) If (V ↠^{G} W, C) is a colored Galois cover and V′ ↠^{G′} W is a refinement with canonical map π: G′ ↠ G, then we can also refine the coloring: by setting C′ := {Q ∈ Psub(G′) | π(Q) ∈ C}, we get X(V′ ↠^{G′}

W, C′) = X(V ↠^{G} W, C).

(2) Similarly if V′′ ↠^{G′′} W′′ is a restriction of f: V ↠^{G} W: in that case, set C′′ :=

{Q ∈ C | Q ⊂ G′′}. Then we get X(V′′ ↠^{G′′} W′′, C′′) = X(V ↠^{G} W, C)∩W′′.

2.3. Galois stratifications.

Definition 2.7. A Galois stratification A of a variety W is a finite family (f_{i}: V_{i} ↠^{G_{i}} W_{i}, C_{i})_{i∈I} of colored Galois covers where the W_{i} form a partition of W into locally closed sub-varieties. We shall say that A defines the following subset A(K) ⊂ W (K), where K ⊧ T is a model:

A(K) := ⋃_{i∈I} X(V_{i} ↠^{G_{i}} W_{i}, C_{i})(K)

The data of a Galois stratification denoted by A will always be denoted by V_{i}, W_{i}, G_{i}, C_{i}, and analogously with primes for A′, A′′, etc. This will not always be explicitely mentioned.

Definition 2.8. Suppose A and A′ are two Galois stratifications. We say that A′ is a refinement of A, if:

• Each W_{i} is a union ⋃_{j∈J_{i}} W′_{j} for some J_{i} ⊂ I′.

• For each i ∈ I and each j ∈ J_{i}, the Galois cover V′_{j} ↠^{G′_{j}} W′_{j} is a refinement

of the restriction of the Galois cover V_{i} ↠^{G_{i}} W_{i} to the set W′_{j}.

• C′_{j} is constructed out of C_{i} as described in Remark 2.6, such that X(V′_{j} ↠^{G′_{j}}

W′_{j}, C′_{j}) = X(V_{i} ↠^{G_{i}} W_{i}, C_{i}) ∩ W′_{j}.

By the third condition, A and A′ define the same set.

One reason for Galois stratifications being handy to use is the following wellknown lemma:

Lemma 2.9. If A and A′ are two Galois stratifications, then there exist refinements Ã and Ã′ of A resp. A′ which differ only in the colorings.

2.4. Quantifier elimination to Galois stratifications. We now can state the version of quantifier elimination which we will use. It is given in [6], Proposition 25.9. Applied to our situation, that proposition reads:

Lemma 2.10. Suppose Gal is a torsion-free pro-cyclic group and k is any field. Then each definable set X of T_{Gal,k} is already definable by a Galois stratification A (over k), i.e. for any K ⊧ T_{Gal,k}, we have X(K) = A(K).

Note that Proposition 25.9 of [6] requires that K is what Fried-Jarden call a “prefect Frobenius field”; this is indeed the case for any model of T_{Gal,k}.

3. Existence of χ_{c}

In this section we prove the existence of the map χ_{c} of Theorem 1.1. For the whole section, we fix a torsion-free pro-cyclic group Gal and a field k of characteristic zero. We also fix the theory T := T_{Gal,k} we will work in.

3.1. Overview of the proof. We follow the ideas of [4]. The proof consists of three parts:

(1) Associate virtual motives to colored Galois covers, and, more generally, to Galois stratifications.

(2) Verify that this only depends on the set defined by the stratification. Using the quantifier elimination result Lemma 2.10, we thus get a map χ_{c} from the definable sets to the virtual motives.

(3) Check that this map χ_{c} has all the required properties: that it is invariant under definable bijections and compatible with disjoint union and products (so it defines a ring homomorphism K_{0}(T_{Gal,k}) → K_{0}(Mot_{k})_{ℚ}) and that it satisfies condition (∗) of Theorem 1.1.

To associate a virtual motive to a colored Galois cover, we use results of [1] where an “equivariant version” of the map χ_{c}: K_{0}(Var_{k}) → K_{0}(Mot_{k}) is given. In this definition we have to be careful if Gal ≠ ℤ̂; otherwise, the motive will really depend on the Galois cover and not only on the set defined by the cover. We will do this in Section 3.4.

In Section 3.5, we will extend the definition of χ_{c} to arbitrary Galois stratifications and check (2). The most important ingredient for this is that different colorings of a fixed Galois cover always define different sets. This is essentially a qualitative version of Chebotarev’s density theorem for models of T = T_{Gal,k}, which we will prove in Section 3.3.

Finally, the properties (3) of χ_{c} will be verified in Section 3.6. The most laborious one is that χ_{c} is invariant under definable bijections. For this, we will first prove an additional lemma in Section 3.2, which replaces the quantitative part of Chebotarev’s density theorem.

3.2. Some fiber sizes.

Lemma 3.1. Suppose f: V ↠^{G} W is a Galois cover, K ⊧ T a model, w ∈ W (K), and C = Ar(w) is the Artin symbol of w. Fix a group Q ∈ C. Then the number

# {v ∈ V (K̃) | f(v) = w ∧ Dec(v) = Q}

of elements in the fiber of w with decomposition group Q is \frac{|G|}{|C|}.

Proof. For Q ∈ C, write F (Q) := {v ∈ V (K̃) | f(v) = w ∧Dec(v) = Q} for the part of the fiber with decomposition group Q. We will show that for any two Q, Q′ ∈ C, F (Q) and F (Q′) have the same cardinality. As the whole preimage of w consists of |G| elements, the lemma follows.

Given Q, Q′ ∈ C, there exists a g ∈ G such that Q′ = Q^{g}. Suppose v ∈ F (Q), i.e. the homomorphism ρ: Gal(K̃/K) → G which satisfies σ(v) = v.ρ(σ) (for all σ ∈ Gal(K̃/K)) has image im ρ = Q. Then σ(v).g^{−1} = v.ρ(σ)g^{−1} = v.g^{−1}ρ(σ)^{g}. Now im(ρ^{g}) = Q′, so v.g^{−1} ∈ F (Q′), which means that g^{−1} maps F (Q) to F (Q′). By the same argument, g maps F (Q′) to F (Q), so we have a bijection. □

Lemma 3.2. Suppose we have the following diagram, where the maps f_{1} : V → W_{1} and f_{2} : V → W_{2} are Galois covers with groups G_{1} resp. G_{2}. Note that we have naturally G_{1} ⊂ G_{2}.

\begin{matrix}V \\ W_{1} & W_{2} \end{matrix}f_{1} f_{2} ϕ

Suppose additionally that C_{1} is a coloring of f_{1} : V ↠^{G_{1}} W_{1} consisting of a single conjugacy class of groups and C_{2} := C_{1}^{G_{2}} is the induced coloring of f_{2} : V ↠^{G_{2}} W_{2}. Then the image under ϕ of X_{1} := X(V ↠^{G_{1}} W_{1}, C_{1}) is X_{2} := X(V ↠^{G_{2}} W_{2}, C_{2}). In addition, ϕ restricts to a bijection X_{1} → X_{2} if and only if \frac{|G_{1}|}{|C_{1}|} = \frac{|G_{2}|}{|C_{2}|}.

