A survey on the model theory of difference fields Notes on lectures given at MSRI (Berkeley, USA), January 98 Zoé Chatzidakis, CNRS - Paris 7

Introduction. Difference field are fields with a distinguished automorphism σ. They were first studied by Ritt in the 30’s. A good reference for the algebraic results is Cohn’s book [C]. Interest in the model theory of difference fields started at the end of the eighties, particularly during the MSRI logic year, because of two questions.

The first question stemmed from the failure of Zil’ber’s conjecture: there is a strongly minimal theory extending the theory of algebraically closed fields of any given characteristic. People were looking at the possibility of finding a non-definable automorphism σ of 𝔽_{p}^{alg} (the algebraic closure of the field 𝔽_{p} with p elements), such that Th(𝔽_{p}^{alg}, +, ·, σ) is strongly minimal. This question so far remains open.

The second problem had to do with the difference fields F_{q} = (𝔽_{p}^{alg}, +, ·, ϕ_{q}), where q is a power of p and ϕ_{q} : x ↦ x^{q} is a power of the Frobenius automorphism x ↦ x^{p}. The hope was to generalise the work of Ax on finite fields to these structures, and in particular to describe the theory of the non-principal ultraproducts of the difference fields F_{q}.

These questions led Macintyre, Van den Dries and Wood to look for a model companion of the theory of difference fields, and to prove various results (decidability, description of the completions, etc …) for this theory, henceforth called ACFA. For details and attribution of results, see Macintyre’s paper [M95]. I should also mention that the second problem was solved recently, by Hrushovski [H96] and Macintyre [M97], showing that non-principal ultraproducts of F_{q}’s are models of ACFA.

In 94, Hrushovski and myself started looking at stability-type properties of the theory ACFA. Our main result is a dichotomy result for types of rank 1 for models of characteristic 0, which was later partially extended to the case of positive characteristic with the help of Peterzil (see [CH] and [CHP]). It has some applications to the description of types of finite rank, and to groups definable in models of ACFA. These results were used by Hrushovski to find explicit bounds in the Manin-Mumford conjecture, see [H95].

The first four chapters of the paper gives a survey of the results obtained to-date for difference fields. In chapter 5 we state the results used by Hrushovski in his proof of the Manin-Mumford conjecture over a number field, and show how he effectively derives from them the bounds. In the last chapter we conclude with the statements of some other applications due to Hrushovski and Scanlon.

I take this opportunity to thank MSRI for their support during the special semester on the Model Theory of Fields, and to show my appreciation for the congenial atmosphere. I would also like to thank D. Haskell and E. Hrushovski for many improvements in this paper.

1. Description and elementary results on the theory ACFA

We work in the language L = {+, −, ·, 0, 1, σ}, where +, −, · are the usual ring operations, 0 and 1 are constants, and σ is a unary function.

(1.1) Some examples.

(1) The shift operator. Consider the field K = ℂ(t), and define σ by

σ|_{ℂ} = id, σ(t) = t + 1.

The name “difference field” originated from this example: an equation of the form P (f(t), f(t + 1), …, f(t + n)) = 0, where f is the unknown function to be found and P is a polynomial over K, is called an algebraic difference equation. One can replace K by other fields, e.g., the field of meromorphic functions on ℂ or on ℝ.

(2) Let K be a field, K^{s} its separable closure and σ ∈ Gal(K^{s}/K). Then (K^{s}, σ) is a difference field. Note that because the algebraic closure K^{alg} of K is purely inseparable over K^{s}, σ extends uniquely to an automorphism of K^{alg}. One often identifies Gal(K^{s}/K) and Aut(K^{alg}/K).

The structures F_{q} described above are a particular example. More generally, we have:

(3) Let K be a perfect field of characteristic p > 0, and q a power of p. Then (K, ϕ_{q}) is a difference field. If the field K is algebraically closed then (𝔽_{p}^{alg}, ϕ_{q}) ≺ (K, ϕ_{q}). This is because for fixed q the map x ↦ x^{q} is definable in the language of fields, and because the theory of algebraically closed fields is model complete.

(1.2) Definitions, notation and some basic algebraic results. In the literature, a difference field is a field K with a distinguished monomorphism σ. If σ is onto, then (K, σ) is called an inversive difference field. However, a simple inductive limit argument shows that every difference field has a unique (up to isomorphism) inversive closure. We will assume in what follows, that all our difference fields are inversive. The references are to Cohn’s book [C].

Let K be a difference field, and let X̄ = (X_{1}, …, X_{n}) be indeterminates. A difference polynomial over K in X_{1}, …, X_{n} is an ordinary polynomial with coefficients in K, in the variables X_{1}, …, X_{n}, σ(X_{1}), …, σ^{i}(X_{j}), . . .. The ring of those difference polynomials is denoted K[X_{1}, …, X_{n}]_{σ}, and σ extends naturally to K[X_{1}, …, X_{n}]_{σ}, in the way suggested by the names of the variables. Note: As defined, σ is not onto. It is sometimes convenient to consider the inversive closure of this ring, namely K[σ^{i}(X_{1}), …, σ^{i}(X_{n})]_{i∈ℤ}, but we will not do this here.

There is a natural notion of σ-ideal, i.e., an ideal closed under σ, and of reflexive σ-ideal (a ∈ I ⟺ σ(a) ∈ I). The analog of a radical ideal is called a perfect σ-ideal: a σ-ideal I is perfect if a ∈ I whenever a^{j}σ^{i}(a) ∈ I for some i, j ∈ ℕ. A prime σ-ideal is a reflexive σ-ideal which is prime. Note that a prime σ-ideal is perfect. K[X_{1}, …, X_{n}]_{σ} does not satisfy the ascending chain condition on σ-ideals; however it satisfies it for perfect σ-ideals, and therefore for prime σ-ideals. This allows one to define σ-closed sets and σvarieties (also called irreducible σ-closed sets) in affine n-spaces. They correspond dually to perfect σ-ideals and prime σ-ideals, and are the basic closed sets of a noetherian topology.