Proof. “ϕ(X_{1}) ⊂ X_{2}”: Suppose K ⊧ T is a model, w_{1} ∈ X_{1}(K) and v ∈ f_{1}^{−1}(w_{1}) is a preimage in V (K̃). Then Dec(v) ∈ C_{1}. But v is also a preimage of ϕ(w_{1}), so Ar(ϕ(w_{1})) contains Dec(v) and is therefore equal to C_{1}^{G_{2}}.

“ϕ(X_{1}) ⊃ X_{2}”: Suppose w_{2} ∈ X_{2}(K) for some model K ⊧ T. As Ar(w_{2}) contains C_{1}, we may choose a preimage v ∈ V (K̃) of w_{2} with Dec(v) ∈ C_{1}. In particular, Dec(v) ⊂ G_{1}, which means that Gal(K̃/K) fixes v.G_{1} ∈ (V/G_{1})(K) ≅ W_{1}(K). So the image f_{1}(v) lies in W_{1}(K). As Ar(f_{1}(v)) contains Dec(v), we have Ar(f_{1}(v)) = C_{1}, so f_{1}(v) is a preimage of w_{2} lying in X_{1}(K).

It remains to check that the fibers of ϕ↾_{X_{1}} have size one if and only if \frac{|G_{1}|}{|C_{1}|} = \frac{|G_{2}|}{|C_{2}|}.

Indeed, we prove that the size of the fibers is \frac{|G_{2}|·|C_{1}|}{|C_{2}|·|G_{1}|}. Fix a group Q ∈ C_{1}. For any model K ⊧ T and any w_{2} ∈ X_{2}(K), consider the set F_{2} := {v ∈ f_{2}^{−1}(w_{2}) | Dec(v) = Q\}. By Lemma 3.1, this set has \frac{|G_{2}|}{|C_{2}|} elements.

For any v ∈ F_{2}, f_{1}(v) lies in X_{1}(K) ∩ ϕ^{−1}(w_{2}) (lying in X_{1}(K) holds true by the same argument as in “ϕ(X_{1}) ⊃ X_{2}”). So the sets F_{1}(w_{1}) := {v ∈ f_{1}^{−1}(w_{1}) | Dec(v) = Q}, w_{1} ∈ X_{1}(K) ∩ ϕ^{−1}(w_{2}), form a partition of F_{2}. By Lemma 3.1, each set F_{1}(w_{1}) has \frac{|G_{1}|}{|C_{1}|} elements, so w_{2} has \frac{|G_{2}|·|C_{1}|}{|C_{2}|·|G_{1}|} preimages in X_{1}(K). □

3.3. A version of Chebotarev’s density theorem. We will need the following (qualitative) version of Chebotarev’s density theorem. Remember that we fixed a torsion-free pro-cyclic group Gal, a field k of characteristic zero, and the theory T = T_{Gal,k}. Concerning the permitted subgroups Psub, remember Definition 2.4.

Lemma 3.3. For any Galois cover V ↠^{G} W and any conjugacy class C ⊂ Psub(G) of permitted subgroups of G, there exists a model K ⊧ T such that X(V ↠^{G} W, C)(K) is Zariski dense in W (K).

Remark 3.4. As K is PAC, X(V ↠^{G} W, C)(K) is even dense in W (K̃).

Proof of Lemma 3.3. First note that the condition “X(V ↠^{G} W, C)(K) is dense in W (K)” can be expressed by an infinite number of first order sentences: for every open subset W_{0} ⊂ W, take the sentence which says that X(V ↠^{G} W, C) ∩ W_{0} is not empty. Let T′ be the theory T with these sentences added. Using the usual density theorem of Chebotarev, we will show that T′ is finitely satisfyable; then, the claim follows by compactness. (In algebraic terms: we will construct a lot of fields satisfying some of the properties we would like to have; then an appropriate ultraproduct of these fields will do the job.)

Fix once and for all a ring R ⊂ k which is finitely generated over ℤ and such that V ↠^{G} W can be defined over R. Choose a version V_{R} ↠^{G} W_{R} of V ↠^{G} W over R, i.e. such that V = V_{R} ⊗_{R} k, etc.

Let A be any finite subset of k, and let W_{0} ⊂ W_{R} ⊗_{R} R[A] be any open subset defined over R[A]. Out of this, we will construct a field K′ = K′_{A,W_{0}} together with a ring homomorphism ϕ: R[A] → K′ with the following properties: K′ is an algebraic extension of a finite field, its Galois group is Gal, and X(V_{R} ↠^{G} W_{R}, C)(K′) ∩ W_{0}(K′) is not empty, where we use ϕ to interpret the parameters of this formula (which lie in R[A]) in K′.

Before we give the construction of K′, let us check that those fields indeed verify any finite part of T′. Of course, the constants for the elements of R[A] in the language of T′ are interpreted using ϕ and the remaining constants k ∖ R[A] are interpreted anyhow.

All fields K′_{A,W_{0}} are perfect and have Galois group Gal. By choosing W_{0} small

enough, we can make K′_{A,W_{0}} satisfy any finite part of “X(V ↠^{G} W, C) is dense in W ”. To get a field K′_{A,W_{0}} which satisfies a finite part of the atomic diagram of k, we just have to choose A big enough; in particular to get ϕ(a) ≠ ϕ(a′) for a, a′ ∈ k, we put \frac{1}{a−a′} into A. To get a finite part of PAC, by the Lang-Weil estimates we just have to ensure that K′_{A,W_{0}} is big enough. This can be done by adding elements of the form \frac{1}{p} to A.

Now the only thing left to do is to construct the field K_{A,W_{0}} for any given A and W_{0} as above. Let us start by specifying the objects we want to apply the usual density theorem of Chebotarev to. Tensor the Galois cover V_{R} ↠^{G} W_{R} with ⊗_{R}R[A], restrict it to W_{0}, and denote the result by V_{0} ↠^{G} W_{0}. This is again a Galois cover, and both V_{0} and W_{0} are finitely generated over ℤ. Define Ĉ := {g ∈ G | Ppart(⟨g⟩) ∈ C}.