Let K be a difference field, a a tuple of elements (in some difference field extending K). We denote by K(a)_{σ} the difference field generated by a over K, by acl_{σ}(Ka) its algebraic closure, and by deg_{σ}(a/K) the transcendence degree of K(a)_{σ} over K. If a is a single element and deg_{σ}(a/K) is infinite, then a is called transformally transcendental. The elements σ^{j}(a), j ∈ ℤ, are then algebraically independent over K. If deg_{σ}(a/K) is finite, then a is called transformally algebraic. There are natural notions of transformal transcendence basis and transformal dimension.

(1.3) An axiomatisation of the theory ACFA.

Consider the theory ACFA, whose models are the L-structures K satisfying:

(i) K is an algebraically closed field, (ii) σ ∈ Aut(K), (iii) if U and V are (affine) varieties defined over K, with V ⊆ U × σ(U) projecting

generically onto U and σ(U), then there is a tuple a in K such that (a, σ(a)) ∈ V. Here, by a variety, we mean an absolutely irreducible Zariski closed set, i.e., a set defined by polynomial equations, and which is not the proper union of two smaller Zariski closed sets. The set σ(U) is the variety obtained from U by applying σ to the coefficients of the defining polynomials of U. When we say that V projects generically onto U, we mean that the image of V under the natural projection U × σ(U) → U is Zariski dense in U (i.e., not contained in any proper Zariski closed subset). Note that (iii) is indeed a conjunction of first-order sentences, since (by classical results on polynomial rings over fields) the fact that polynomials f_{1}(X̄), …, f_{n}(X̄) generate a prime ideal of K[X̄] is an elementary condition on the coefficients of f_{1}, …, f_{n}. Similarly for the inclusion of ideals in K[X̄].

Theorem. ACFA is the model companion of the theory of difference fields. Proof. (Sketch) We first need to show that every difference field embeds in a model of ACFA. Axioms (i) and (ii) pose no problem, as every automorphism of a field extends to its algebraic closure. Let U and V be as in (iii). Choose a generic point (a, b) of V over K (i.e., the ideal of polynomials over K vanishing at (a, b) is exactly the ideal of polynomials vanishing at all points of V), in some field containing K. Then a is a generic of U, and b is a generic of σ(U). By elementary properties of algebraically closed fields, the isomorphism τ : K(a) → K(b) which extends σ and sends a to b, extends to an automorphism of the algebraic closure of K(a, b).

This shows that every difference field embeds in a model of ACFA. It remains to show that the models of ACFA are existentially closed. Let (K, σ) ⊧ ACFA, let φ(x), x a tuple of variables, be a quantifier-free formula with parameters in K, and assume that φ(x) has a solution in some difference field (L, σ) extending K. The usual trick of replacing the inequality y ≠ 0 by ∃z yz − 1 = 0, shows that one can assume that φ(x) is a conjunction of σ-equations. Let a ∈ L satisfy φ. For n large enough, the σ-ideal I generated by the set

{f(X, σ(X), …, σ^{n}(X)) | f(Y, Y_{1}, …, Y_{n}) ∈ K[Y, Y_{1}, …, Y_{n}], f(a, σ(a), …, σ^{n}(a)) = 0}

is precisely the prime σ-ideal of difference polynomials over K annulled by a. Thus any point satisfying these equations will satisfy φ(x).

Let U be the variety defined over K with generic (a, σ(a), …, σ^{n−1}(a)), and V the variety defined over K with generic (a, σ(a), …, σ^{n−1}(a), σ(a), …, σ^{n}(a)). Then U and V satisfy the hypotheses of axiom (iii), and therefore there is a tuple b in K such that (b, σ(b)) ∈ V. Then b = (c, σ(c), …, σ^{n−1}(c)) for some c, and K ⊧ φ(c).

(1.4) The Frobenius automorphisms. Before continuing with the elementary properties of ACFA, we will state precisely the result of Hrushovski, from which follows that non-principal ultraproducts of F_{q}’s are models of ACFA. It is then a consequence of Tchebotarev’s theorem on the distribution of primes that ACFA is exactly the theory of all non-principal ultraproducts of F_{q}’s, see [M95].

Theorem ([H96]). Let U, V be varieties with V ⊆ U × σ(U), and assume that the projections are onto and have finite fibers. Let d_{1} = [K(V) : K(U)], d_{2} = [K(V) : K(σ(U))]_{i} (purely inseparable degree); let c = d_{1}/d_{2} and d = dim(V). Then for some constant C > 0, depending on the two varieties U and V, and which remains bounded when U and V move inside an algebraic family of varieties,

|Card({a ∈ (𝔽_{p}^{alg})^{n} | (a, a^{q}) ∈ V }) − cq^{d}| ≤ Cq^{d−1/2}.

(1.5) Proposition. If (K, σ) ⊧ ACFA, then the subfield F ix(σ) of K fixed by σ is a pseudo-finite field. Proof. By Ax’s results [A], one needs to show that: F ix(σ) is perfect; F ix(σ) has exactly one algebraic extension of each degree; every (absolutely irreducible) variety defined over F ix(σ) has an F ix(σ)-rational point.

The first assertion is obvious, and the third one follows easily from axiom (iii). For the second assertion, it suffices to show that that for each n > 1, the system

σ^{n}(x) = x, σ^{j}(x) ≠ x, j = 1, …, n − 1,

has a solution in K. Since K is existentially closed, it suffices to find a difference field extending K in which this system has a solution. Consider the field K(X_{1}, …, X_{n}) in n indeterminates, and extend σ by defining σ(X_{j}) = X_{j+1} for j < n and σ(X_{n}) = X_{1}. Then X_{1} is a solution of the system.