We apply Chebotarev’s density theorem (in the version of [12], Theorem 7) to V_{0} ↠^{G} W_{0} and Ĉ. This yields the existence of a finite field F and an element w ∈ W_{0}(F) such that the usual Artin symbol of w lies in Ĉ. Note that here W_{0} is interpreted as a scheme over ℤ, so “w ∈ W_{0}(F)” means that we get a map R[A] → F, and W_{0}(F) has to be interpreted using this map.

To get our desired field K′, we identify Gal with a subgroup of ℤ̂ ≅ Gal(F̃/F) in such a way that ℤ̂/Gal is torsion-free. Then we define K′ to be the subfield of F̃ fixed by Gal ⊂ Gal(F̃/F). As promised it is an algebraic extension of a finite field, it has Galois group Gal, and it comes with a map ϕ: R[A] → F ↪ K′. It remains to check that the element w, interpreted as an element of W_{0}(K′), has Artin symbol C in our sense.

Denote the usual Frobenius automorphism x ↦ x^{|F |} in Gal(F̃/F) by σ_{0}. Choose a preimage v ∈ V_{0}(F̃) of w and let ρ: Gal(F̃/F) → G be the group homomorphism satisfying σ(v) = v.ρ(σ) for all σ ∈ Gal(F̃/F). Then the image of ρ is ⟨ρ(σ_{0})⟩, and the image of ρ↾_{Gal} is ρ(Gal) = Ppart(⟨ρ(σ_{0})⟩) (see the remark after Definition 2.4). As ρ(σ_{0}) lies in Ĉ, im(ρ↾_{Gal}) lies in C, which is what we had to show. □

3.4. Assigning a virtual motive to a colored Galois cover. Gillet and Soulé [7] and Guillén and Navarro Aznar [8] have defined a map χ_{c}: K_{0}(Var_{k}) → K_{0}(Mot_{k}) assigning a virtual motive to any variety. As in [4], we use the following “equivariant version” of this result from [1].

For a finite group G, denote by R(G) the group of virtual characters of G (i.e. ℤ-linear combinations of irreducible characters of G) and by K_{0}(G, Var_{k}) the Grothendieck ring of G-varieties. (As for the Galois covers, we let G act from the right.) The following is Theorem 6.1 of [1]:

Theorem 3.5. There exists a unique map χ_{c} which associates to each finite group G, each G-variety V and each character α ∈ R(G) a virtual motive χ_{c}(G ↬ V, α) ∈ K_{0}(Mot_{k}) such that:

(1) For any G and α, the induced map χ_{c}(G ↬ , α): K_{0}(G, Var_{k}) → K_{0}(Mot_{k}) is a group homomorphism.

(2) For any G and V, χ_{c}(G ↬ V, ): R(G) → K_{0}(Mot_{k}) is a group homomorphism.

(3) If α_{reg} is the character of the regular representation of G, then χ_{c}(G ↬ V, α_{reg}) = χ_{c}(V).

(4) Suppose V is projective and smooth and α ∈ R(G) is irreducible and of dimension n_{α}. Then χ_{c}(G ↬ V, α) is the class of the motive (V, p_{α}), where p_{α} := \frac{n_{α}}{|G|} ∑_{g∈G} α(g^{−1})[g] ∈ ℚ[G] is the idempotent corresponding to α.

We shall also need the following properties of χ_{c}.

Lemma 3.6. Let G be a finite group, H ⊂ G be a subgroup and V be a G-variety.

(1) Suppose H is normal in G and denote the projection G ↠ G/H by π. Then, for any character α ∈ R(G/H),

χ_{c}(G/H ↬ V/H, α) = χ_{c}(G ↬ V, α ∘ π) .

(2) For any character α ∈ R(H),

χ_{c}(G ↬ V, Ind_{H}^{G} α) = χ_{c}(H ↬ V, α) .

(3) Suppose V is isomorphic as a G-variety to ˙⋃_{s∈G/H} Us for a H-variety U. Then, for any character α ∈ R(G),

χ_{c}(G ↬ V, α) = χ_{c}(H ↬ U, Res_{H}^{G} α) .

(4) Suppose that for i = 1, 2 we have a finite group G_{i}, a G_{i}-variety V_{i}, and a character α_{i} of G_{i}. Then

χ_{c}(G_{1} × G_{2} ↬ V_{1} × V_{2}, α_{1} ⊗ α_{2}) = χ_{c}(G_{1} ↬ V_{1}, α_{1}) ⊗ χ_{c}(G_{2} ↬ V_{2}, α_{2}) .

Proof. (1) to (3) are Proposition 3.1.2 of [4]. (4) is a straightforward computation using the definition of χ_{c} in [1]. □

χ_{c}(G ↬ V, ) naturally extends to a map R(G)⊗_{ℤ}ℚ → K_{0}(Mot_{k})_{ℚ} = K_{0}(Mot_{k})⊗_{ℤ} ℚ with the same properties.

Using this map χ_{c} we will now associate to any colored Galois cover (V ↠^{G} W, C) a motive χ_{c}(V ↠^{G} W, C). This will be done by associating a virtual character α_{C} ∈ R(G) ⊗_{ℤ} ℚ to any coloring C.

In the case of pseudo-finite fields, in [4] Denef-Loeser defined α_{C} to be 1 on the set {g ∈ G | ⟨g⟩ ∈ C} and 0 elsewhere. Just copying this definition does not work when the Galois group is not ℤ̂. The reason is that in a certain sense, the meaning of “Q ∈ C” depends on the Galois group. For example, “{1} ∈ C” means “just a little part of W ” when Gal = ℤ̂, whereas when Gal is trivial, it means “the whole of W ”.

To get a working definition for α_{C} in the non-ℤ̂-case, one has to remember that the Artin symbol is the image of a certain map ρ: Gal → G. Then one views Gal as a subgroup of ℤ̂ and considers extensions of ρ to ℤ̂, as described in the remark after Definition 2.4 and as it was done in the proof of Lemma 3.3. In this way one naturally gets the following definition:

α_{C}(g) := \begin{cases}1 & if Ppart(⟨g⟩) ∈ C \\ 0 & otherwise . \end{cases}

The first thing we have to check is that indeed α_{C} lies in R(G) ⊗_{ℤ} ℚ (and not just in R(G) ⊗_{ℤ} ℚ̃). This follows from the fact that α_{C}(g) = α_{C}(g′) if ⟨g⟩ = ⟨g′⟩: we get that α_{C} can be written as ℚ-linear combination of characters induced by trivial representations of cyclic subgroups of G.