In characteristic p > 0 one shows similarly that if m ≠ 0 and n are integers, then the set of elements of K satisfying σ^{m}(x) = x^{p^{n}} is a pseudo-finite field.

(1.6) It turns out that many of the proofs given by J. Ax [A] for pseudo-finite fields generalise to models of ACFA. The results (1) – (5) appear in [M95].

Proposition.

(1) Let (K_{1}, σ_{1}) and (K_{2}, σ_{2}) be models of ACFA, and let E be a common difference subfield. Then

(K_{1}, σ_{1}) ≡_{E} (K_{2}, σ_{2}) ⟺ (E^{alg}, σ_{1}|_{E^{alg}}) ≃_{E} (E^{alg}, σ_{2}|_{E^{alg}}).

(2) From the above one deduces immediately that the completions of ACFA are obtained by describing the action of σ on the algebraic closure of the prime field (ℚ^{alg} or 𝔽_{p}^{alg}). This then entails the decidability of the theory ACFA, as well as of its extensions ACFA_{0} and ACFA_{p} obtained by specifying the characteristic of the field.

(3) It also gives a description of the types. Let E be a difference field, a and b two tuples from a model K of ACFA containing E. Then tp(a/E) = tp(b/E) if and only if there is an isomorphism φ from the difference field acl_{σ}(Ea) =_{def} E(a)_{σ}^{alg} onto the difference field acl_{σ}(Eb) which is the identity on E and sends a to b.

(4) If E is an algebraically closed difference field, then ACFA ∪ qftp(E) ⊢ tp(E), where qftp(E) denotes the quantifier-free type of E.

(5) The algebraic closure (in the model-theoretic sense) of a set A coincides with the algebraic closure (in the ordinary field sense) of the difference field generated by A (which we denote by acl_{σ}(A)).

(6) Let K ⊧ ACFA, let U be a variety, ℓ ≥ 1, and V a subvariety of U × σ(U) × ⋯ × σ^{ℓ}(U). Let π_{1} : U × σ(U) × ⋯ × σ^{ℓ}(U) → U × σ(U) × ⋯ × σ^{ℓ−1}(U) and π_{2} : U × σ(U) × ⋯ × σ^{ℓ}(U) → σ(U) × ⋯ × σ^{ℓ}(U) be the two canonical projections, and assume that σπ_{1}(V) and π_{2}(V) have the same generics. Then the set of points Ṽ = {x ∈ U(K) | (x, σ(x), …, σ^{ℓ}(x)) ∈ V } is Zariski dense in U.

(7) If (K, σ) ⊧ ACFA and m ≥ 1, then (K, σ^{m}) ⊧ ACFA.

Proof. (1) The left to right implication is almost immediate. For the other one, moving K_{2} by some E-isomorphism, we may assume that E = E^{alg} and that K_{1} and K_{2} are linearly disjoint over E. This implies that the ring K_{1} ⨂_{E} K_{2} is a domain. Define σ(a ⊗ b) = σ_{1}(a) ⊗ σ_{2}(b) for a ∈ K_{1} and b ∈ K_{2}; then σ extends to an automorphism of the quotient field L of K_{1} ⨂_{E} K_{2}, which agrees with σ_{1} on K_{1} and σ_{2} on K_{2}. Now, (L, σ) embeds in a model (M, σ) of ACFA, and by model-completeness we have (K_{1}, σ_{1}) ≺ (M, σ), (K_{2}, σ_{2}) ≺ (M, σ).

The first part of (2), (3) and (4) are immediate, applying compactness to (1). The decidability follows from the recursive axiomatisation of ACFA, together with the effective computability of Galois groups of the splitting fields over ℚ and 𝔽_{p} of polynomials of ℤ[T].

(5) Let A = acl_{σ}(A) ⊆ K ⊧ ACFA and b ∈ K \ A, B = acl_{σ}(Ab); let B_{1} be an A-isomorphic copy of B, linearly disjoint over A. As in (1), there is a model of ACFA containing the difference fields B and B_{1}. By (3), tp(B_{1}/A) = tp(B/A), which shows that tp(b/A) is not algebraic.

(6) We may assume that U and V are affine. Let W ⊆ U × σ(U) × ⋯ × σ^{ℓ−1}(U) be the Zariski closure of π_{1}(V). By assumption, π_{2}(V) is Zariski dense in σ(W), and we may therefore assume that ℓ = 1. The proof that every difference field embeds in a model of ACFA shows that if K is sufficiently saturated, then K contains a point a such that (a, σ(a)) is a generic of V. This shows that Ṽ is dense in U.

2. Independence and rank

(2.1) Definition of independence. Let A, B and C be subsets of a model K of ACFA. We say that A and B are independent over C (A | ⌣_{C}B), if acl_{σ}(CA) and acl_{σ}(CB) are linearly disjoint over acl_{σ}(C). This notion has all the usual properties of independence in algebraically closed fields. Recall that by (1.6)(5), acl_{σ}(A) is the model-theoretic algebraic closure of the set A in the model K.

(2.2) Definition of the SU-rank. We define a rank based on independence in the usual way, that is, for p a type over E, realised by a tuple a:

— SU(p) = SU(a/E) ≥ 0,

— SU(p) ≥ α for α a limit ordinal, if and only if SU(p) ≥ β for every β < α,

— SU(p) ≥ α + 1 if and only if there is F ⊇ E such that a | ⌣ / _{E}F and SU(a/F) ≥ α.

Then, SU(p) is the least ordinal α such that SU(p) ≱ α + 1. If φ(x) is a formula with parameters in E = acl_{σ}(E), one also defines SU(φ) = max{SU(a/E) | a satisfies φ}.

(2.3) The SU-rank shares the properties of the usual U-rank, and in particular, the Lascar rank inequality: if a, b are tuples and E a set, then SU(a/Eb) + SU(b/E) ≤ SU(a, b/E) ≤ SU(a/Eb) ⊕ SU(b/E), where ⊕ denotes the natural sum on ordinal numbers. (Recall that 1 + ω = ω, while 1 ⊕ ω = ω + 1.)

(2.4) Some examples. Let E be a difference subfield of a model K of ACFA and a a tuple in K. From the definition of the SU-rank, it is clear that

— SU(a/E) = 0 ⟺ a ∈ acl_{σ}(E).

— SU(a/E) = 1 ⟺ a ∉ acl_{σ}(E) and for every F ⊇ E, either a | ⌣_{E}F or a ∈ acl_{σ}(F). Earlier we defined deg_{σ}(a/E), which is also an invariant of tp(a/E). It has some

relation with SU-rank, since independence is defined in terms of non-forking in algebraically closed fields. For instance, one has, for E ⊆ F difference fields and a a tuple with deg_{σ}(a/E) < ∞,

a | ⌣ / _{E}F ⟺ deg_{σ}(a/F) < deg_{σ}(a/E),

and this implies

SU(a/E) ≤ deg_{σ}(a/E).

Thus in particular, every non-algebraic type containing the equation σ(x) = x^{2} + 1 has SU-rank 1. This inequality can be strict, see the example in (2.6) below.

(2.5) One can also show that the SU-rank of an element transformally transcendental over the difference field E is ω: let a be such an element, and consider the sequence (b_{i}), i ∈ ℕ, defined by b_{0} = a, b_{i+1} = σ(b_{i}) − b_{i}. Then the fields L_{i} = E(b_{i})_{σ} form a decreasing sequence of subfields of E(a)_{σ}, with tr.deg(L_{i}/L_{i+1}) = 1. By additivity of rank, we obtain SU(a/L_{i}) = i, which implies that SU(a/E) ≥ ω. On the other hand, SU(a/E) ≱ ω + 1: if a | ⌣ / _{E}F then deg_{σ}(a/F) < ∞, which implies that SU(a/F) < ω. Hence SU(a/E) = ω.

Note that this gives an example of the left equality in (2.3): SU(a/Eb_{1}) = 1, SU(b_{1}/E) = ω, and SU(a/E) = ω. For a tuple b in K this also yields: SU(b/E) < ω ⟺ deg_{σ}(b/E) < ∞.

(2.6) Consider the formula φ(x) : σ^{−2}(x) = x^{2} + 1 (char. ≠ 2). Then SU(φ) = 1. Proof. By (2.4), we want to show that if E is any difference field and a is any solution of σ^{−2}(x) = x^{2} + 1, then either a ∈ acl_{σ}(E), or a | ⌣E, i.e.: deg_{σ}(a/E) = 2. Let E = acl_{σ}(E) be a field, and a a realisation of φ, a ∉ E. We need to show that deg_{σ}(a/E) = 2. Since E is an arbitrary algebraically closed difference field, this will imply: if F = acl_{σ}(F) contains E and tp(a/F) forks over E, then a ∈ F, and therefore that SU(φ) = 1.

Suppose by way of contradiction that deg_{σ}(a/E) = 1, and let K = E(a, σ(a)), m = [K : E(a)] and n = [K : E(σ(a))]. Observe that K contains all σ^{−j}(a) for j ≥ 0 (because σ^{−2j}(a) ∈ E(a)). Since E(σ^{2}(a)) is a Galois extension of E(a), we have that [K(σ^{2}(a)) : K] divides [E(σ^{2}(a)) : E(a)] = 2.

Assume first that [K(σ^{2}(a)) : K] = 1. Then σ(K) = K, which implies that K = E(a)_{σ}. On the other hand, E(a)_{σ} contains the infinite algebraic extension E(σ^{2j}(a))_{j∈ℕ} of E(a), which gives us a contradiction.

Thus [K(σ^{2}(a)) : K] = 2, and therefore E(σ^{2}(a)) ∩ K = E(a). So we have:

\begin{aligned}[E(σ(a), σ^{2}(a)) : E(a)] & = [E(σ(a), σ^{2}(a)) : K][K : E(a)] = 2m \\ & = [E(σ(a), σ^{2}(a)) : E(σ^{2}(a))][E(σ^{2}(a)) : E(a)] = 2n \end{aligned}

since [E(σ(a), σ^{2}(a)) : E(σ^{2}(a))] = [K : E(σ(a))]. This implies m = n. On the other hand,

\begin{aligned}[E(σ(a), σ^{2}(a)) : E(σ(a))] & = [K : E(a)] = m \\ & = [E(σ(a), σ^{2}(a)) : K][K : E(σ(a))] = 2n \end{aligned}

which gives m = 2n and the desired contradiction.

(2.7) The independence theorem. Let E = acl_{σ}(E) ⊆ K, let a, b, c_{1} and c_{2} be tuples from K such that a, b, c_{1} and c_{2} are independent over E and tp(c_{1}/E) = tp(c_{2}/E). Then there is c (in some elementary extension of K) independent from (a, b) over K, and realising tp(c_{1}/acl_{σ}(Ea)) ∪ tp(c_{2}/acl_{σ}(Eb)).