Now we can associate a virtual motive to any colored Galois cover:

Definition 3.7. Given a colored Galois cover (V ↠^{G} W, C), define

χ_{c}(V ↠^{G} W, C) := χ_{c}(G ↬ V, α_{C}) .

3.5. Assigning a virtual motive to a definable set. We define the motive associated to a Galois stratification A in the following way:

χ_{c}(A) := ∑_{i∈I} χ_{c}(V_{i} ↠^{G_{i}} W_{i}, C_{i})

Lemma 3.8. If two Galois stratifications A and A′ of the same variety W define the same subset of W, then the corresponding motives χ_{c}(A) and χ_{c}(A′) are equal.

Proof. By Lemma 2.9, there exist refinements Ã resp. Ã′ of A resp. A′ which differ only in the colorings, and the refinements define the same sets as the originals. So it is enough to check that the associated motives are invariant under refinement of the stratification and that if A and A′ differ only in the colorings and define the same set, then even the colorings are the same.

Suppose first that A = (f_{i}: V_{i} ↠^{G_{i}} W_{i}, C_{i})_{i∈I} and A′ = (f_{i}: V_{i} ↠^{G_{i}} W_{i}, C′_{i})_{i∈I} differ

only in the colorings and that both define the same set. Consider a conjugacy class D of permitted subgroups of one of the groups G_{i}. By Lemma 3.3 there exists a model K ⊧ T and an element w ∈ W_{i}(K) such that Ar(w) = D. So as A(K) and A′(K) do not differ in w, C_{i} and C′_{i} do not differ in D.

To check that the virtual motive associated to a Galois stratification does not change under refinement, it is enough to verify the two following special cases:

(1)

χ_{c}(f: V ↠^{G} W, C) = ∑_{i∈I} χ_{c}(V_{i} ↠^{G_{i}} W_{i}, C_{i})

where (f: V ↠^{G} W, C) is a colored Galois cover, (W_{i}) a decomposition of W into finitely many locally closed subsets, and (V_{i} ↠^{G_{i}} W_{i}, C_{i}) is the restriction

of (V ↠^{G} W, C) to W_{i}.

(2)

χ_{c}(V ↠^{G} W, C) = χ_{c}(V′ ↠^{G′} W, C′)

where (V ↠^{G} W, C) is a colored Galois cover and (V′ ↠^{G′} W, C′) is a refinement.

Proof of (1):

χ_{c}(V ↠^{G} W, C) = χ_{c}(G ↬ V, α_{C})

= ∑_{i} χ_{c}(G ↬ f^{−1}(W_{i}), α_{C}) by Theorem 3.5 (1)

= ∑_{i} χ_{c}(G_{i} ↬ V_{i}, Res_{G_{i}}^{G} α_{C}) by Lemma 3.6 (3)

= ∑_{i} χ_{c}(G_{i} ↬ V_{i}, α_{C_{i}}) = ∑_{i} χ_{c}(V_{i} ↠^{G_{i}} W_{i}, C_{i})

Proof of (2): By Lemma 3.6 (1), it is enough to check that α_{C′} = α_{C} ∘ π. But indeed we have, for any g′ ∈ G′:

α_{C′}(g′) = 1 ⟺ Ppart(⟨g′⟩) ∈ C′ ⟺ π(Ppart(⟨g′⟩)) ∈ C
⟺ Ppart(⟨π(g′)⟩) ∈ C ⟺ α_{C}(π(g′)) = 1 .
□

Using this lemma (and Lemma 2.10 about the quantifier elimination), we are now able to associate virtual motives to definable sets:

Definition 3.9. Let X be a definable set of T. Then define the associated virtual motive χ_{c}(X) by choosing any Galois stratification A defining X and by letting χ_{c}(X) := χ_{c}(A).

3.6. The properties of χ_{c}. We now check all the properties of χ_{c} we need for Theorem 1.1: χ_{c} should induce a ring homomorphism K_{0}(T_{Gal,k}) → K_{0}(Mot_{k})_{ℚ}, i.e. it has to be invariant under definable bijections and compatible with disjoint unions and products; and we want χ_{c} to satisfy condition (∗) of the theorem.

Compatibility with disjoint unions is clear. (Use a Galois stratification which is sufficiently fine to define all involved sets.) Let us now quickly verify compatibility with products and (∗), so that afterwards the only thing remaining is compatibility with definable bijections.

Lemma 3.10. Suppose X_{1} and X_{2} are two definable sets. Then χ_{c}(X_{1} × X_{2}) = χ_{c}(X_{1}) ⊗ χ_{c}(X_{2}).

Sketch of proof. It is enough to verify the statement if X_{i} = X(V_{i} ↠^{G_{i}} W_{i}, C_{i}) for i = 1, 2.

Using the definition of the Artin symbol and by choosing a generator of Gal, one easily verifies that

X_{1} × X_{2} = X(V_{1} × V_{2} ↠^{G_{1}×G_{2}} W_{1} × W_{2}, C) ,

where C = {⟨(g_{1}, g_{2})⟩ ⊂ G_{1} × G_{2} | ⟨g_{1}⟩ ∈ C_{1}, ⟨g_{2}⟩ ∈ C_{2}}.

One then computes α_{C}((g_{1}, g_{2})) = α_{C_{1}}(g_{1}) · α_{C_{1}}(g_{1}) for g_{1} ∈ G_{1} and g_{2} ∈ G_{2}. For this, the following formula is helpful. Let m be the product of all prime factors of |G_{1} × G_{2}| (with multiplicity) not lying in P. Then for any g ∈ G_{1} × G_{2} we have Ppart(⟨g⟩) = ⟨g^{m}⟩.

Finally, apply Lemma 3.6 (4). □

Lemma 3.11. Suppose V ↠^{G} W is a Galois cover such that all prime factors of |G| lie in P (where P is the set of primes such that Gal = ∏_{p∈P} ℤ_{p}). Then the map χ_{c} of Definition 3.9 satisfies

(∗) χ_{c}(X(V ↠^{G} W, \{1\})) = \frac{1}{|G|}χ_{c}(V) .

Proof. Remember the definition: χ_{c}(X(V ↠^{G} W, {1})) = χ_{c}(G ↬ X, α_{{1}}) where α_{{1}}(g) = 1 if Ppart(⟨g⟩) = {1} and α_{{1}}(g) = 0 otherwise. As all prime factors of |G| lie in P, we have Ppart(⟨g⟩) = ⟨g⟩, so α_{\{1\}} = \frac{1}{|G|}α_{reg}, where α_{reg} is the character of the regular representation of G. By Theorem 3.5 (2) and (3), we get

χ_{c}(X(V ↠^{G} W, C)) = \frac{1}{|G|}χ_{c}(V). □

The last piece of work is compatibility with definable bijections:

Lemma 3.12. Suppose X ⊂ 𝔸^{n} and X′ ⊂ 𝔸^{n′} are two definable sets and suppose there is a definable bijection ϕ from X to X′. Then the associated virtual motives χ_{c}(X) and χ_{c}(X′) are the same.