A generalised version of this theorem holds: let n ≥ 3, let x_{1}, …, x_{n} be tuples of variables, and let W be a set of proper subsets of {1, …, n} closed under intersection. Assume that for each w ∈ W we are given a complete type p_{w}(x_{w}) over E = acl_{σ}(E), in the variables x_{w} = {x_{i} | i ∈ w}, which can be realised by some (a_{i} | i ∈ w) such that the elements a_{i}, i ∈ w, are independent over E (i.e., for each j ∈ w, the tuple a_{j} is independent from the set {a_{i} | i ∈ w, i ≠ j} over E). Assume moreover that if v ⊂ w are in W then p_{v}(x_{v}) ⊂ p_{w}(x_{w}). Then the type ⋃_{w∈W} p_{w}(x_{w}) can be realised by some tuple a_{1}, …, a_{n}, with a_{1}, …, a_{n} independent over E. The independence theorem corresponds to the case n = 3, W = {{1, 2}, {1, 3}, {2, 3}}.

(2.8) Independence and non-forking. Using the independence theorem, one proves that independence as defined above coincides with the usual notion of non-forking. Namely, assume that a and F are independent over E = acl_{σ}(E), and let p(x) = tp(a/F). Assume that (F_{i})_{i∈ℕ} is an E-indiscernible sequence of realisations of tp(F/E), and let p_{i}(x) be the type over F_{i} which is the image of p(x) by an E-automorphism mapping F to F_{i}. Then ∪_{i}p_{i}(x) is consistent. Thus any completion of ACFA is simple in the sense of Shelah [Sh].

The connections between the independence theorem and simplicity were first observed by Hrushovski in the context of pseudofinite fields (and more generally bounded PAC fields), see [H91], and also [HP94]. The case n > 3 of the generalised independence theorem goes beyond simplicity, and its model theoretic meaning remains to be clarified.

Recall that any PAC field which is not separably closed is unstable by a result of Duret [Du]. Hrushovski recognised the usefulness of the independence theorem for studying definable groups and generalising the techniques of stability theory to the context of pseudofinite fields, and more generally, to models of S_{1}-theories (an S_{1}-theory has finite SU-rank and some definability property of the SU-rank).

The independence theorem is indeed a good substitute for “uniqueness of non-forking extension” which is true in stable theories, and allows one to generalise the concepts of generic type of a group and of stabilisers of types to groups definable in finite fields, and later, to groups definable in models of ACFA.

The independence theorem was later generalised by B. Kim and A. Pillay to Lascar types in simple theories [KP]. Moreover their result gives a nice characterisation of nonforking. The results on definable groups were also generalised to the context of simple theories, see [P98] and [Wa].

(2.9) The independence theorem is also used in the proof of the following two results: Let K be a model of ACFA.

(1) Th(K) has elimination of imaginaries.

(2) Let S ⊆ F ix(σ)^{n} be definable in K. Then S is definable in the pure field F ix(σ) (maybe with additional parameters from F ix(σ)).

(2.10) Groups of finite SU-rank. Using the techniques developed in [HP94] and [HP95], one obtains for instance a generalisation of a well-known result of algebraic geometry:

Proposition. Let G be a group of finite SU-rank defined over a model K of ACFA, and let {X(i) | i ∈ I} be a family of definable subsets of G. There is a definable group H contained in the subgroup of G generated by the X(i), i ∈ I, such that X(i)H/H is finite for every i ∈ I.

Note that no uniformity is assumed in the family X(i), just that each of them is definable by some formula. The proof gives more information. Without loss of generality we will assume that for every i ∈ I there is j ∈ I such that X(j) = X(i)^{−1}.

(1) There are elements i_{1}, …, i_{n} ∈ I such that H ⊆ X(i_{1}) ⋯ X(i_{n}) and n ≤ 2SU(G).

(2) Assume that G is a subgroup of an algebraic group defined over K, that G and the sets X(i) are irreducible σ-closed sets, and that the identity element of the group belongs to all X(i)’s. Then H is the subgroup generated by all the X(i)’s, and the number n in (1) is ≤ SU(G).

(2.11) Finite simple groups. One can then use Hrushovski’s result on the Frobenius automorphisms ϕ_{q} (1.4) to get information about certain classes of finite simple groups. With the exception of the sporadic groups and the alternating groups, finite simple groups are defined in terms of algebraic groups, and form families (e.g., P SL_{n}(𝔽_{q}) for fixed n and q ranging over prime powers). All but the Suzuki and Ree families are already definable over finite fields in the language of fields {+, −, ·, 0, 1}. The Suzuki and Ree families become uniformly definable in the structures F_{p^{m+1}}, p = 2 or 3, as m varies over the positive integers. Indeed, these groups are defined as follows: we have some algebraic group G (in the family B_{2}, G_{2} or F_{4}), and an algebraic automorphism φ of G whose square induces the Frobenius map ϕ_{p} on G(𝔽_{p}^{alg}). Then the subgroup^{2}G(p^{2m+1}) is the subgroup of G(𝔽_{p^{2m+1}}) left fixed by φϕ_{p^{m+1}}^{−1} (see [Su] p. 388). This implies that:

2G(p^{2m+1}) is the subgroup of G(𝔽_{p}^{alg}) defined by the equation σ(g) = φ(g) in the structure F_{p^{m+1}}.

The results in the previous paragraph apply and give for instance: if m is large enough, then any non-trivial conjugacy class of ^{2}G(p^{2m+1}) generates the whole group in at most dim(G) + 1 steps (dim(B_{2}) = 10, dim(G_{2}) = 14, dim(F_{4}) = 52).

(2.12) We will conclude this chapter with:

Proposition. Let G be a group of finite SU-rank defined over a model K of ACFA. There is an algebraic group H defined over K, and a definable group homomorphism f from some definable subgroup G_{0} of G of finite index in G into H(K), with ker(f) finite central.

Note that f(G_{0}) has infinite index in H(K), since the latter has SU-rank ωdim(H). However, if H_{0} is the smallest quantifier-free definable subgroup of H containing f(G_{0}), then f(G_{0}) has finite index in H_{0}, and SU(G) = SU(H_{0}).

3. Study of types of finite rank In this chapter we will study types of finite SU-rank. First a reduction to types of

SU-rank 1:

(3.1) Proposition. Let E = acl_{σ}(E) and a a tuple with 0 < SU(a/E) < ω. Then there is a tuple b independent from a over E, and an element c ∈ acl_{σ}(Eab) \ acl_{σ}(Eb) such that SU(c/Eb) = 1.

(3.2) Orthogonality. Recall that two types p and q are orthogonal (denoted by ⊥), if for every set E containing the sets over which p and q are defined, if a and b realise non-forking extensions of p and q respectively to E, then a | ⌣_{E}b. A type is orthogonal to a formula if it is orthogonal to any type containing this formula. Two formulas φ(x) and ψ(y) are orthogonal if and only if, for every F = acl_{σ}(F) containing the parameters needed to define φ and ψ, and any tuples a and b satisfying φ and ψ respectively, a and b are independent over F.

Rephrased in terms of orthogonality, Proposition 3.1 says: every type of finite SU-rank is non-orthogonal to a type of SU-rank 1.

(3.3) Modularity. Let E = acl_{σ}(E) be a subset of the model K of ACFA, and let R ⊆ K^{n} be the set of realisations of a set of types over E (so, for instance, a subset of K^{n} which is definable over E). We say that R is modular (over E) if and only if any two subsets A and B of R are independent over acl_{σ}(EA) ∩ acl_{σ}(EB). We say that a (maybe incomplete) type over E, or a formula, is modular (over E) if its set of realisations is modular over E.

Remarks and comments.

(1) The definition of modularity first appears in an unstable context in a paper by Cherlin and Hrushovski [ChH], where it is given in terms of acl^{eq}. This agrees with our definition because ACFA eliminates imaginaries. This notion of modularity generalises several notions introduced in the eighties: locally modular, one-based, module-like. All three were defined in a stable context, and some required the underlying set to be a set of realisations of rank 1 types, or of regular types.

(2) It suffices to check modularity for finite sets A and B.

(3) A modular set satisfies the stronger property: if A ⊆ R and B ⊆ K, then A and B are independent over acl_{σ}(EA) ∩ acl_{σ}(EB).

(4) The set of realisations of (a set of) modular types of SU-rank 1 is modular. A subset of a modular set is modular. Any trivial type of SU-rank 1 is modular (a type p over E is trivial if acl_{σ}(EA) = ⋃_{a∈A} acl_{σ}(Ea) for any set A of realisations of p).

(5) Assume that the elements of R have SU-rank 1 over E. Then the modularity of R can be rephrased as follows: there is no rank-2 family of definable curves on R^{2}.

(6) If p and q are non-orthogonal types of SU-rank 1, and if p is modular then so is q.

(7) Assume that R is the set of realisations of a type of SU-rank 1 over E, and that R is modular and stable, stably embedded. Then R satisfies the stronger property: any two subsets A and B of R are independent over acl_{σ}(EA)∩acl_{σ}(EB)∩R, provided this intersection is non-empty. This coincides with the classical notion of local modularity known to model theorists.

(3.4) Additional remarks on modularity. (1) Modularity is a very strong property. In particular it implies that no field is interpretable. As we will see below, if a stable group G is modular, then there is essentially only one possible group law on G (see Proposition (4.2) below). Modular stable groups are abelian by finite.

Let me show by an example that an algebraically closed field k cannot be modular (we work in the language of rings {+, −, ·, 0, 1}). Indeed, consider three (algebraically) independent elements a, b, c in some algebraically closed field containing k, and let d = ac + b. Then the algebraic closures of the fields k(a, b) and k(c, d) intersect in k; but clearly (a, b) and (c, d) are not independent over k since e.g., tr.deg(k(a, b, c, d)/k) = 3 < tr.deg(k(a, b)/k) + tr.deg(k(c, d)/k) = 4. The failure of modularity is of course due to the existence of the two-dimensional family C_{a,b} of curves y = ax + b in the plane.

(2) Let K ⊧ ACFA, let R ⊆ K^{n} be definable over E = acl_{σ}(E), and assume that R is modular. This gives us information about the field of definition of the σ-closure R̄ of R: if a ∈ R is a generic of an irreducible component Z of R̄, then Z is defined over acl_{σ}(Ea). When R is quantifier-free definable, then R̄ is what we could call a “good approximation” of R, because deg_{σ}(R̄ \ R) < deg_{σ}(R). When R is not quantifier-free definable, then usually deg_{σ}(R̄ \ R) = deg_{σ}(R), and in an unstable context it may happen that any set S containing R and satisfying deg_{σ}(S \ R) < deg_{σ}(R) “needs” parameters from outside the algebraic closure of the field of definition of R̄.

(3.5) Proposition. Let K ⊧ ACFA, let E = acl_{σ}(E) ⊆ K and let p be a non-trivial modular type over E, of SU-rank 1. Then p is non-orthogonal to the generic of a definable subgroup of some (simple) commutative algebraic group, i.e., a simple abelian variety, or the multiplicative group 𝔾_{m}, or the additive group 𝔾_{a}; the latter case can only occur in positive characteristic.

(3.6) Zil’ber’s conjecture and the dichotomy. Zil’ber’s conjecture states: Let T be a strongly minimal theory. Then either all types of T are modular, or T interprets a (pure) algebraically closed field.

This conjecture was disproved by Hrushovski. However, the philosophy behind Zil’ber’s conjecture remains true: in most natural situations, the conjecture should be valid. An axiomatic system of such “natural situations” is given in [HZ] (Zariski geometries).