Proof. We have definable bijections between the graph of ϕ and X and X′, induced by the projections 𝔸^{n+n′} ↠ 𝔸^{n} and 𝔸^{n+n′} ↠ 𝔸^{n′}. So it is enough to check the statement when the bijection itself is induced by a projection.

Choose a Galois stratification A = (V_{i} ↠^{G_{i}} W_{i})_{i∈I} defining X. By compatibility of χ_{c} with disjoint union, it is enough to show the statement for all the restricted bijections W_{i} → ϕ(W_{i}), so we can suppose that X is a subset of a variety W and defined by a single colored Galois cover: X = X(f: V ↠^{G} W, C).

Now choose a Galois stratification A′ of X′. Again it is enough to show the statement for all restricted bijections ϕ^{−1}(W′_{i}) ∩ W → W′_{i}, so we can suppose that

X′ is defined by a single colored Galois cover: X′ = X(f′: V′ ↠^{G′} W′, C′). Note that X is still defined by a single Galois cover as we can simply restrict the previous one to ϕ^{−1}(W′_{i}) ∩ W. ϕ will now denote the map from W to W′.

By induction, we can remove parts of W′ of smaller dimension (and restrict the two Galois covers appropriately). We will do this in the next paragraph.

The next thing we claim is that we can suppose V = V′. By Lemma 3.3 (our version of Chebotarev) and Remark 3.4, there exists a model K′ ⊧ T such that X′(K′) is dense in W′(K̃′). As the image of ϕ contains X′(K′), ϕ is dominant. Similarly there is a model K ⊧ T such that X(K) is dense in W (K̃); as ϕ is bijective on X(K), we get dim W = dim W′, and ϕ is generically finite. Therefore, after cutting away parts of smaller dimension, we can refine both Galois covers such that we get V = V′ and G ⊂ G′.

Decompose the set C into conjugacy classes C_{i}. To each X(V ↠^{G} W, C_{i}) ⊂ X we apply Lemma 3.2. The result is that ϕ restricted to X(V ↠^{G} W, C_{i}) has image

X(V ↠^{G′} W′, C_{i}^{G′}), so we may again cut ϕ into parts and suppose that C consists of a single conjugacy class and C′ = C^{G′}. Lemma 3.2 also yields \frac{|C|}{|G|} = \frac{|C′|}{|G′|}. This equation in particular implies that for any Q ∈ C we have N_{G′}(Q) ⊂ G.

In view of Lemma 3.6 (2), we now only have to check that α_{C′} = Ind_{G}^{G′} α_{C}. Set

Ĉ := {⟨g⟩ ⊂ G | α_{C}(g) = 1} = {⟨g⟩ ⊂ G | Ppart(⟨g⟩) ∈ C} and
Ĉ′ := {⟨g′⟩ ⊂ G′ | α_{C′}(g′) = 1} = {⟨g′⟩ ⊂ G | Ppart(⟨g′⟩) ∈ C′} .

We want to understand the relation between Ĉ and Ĉ′. For this, consider the map η: Ĉ′ → C′, Q ↦ Ppart(Q). It maps Ĉ to C. We claim that Ĉ is exactly the preimage of C under η. For this, we have to verify that for any group Q ∈ Ĉ′ with Ppart(Q) ∈ C, we already have Q ⊂ G. Indeed: Q is abelian, so it is contained in N_{G′}(Ppart(Q)), and N_{G′}(Ppart(Q)) is contained in G.

Now using that C consists of a single conjugacy class and that η commutes with conjugation, we arrive at two conclusions: Ĉ′ = Ĉ^{G′} and \frac{|Ĉ′|}{|C′|} = fibersize of η = \frac{|Ĉ|}{|C|}.

Using this, we can finally compute Ind_{G}^{G′} α_{C}. For any g′ ∈ G′, we have

Ind_{G}^{G′} α_{C}(g′) = \frac{1}{|G|}#\{h ∈ G′ | ⟨hg′h^{−1}⟩ ∈ Ĉ\} .

This is zero if ⟨g′⟩ ∉ Ĉ^{G′} = Ĉ′. Otherwise:

⋯ = \frac{1}{|G|} · |Ĉ| · | N_{G′}(⟨g′⟩)| = \frac{|Ĉ|}{|G|} · \frac{|G′|}{|Ĉ′|} = 1 .

(In the last equality, we combine \frac{|C|}{|G|} = \frac{|C′|}{|G′|} and \frac{|Ĉ′|}{|C′|} = \frac{|Ĉ|}{|C|}.) □

4. Other proofs on χ_{c}

In this section we prove the remaining statements on the map χ_{c}, namely the uniqueness statement of Theorem 1.1 in the pseudo-finite case and compatibility of the map χ_{c} with change of the base field k.

4.1. The uniqueness statement. We will use following properties of χ_{c}: it extends the known map χ_{c}: K_{0}(Var_{k}) → K_{0}(Mot_{k}), it is invariant under definable bijections, it is compatible with disjoint unions, and for any Galois cover V ↠^{G} W, the equality

(∗) χ_{c}(X(V ↠^{G} W, \{1\})) = \frac{1}{|G|}χ_{c}(V)

holds.

Note that we will not need that χ_{c} is compatible with products.

Proof of uniqueness in Theorem 1.1. By the almost quantifier elimination (Lemma 2.10) and compatibility with disjoint unions, it is enough to prove uniqueness for definable sets of the form X(V ↠^{G} W, C), where (V ↠^{G} W, C) is a colored Galois cover and C = Q^{G} consists of a single conjugacy class of cyclic subgroups of G.

We proceed by induction on |G| and |Q|. (We will suppose that the statement is true for G of the same size and Q smaller and vice versa.)

Suppose first that Q is not normal in G. Let G′ := N_{G}(Q) be its normalizer

and W′ := V/G′. Note that C′ := Q^{G′} = {Q}. By induction, we know χ_{c}(X(V ↠^{G′} W′, C′)). We have \frac{|G′|}{|C′|} = \frac{|G|}{|C|}, so Lemma 3.2 implies that the map W′ → W induces

a bijection X(V ↠^{G′} W′, C′) → X(V ↠^{G} W, C). So by assumption χ_{c}(X(V ↠^{G}

W, C)) = χ_{c}(X(V ↠^{G′} W′, C′)).