The dichotomy “modular/field” was proved for strongly minimal types in differentially closed fields (see [HS]) and for minimal types in separably closed fields (see [H93] and [De]). Its interest lies in the fact that there is a complete characterisation of the fields of rank 1 interpretable in the theory of differentially closed fields or in the theory of separably closed fields of positive degree of imperfection: they are definably isomorphic to, respectively, the field of constants and the field of elements which are q-th powers for all power q of the characteristic. This is an important tool in Hrushovski’s proof of the Geometric Mordell-Lang conjecture.

4. The dichotomy theorems As explained in the previous chapter, our goal was to prove the following dichotomy:

a type of SU-rank 1 is either non-orthogonal to one of the fixed fields, or it is modular. We first proved the characteristic 0 case, in a stronger form. The proof in that case is very algebraic and uses ramification theory. We were then able to establish the dichotomy in positive characteristic by completely different methods, see (4.4) for some details.

The dichotomy result allows us to get a good description of certain definable sets in the modular case (see (4.2) and (4.7)) and a semi-minimal analysis of types of finite rank (4.6).

(4.1) Theorem (The dichotomy in characteristic 0). Let p be a type of SU-rank 1 over E = acl_{σ}(E). Then either p ⊥̸ (σ(x) = x), or p is modular, stable, stably embedded, and has a unique non-forking extension to any set containing E. Also, p ⊥̸ (σ(x) = x) if and only if deg_{σ}(p) = 1 and there is an integer N such that [E(a, σ^{k}(a)) : E(a)] ≤ N for every k ∈ ℤ. Stably embedded means (n the arity of p, P the set of realisations of p): if S ⊆ K^{nm} is definable, then S ∩ P^{m} = S′ ∩ P^{m} for some S′ definable with parameters from P.

Note that a type can be stably embedded even if it is unstable. Indeed, one can show that if P is the set of realisations of a type p containing the formula σ(x) = x, then the field generated by P is all of F ix(σ). Thus, by (2.9)(2), p is stably embedded.

(4.2) The result of (4.1) extends to formulas: if φ(x) ⊥ (σ(x) = x), then the set of elements satisfying φ, with the structure inherited from K, is stable and modular. In the case of groups, this has the following striking consequence, by a theorem of Hrushovski-Pillay [HP85]: Proposition. (char. 0) Let G be a group of finite SU-rank definable in K ⊧ ACFA, and assume that the formula defining G is orthogonal to (σ(x) = x), and has its parameters in E = acl_{σ}(E). Let S ⊆ G^{m} be definable. Then S is a Boolean combination of cosets of E-definable subgroups of G^{m}.

(4.3) Theorem (The dichotomy in characteristic p > 0). Let q be a type of SU-rank 1. Then either q is modular, or q is non-orthogonal to the formula σ^{m}(x) = x^{p^{n}} for some m > 0 and n ∈ ℤ. Remarks

(1) The Frobenius automorphism x ↦ x^{p} is definable. Hence, for m > 0 and n ∈ ℤ, the formula σ^{m}(x) = x^{p^{n}} defines a pseudofinite subfield of K. We will refer to these fields as fixed fields.

(2) The result obtained in characteristic 0 does not generalise to characteristic p > 0. For instance, one can show that the set of realisations of σ(x) = x^{p} −x is unstable, and not stably embedded either. However, any complete type containing this formula is modular. We will see below that this is enough for some applications.

(3) There is a criterion analogous to the one given in characteristic 0 for types nonorthogonal to (σ(x) = x^{p^{n}}): one replaces algebraic degree by separable degree. If the field is defined by the equation σ^{m}(x) = x^{p^{n}} with m > 1, then the criterion has to be suitably modified.

(4.4) The proof of the dichotomy in characteristic p > 0 is quite different from the one in zero characteristic. An essential ingredient of the proof is the central role played by certain reducts of the structure. If M = (K, σ) is a model of ACFA, we let M[n] be the structure (K, σ^{n}), which is also a model of ACFA by (1.6)(7). While M[n] is a reduct of M, certain definable sets appear to attain more structure. It turns out that M[n] behaves more and more smoothly as n approaches infinity (a phenomenon which already showed up in the proof of the characteristic 0 case). In the characteristic p > 0, the proof begins by defining a certain limit structure M[∞] of the sequence M[n] (the “virtual structure”). This limit structure is shown to be very well-behaved, and some of its properties are translated back to the reducts M[n] and to M. This role for reducts and the type of limit taken, appear to be new in model theory.

We put a topology on some definable subsets of M[∞], and show that it satisfies an adapted version of the axioms of Zariski geometries. Then, given a non-modular definable subset X of M[∞], we reproduce the proof of [HZ] to obtain a field F of rank 1 interpretable in M[∞] and non-orthogonal to X, and show that this field F is algebraically closed. The proof that this gives the theorem uses the following result, of independent interest:

(4.5) Proposition. Let H be a simple algebraic group, and let G be a Zariski dense subgroup of H(K) definable in K ⊧ ACFA. If SU(G) is infinite, then G = H(K). If SU(G) is finite, then the generics of G are non-orthogonal to some fixed field F. Moreover, some subgroup of finite index of G is conjugate to a subgroup of H(F).

(4.6) Semi-minimal analysis. Let E = acl_{σ}(E), and a a tuple with SU(a/E) < ω. There are a_{1}, …, a_{n} ∈ acl_{σ}(Ea), such that a ∈ acl_{σ}(Ea_{1}, …, a_{n}), and for every i, either tp(a_{i+1}/E(a_{i})_{σ}) is modular of SU-rank 1, or there is some finite set B, such that the set of realisations of tp(a_{i+1}/E(a_{i})_{σ}) is contained in the perfect closure of the difference field generated by E(a_{i}) ∪ B ∪ F, where F is some fixed field.