Now suppose Q is normal in G (and in particular C = {Q}). Let G′ := G/Q

and V′ := V/Q. We know χ_{c}(X(V′ ↠^{G′} W, {1})) by (∗), and we have X(V′ ↠^{G′} W, {1}) = X(V ↠^{G} W, C_{1}), where C_{1} = {Q_{1} ∈ Psub(G) | Q_{1} ⊂ Q} consists of all (cyclic) subgroups of G contained in Q. But for any strict subgroup Q_{1} ⊊ Q, we know χ_{c}(X(V ↠^{G} W, Q_{1}^{G})) by induction. So χ_{c}(X(V ↠^{G} W, Q)) is the only (up to now) unknown term in the equation

χ_{c}(X(V ↠^{G} W, C_{1})) = ∑_{C_{2}⊂C_{1} C_{2} one conjugacy class} χ_{c}(X(V ↠^{G} W, C_{2})) .

□

4.2. Change of the base field k. In this section, we prove Proposition 1.2, i.e. that our map χ_{c} is essentially independent of the base field k. So suppose we have a field k′ containing k and remember that we have canonical homomorphisms ⊗_{k}k′: K_{0}(T_{Gal,k}) → K_{0}(T_{Gal,k′}) and ⊗_{k}k′: K_{0}(Mot_{k})_{ℚ} → K_{0}(Mot_{k′})_{ℚ}.

Proof of Proposition 1.2. The claim of the proposition is χ_{c}(X ⊗_{k} k′) = χ_{c}(X)⊗_{k} k′ for any definable set X of T_{Gal,k}. It is enough to check the statement for generators of K_{0}(T_{Gal,k}), i.e. we may suppose that X is of the form X = X(V ↠^{G} W, C) where (V ↠^{G} W, C) is a colored Galois cover defined over k.

By tensoring with k′ and choosing one irreducible component V′ of V ⊗_{k} k′, we get a Galois cover V′ ↠^{G′} W′ over k′, where G′ is a subgroup of G. One easily verifies

X ⊗_{k} k′ = X(V′ ↠^{G′} W′, C′) where C′ := {Q ∈ C | Q ⊂ G′}.

Using Lemma 3.6 we get

χ_{c}(X(V′ ↠^{G′} W′, C′)) = χ_{c}(G′ ↬ V′, α_{C′}) = χ_{c}(G′ ↬ V′, Res_{G′}^{G} α_{C})

= χ_{c}(G ↬ V ⊗_{k} k′, α_{C}) = χ_{c}(G ↬ V, α_{C}) ⊗_{k} k′

= χ_{c}(X(V ↠^{G} W, C)) ⊗_{k} k′ .

□

5. Maps between Grothendieck rings

In this section we first prove the existence of the map θ_{ι} between the different Grothendieck rings K_{0}(T_{Gal,k}) (Theorem 1.3) and then apply this to get Proposition 1.4. Finally we check a compatibility between the maps θ_{ι} and the maps χ_{c}.

5.1. Existence of the maps θ_{ι}. Remember the statement of the theorem. We have a field k and an inclusion of torsion-free pro-cyclic groups ι: Gal_{2} ↪ Gal_{1}. For simplicity, we will now identify Gal_{2} with ι(Gal_{2}) ⊂ Gal_{1}. Denote by T_{i} := T_{Gal_{i},k} the theory of perfect PAC fields with Galois group Gal_{i} and which contain k.

The map θ := θ_{ι}: K_{0}(T_{2}) → K_{0}(T_{1}) was defined as follows. Any model K_{1} of T_{1} yields a model K_{2} := K̃_{1}^{Gal_{2}} of T_{2}. For any X_{2} ⊂ 𝔸^{n} definable in T_{2}, we defined θ(X_{2})(K_{1}) = X_{2}(K_{2}) ∩ K_{1}^{n}.

What we have to check is:

(1) X_{2}(K_{2}) ∩ K_{1}^{n} is definable (uniformly for all K_{1}).

(2) If there is a definable bijection X_{2} → X′_{2} in T_{2}, then there is also a definable bijection θ(X_{2}) → θ(X′_{2}) in T_{1}.

(3) θ is a ring homomorphism, i.e. compatible with disjoint unions and products.

The third statement is clear by definition.

(1) Any definable set X_{2} of T_{2} can be written as disjoint union of sets of the form X(f: V ↠^{G} W, C_{2}), where C_{2} is a conjugacy class of permitted subgroups of G, so it is enough to prove that θ maps such sets to definable ones. We claim: θ(X(V ↠^{G} W, C_{2})) = X(V ↠^{G} W, C_{1}), where C_{1} is defined as follows: Let M be the set of homomorphisms ρ_{1} : Gal_{1} → G such that ρ_{1}(Gal_{2}) ∈ C_{2}. Then C_{1} is the set of images of these homomorphisms M. In a formula:

C_{1} = {im ρ_{1} | ρ_{1} : Gal_{1} → G, ρ_{1}(Gal_{2}) ∈ C_{2}} .

We have to check: For any model K_{1} of T_{1} and any element w ∈ W (K_{1}), we have

w ∈ X(V ↠^{G} W, C_{1})(K_{1}) if and only if w ∈ X(V ↠^{G} W, C_{2})(K_{2}), where K_{2} = K̃_{1}^{Gal_{2}}

as above.

Choose an element v ∈ V (K̃_{1}) with f(v) = w. We get a homomorphism ρ_{1} : Gal_{1} → G defined by σ(v) = v.ρ_{1}(σ) for any σ ∈ Gal_{1}. Of course the restriction ρ_{2} := ρ_{1}↾_{Gal_{2}} satisfies the same property. By definition, we have w ∈ X(V ↠^{G} W, C_{1})(K_{1}) if and only if im ρ_{1} ∈ C_{1} and w ∈ X(V ↠^{G} W, C_{2})(K_{2}) if and only if im ρ_{2} = ρ_{1}(Gal_{2}) ∈ C_{2}. So we have to check that for any ρ_{1} : Gal_{1} → G we have im ρ_{1} ∈ C_{1} if and only if ρ_{1}(Gal_{2}) ∈ C_{2}.

“⇐” is clear by the definition of C_{1}.

“⇒”: Suppose Q_{1} := im ρ_{1} ∈ C_{1}. By the definition of C_{1}, there is a homomorphism ρ′_{1} ∈ M with im ρ′_{1} = Q_{1}. As Gal_{1} is pro-cyclic, homomorphisms Gal_{1} → Q_{1} are determined by the image of a generator, so we can write ρ_{1} = α ∘ ρ′_{1} for some automorphism α ∈ Aut(Q_{1}). As Q_{1} is cyclic, all its subgroups are characteristic subgroups, so ρ_{1}(Gal_{2}) = α(ρ′_{1}(Gal_{2})) = ρ′_{1}(Gal_{2}) ∈ C_{2}. This implies ρ_{1} ∈ C_{1}.

(2) Suppose X_{2} ⊂ 𝔸^{n} and X′_{2} ⊂ 𝔸^{n′} are two definable sets in T_{2} and f: X_{2} → X′_{2} is a definable bijection. We have to show that there is a T_{1}-definable bijection θ(X_{2}) → θ(X′_{2}). Indeed, we will check that θ(f) is such a bijection. In other words we have to verify the following statement:

Let K_{1} be any model of T_{1} and K_{2} = K̃_{1}^{Gal_{2}}. Then for any x ∈ X_{2}(K_{2}) and x′ := f(x) ∈ X′_{2}(K_{2}), we have x ∈ K_{1}^{n} if and only if x′ ∈ K_{1}^{n′}.

Suppose x ∉ K_{1}^{n}. Then there exists a σ ∈ Gal(K_{2}/K_{1}) moving x. But σ(X_{2}(K_{2})) = X_{2}(K_{2}), so σ(x) ∈ X_{2}. As f is injective on X_{2}(K_{2}), this implies σ(f(x)) = f(σ(x)) ≠ f(x), so f(x) ∉ K_{1}^{n′}.

The other direction works analogously. □

5.2. χ_{c} is not injective. As an example application of the maps θ_{ι}, we will now prove Proposition 1.4. To this end, we will construct a pair of definable sets X_{1} and X_{2} such that χ_{c}(X_{1}) = χ_{c}(X_{2}) but χ_{c}(θ_{ι}(X_{1})) ≠ χ_{c}(θ_{ι}(X_{2})) for a suitable map ι: Gal ↪ Gal. (In fact, we will construct a whole bunch of such pairs.)

Proof of Proposition 1.4. Remember that Gal is a non-trivial subgroup of ℤ̂, i.e. Gal = ∏_{p∈P} ℤ_{p}, where P is a non-empty set of primes.

For n ∈ ℕ_{≥1}, consider the group homomorphism ι: Gal ↪ Gal, σ ↦ σ^{n}. Applying Theorem 1.3 to this map gives an endomorphism θ_{n} of K_{0}(T_{ℤ̂,k}), which can be

explicitly computed on sets defined by Galois covers as follows. Let (V ↠^{G} W, C_{2}) be a colored Galois cover. The computation in the proof of Theorem 1.3 shows

that θ_{n}(X(V ↠^{G} W, C_{2})) = X(V ↠^{G} W, C_{1}), where C_{1} = {Q ∈ Psub(G) | Q^{n} ∈ C_{2}}

consists of those permitted subgroups of G whose subgroups of n-th powers lie in C_{2}.

Note that θ_{n} is interesting only if n has prime factors which lie in P; otherwise, n and |Q| are coprime for any permitted subgroup Q ⊂ G, which implies Q = Q^{n}, C_{1} = C_{2}, and θ_{n} = id.

Now let V ↠^{G} W be any non-trivial Galois cover such that all prime factors of |G| lie in P, and define X := X(V ↠^{G} W, {id}). By condition (∗) of Theorem 1.1, we have χ_{c}(X) = \frac{1}{G}χ_{c}(V), so χ_{c}(X × G) = χ_{c}(V). (Here G is interpreted as a discrete set.) However, we will see that for n = |G|, we have χ_{c}(θ_{n}(X × G)) ≠ χ_{c}(θ_{n}(V)).

As θ_{n} is the identity on K_{0}(Var_{k}), we have θ_{n}(V) = [V]. On the other hand, the subgroup of n-th powers of any cyclic subgroup of G is trivial, so θ_{n}(X) = [X(V ↠^{G} W, Psub(G))] = [W] and θ_{n}(X × G) = [W × G]. But V and W × G are two varieties with a different number of irreducible components, so χ_{c}(θ_{n}(X×G)) ≠ χ_{c}(θ_{n}(V)). □

5.3. Compatibility of χ_{c} and θ_{ι}. We prove the following compatibility statement:

Proposition 5.1. Suppose k is a field of characteristic zero and Gal_{2} ⊂ Gal_{1} are two torsion-free pro-cyclic groups such that Gal_{1} / Gal_{2} is torsion-free, too. We use the following notation: T_{i} := T_{Gal_{i},k} (for i = 1, 2) are the corresponding theories, χ_{c}^{i}: K_{0}(T_{i}) → K_{0}(Mot_{k})_{ℚ} are the maps of Theorem 1.1, and θ: K_{0}(T_{2}) → K_{0}(T_{1}) is the map provided by Theorem 1.3 applied to the inclusion Gal_{2} ⊂ Gal_{1}. Then we have:

χ_{c}^{2} = χ_{c}^{1} ∘ θ .

Proof. For i = 1, 2 let P_{i} be the set of primes such that Gal_{i} = ∏_{p∈P_{i}} ℤ_{p}. We have P_{2} ⊂ P_{1}, and Gal_{2} is just the factor of Gal_{1} corresponding to P_{2}. We will write Psub_{i} resp. Ppart_{i} for the permitted subgroups and the permitted part to distinguish between the two Galois groups.

We only have to verify the statement for sets of the form X(V ↠^{G} W, C_{2}), where (V ↠^{G} W, C_{2}) is a colored Galois cover for T_{2}. By the proof of Theorem 1.3, we have θ(X(V ↠^{G} W, C_{2})) = X(V ↠^{G} W, C_{1}), where C_{1} consists of the images of those maps ρ: Gal_{1} → G which satisfy ρ(Gal_{2}) ∈ C_{2}. As Gal_{2} is a direct factor of Gal_{1}, we get C_{1} = {Q ∈ Psub_{1}(G) | Ppart_{2}(Q) ∈ C_{2}}.

Now remember Definition 3.7: χ_{c}^{i}(X(V ↠^{G} W, C_{i})) = χ_{c}(G ↬ V, α_{C_{i}}), where

α_{C_{i}}(g) := \begin{cases}1 & if Ppart_{i}(⟨g⟩) ∈ C_{i} \\ 0 & otherwise. \end{cases}

But Ppart_{1}(⟨g⟩) ∈ C_{1} if and only if Ppart_{2}(Ppart_{1}(⟨g⟩)) = Ppart_{2}(⟨g⟩) ∈ C_{2}, so α_{C_{1}} = α_{C_{2}}, and the claim is proven. □

6. Open problems

6.1. Uniqueness of χ_{c}. In the case of pseudo-finite fields, the conditions given in Theorem 1.1 are enough to render χ_{c} unique. One would like to have a similar uniqueness statement in the other cases. Unfortunately, the condition

(∗) χ_{c}(X(V ↠^{G} W, \{1\})) = \frac{1}{|G|}χ_{c}(V)

is false in general if |G| has prime factors not in P (where Gal = ∏_{p∈P} ℤ_{p}). For

algebraically closed fields for example, we have χ_{c}(X(V ↠^{G} W, {1})) = χ_{c}(W), which is not equal to \frac{1}{|G|}χ_{c}(V) unless G is trivial.

The first question is: is the weak version of (∗) (when one requires all prime factors of |G| to lie in P) enough to get uniqueness? And if not: is there some other nice condition rendering χ_{c} unique? One fact suggesting that the weak condition could already be strong enough is that this is true indeed for algebraically closed fields.

6.2. From motives to measure. The parallels between the definitions of the virtual motive associated to a definable set and the measure of such a set ([2], [9]) suggest that one should be able to extract the measure from the motive. More precisely, fix a perfect PAC field K of characteristic zero with pro-cyclic Galois group Gal. Note that there are two theories around now: T_{Gal,K}, the theory of pseudo-finite fields containing K (which is not complete) and Th(K), the (complete) theory of K itself.

Denote by dim: K_{0}(Th(K)) → ℕ the dimension of [3] (which needs not coincide with the usual dimension for varieties: only components “visible over K” are considered) and by μ: K_{0}(Th(K)) → ℚ the measure of [9]. The question is whether a dotted map in the following diagram exists making the diagram commutative.

K_{0}(T_{K})  K_{0}(Th(K))
K_{0}(Mot_{K})_{ℚ}  ℕ × ℚ

χ_{c} (dim, μ)

If K is algebraically closed, then this is obviously true: In this case μ(V) is just the number of irreducible components of V, and both this and the dimension of V (which is the usual one in this case) can be seen in the corresponding motive.

If K is pseudo-finite, this is true, too: Let X be a definable set of T_{Gal,K}. Then it makes sense to speak about X(F) for finite fields F of almost all characteristics.

Lemma 3.3.2 of [4] states that for almost all characteristics, the number of points |X(F)| is encoded in the motive. (Not very surprisingly, it is the trace of the Frobenius automorphism on the motive.) The dimension and the measure of X in K can be computed from these cardinalities.

The way one extracts the dimension and the measure from the motive seems quite different in the two above cases. This suggests that one might get interesting new insights by generalizing this to arbitrary procyclic Galois groups.

6.3. Larger Galois groups for the maps θ_{ι}. The quantifier elimination result of [6] does not only work for fields with pro-cyclic Galois groups, but for some larger Galois groups as well. (The Galois group has to satisfy what Fried-Jarden call the “embedding property”.) It seems plausible that Theorem 1.3 should be generalizable to this context as well. However the proof will need some modifications. Indeed for Gal_{1} = ℤ̂ ∗ ℤ̂ = ⟨a, b⟩ and Gal_{2} = ⟨a⟩ ⊂ Gal_{1}, one can construct a T_{2}-definable set X = X(V ↠^{G} W, C) such that θ(X) is not definable using the same Galois cover V ↠^{G} W.

6.4. Larger Galois groups for the maps χ_{c}. Another natural question is whether the map χ_{c} can also be defined for fields with larger Galois group. However, in [9] we already showed that the measure of [2] does not extend to this generality. Indeed, no measure exists for example if the Galois group is ℤ̂ ∗ ℤ̂. This suggests that it is neither possible to associate motives to definable sets of such theories. Probably, T_{ℤ̂∗ℤ̂,k} contains too many definable bijections so that the corresponding Grothendieck ring gets too small. One might even hope to show that K_{0}(T_{ℤ̂∗ℤ̂,k}) is trivial.

6.5. What exactly do we know about K_{0}(T_{Gal,k})? We showed that the maps χ_{c} do not yield the full information about the definable sets and we showed how additional information can be obtained using the maps θ_{ι}. The question is now: do we get all additive information using both maps together? More precisely, a strong version of this would be:

Question 6.1. Suppose X_{1} and X_{2} are two definable sets in T_{Gal,k}, and suppose that for any injective endomorphism ι: Gal ↪ Gal we have χ_{c}(θ_{ι}(X_{1})) = χ_{c}(θ_{ι}(X_{2})). Does this imply [X_{1}] = [X_{2}] in K_{0}(T_{Gal,k})?

This is of course a very difficult question (and I am not so sure if I really believe that the answer is positive), as in particular it would imply that χ_{c} is injective for algebraically closed fields. In fact the question I would prefer to ask is: is Question 6.1 true “up to varieties”, or, “up to whatever is missing in the case of algebraically closed fields”? The first open problem here is to give a precise meaning to this last question.

References

[1] S. del Baño Rollin & V. Navarro Aznar, “On the motive of a quotient variety”, Collect. Math. 49 (1998), no. 2-3, p. 203-226, Dedicated to the memory of Fernando Serrano.

[2] Z. Chatzidakis, L. van den Dries & A. Macintyre, “Definable sets over finite fields”, J. Reine Angew. Math. 427 (1992), p. 107-135.

[3] Z. Chatzidakis & E. Hrushovski, “Perfect pseudo-algebraically closed fields are algebraically bounded”, J. Algebra 271 (2004), no. 2, p. 627-637.

[4] J. Denef & F. Loeser, “Definable sets, motives and p-adic integrals”, J. Amer. Math. Soc. 14 (2001), no. 2, p. 429-469 (electronic).

[5] ——— , “On some rational generating series occurring in arithmetic geometry”, in Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, p. 509-526.

[6] M. D. Fried & M. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986, xviii+458 pages.

[7] H. Gillet & C. Soulé, “Descent, motives and K-theory”, J. Reine Angew. Math. 478 (1996), p. 127-176.

[8] F. Guillén & V. N. Aznar, “Un critère d’extension d’un foncteur défini sur les schémas lisses”, IHES Publ. Math. (2002), no. 95, p. 1-91.

[9] I. Halupczok, “A measure for perfect PAC fields with pro-cyclic Galois group.”, J. Algebra (2007), no. 310, p. 371-395.

[10] E. Hrushovski, “Pseudo-finite fields and related structures”, in Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, p. 151-212.

[11] A. J. Scholl, “Classical motives”, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, p. 163-187.

[12] J.-P. Serre, “Zeta and L functions”, in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, p. 82-92.

DMA, Ecole Normale Supérieure, 45, rue d’Ulm, 75230 Paris Cedex 05 – France

E-mail address: math@karimmi.de