(4.7) One can show easily that if a formula φ(x) is orthogonal to all fixed fields then φ(x) is modular. While the full theory of the set of realisations of φ(x) may be unstable, we have what is called quantifier-free ω-stability. Thus, in the case of groups, we get the analogue of (4.2) but only for subsets of G^{m} defined by quantifier-free formulas (within G^{m}).

I thought it was worthwhile to give a proof of this result, for two reasons. The first is the reaction of the audience during my talk at MSRI: they weren’t surprised by the dichotomy results but by their corollaries. The second is that in the particular context of a quantifier-free definable set X of an algebraic group H, the classical proof of [HP85] becomes very short, and still retains many of the ingredients which demonstrate the strength of modularity.

Proposition. Let K ⊧ ACFA, H an algebraic group defined over K, and let G be a definable subgroup of finite SU-rank of H(K). Assume that the formula defining G is orthogonal to all fixed fields, and has its parameters in E = acl_{σ}(E). Let X ⊆ H(K)^{m} be a quantifier-free definable set. Then X ∩ G^{m} is a Boolean combination of cosets of subgroups of G^{m} which are defined in G^{m} by a quantifier-free formula with parameters in E. Proof. The group G has finite index in the smallest quantifier-free definable group Ḡ containing it. By (4.3), the group G is modular, which implies that Ḡ is also modular. We may therefore assume that G is quantifier-free definable. By an easy reduction, we may also assume that m = 1 (work in G^{m}), and that X is an irreducible σ-closed set contained in G. We then want to show that X is the coset of a σ-closed subgroup S of G, and that S is defined over E.

We will assume that the difference field (K, σ) has sufficiently many automorphisms. If Z is a σ-closed set defined over some difference field F, we define deg_{σ}(Z) = max{deg_{σ}(a/F) | a ∈ Z}. Note that deg_{σ} is invariant under translation, i.e.: deg_{σ}(Z) = deg_{σ}(aZ) for any a ∈ G. In analogy with algebraic varieties, if Z is an irreducible σ-closed set defined over some field F, we will say that a is a generic of Z over F if a ∈ Z and every difference polynomial over F which vanishes at a, vanishes on Z. Equivalently, if a ∈ Z and deg_{σ}(a/F) = deg_{σ}(Z). One can show that the generics of the group G in the stability theoretic sense (i.e., of maximal SU-rank) are precisely the generics of G in this sense.

Let F be the smallest algebraically closed difference field containing E and over which X is defined. Let S = {h ∈ G | hX = X}, fix a generic a of X over F, and a generic g of G over F (a)_{σ}. Then S is a σ-closed subgroup of G defined over F, and b = ga is a generic of G over acl_{σ}(F a). Consider the set Y = gX; then b is a generic of Y over acl_{σ}(F g). Claim 1. Let τ be an automorphism of the difference field (K, σ), which is the identity on F. Then τ(Y) = Y ⟺ τ(gS) = gS.

Indeed, using the fact that X and S are defined over F, we get: τ(Y) = Y ⟺ τ(gX) = gX ⟺ τ(g)X = gX ⟺ g^{−1}τ(g) ∈ S ⟺ τ(gS) = gS, which gives the result. Claim 2. The fields of definition of Y and of gS are equi-algebraic over F.

This follows from Claim 1 and the following two observations: (1) if τ does not fix the field of definition of an irreducible σ-closed set Z, then τ(Z) ≠ Z; (2) Let k_{0} ⊆ k_{1} ⊆ K be fields, with k_{1} ⊈ acl_{σ}(k_{0}). Then there is some automorphism τ of (K, σ) which fixes acl_{σ}(k_{0}) and does not fix k_{1}.

By modularity and because b is a generic of Y over acl_{σ}(F g), the field of definition of Y is contained in acl_{σ}(F b) ∩ acl_{σ}(F g). Choose c ∈ acl_{σ}(F b) ∩ acl_{σ}(F g) which generates the field of definition of Y (and the one of gS by Claim 2). Using b = ga and the fact that g and b are independent from a over F, and hence over F (c)_{σ}, we obtain that deg_{σ}(Y) = deg_{σ}(b/F (c, a)_{σ}) = deg_{σ}(g/F (c, a)_{σ}) = deg_{σ}(gS). Hence deg_{σ}(S) = deg_{σ}(X), and from Sa ⊆ X and the irreducibility of the σ-closed set X, we deduce that S = Xa.

Because g is a generic of the coset gS and by modularity, gS is defined over acl_{σ}(Eg), which implies that S = g^{−1}(gS) is also defined over acl_{σ}(Eg). So, S is defined over acl_{σ}(Eg) ∩ F = E.

5. Application: the Manin-Mumford conjecture over a number field The result of Raynaud [R] (which implies the Manin-Mumford conjecture) states that

if A is an abelian variety and X a subvariety of A, then T or(A) ∩ X is a finite union of sets of the form a_{i} + T or(A_{i}), with A_{i} a group subvariety of A. (Here, T or(A) denotes the set of torsion points of A.) This result was later extended by Hindry and McQuillan. It is a particular case of a conjecture of Lang, for details see [L91] p. 37.

Hrushovski saw that the results on difference fields could be used to obtain a new proof of this theorem, for A a commutative algebraic group defined over a number field K. His proof gives an explicit bound on the number of cosets, of the form M = c deg(X)^{e}, where c and e depend on A but not on X, and deg(X) is the degree of the variety X with respect to a fixed embedding of A into projective space. His result appears in [H95], see also Pillay’s survey article in [P97]. His bound is explicit modulo the choice of two primes of good reduction for A, see (5.10) for the definition of good reduction. (If A is semi-abelian, let h(A) denote the height of A in the sense of Faltings; according to some specialist, the order of magnitude of a bound for a prime of good reduction is likely to be h(A).)

The strategy is very simple. Suppose we are given a commutative algebraic group A, a subvariety X of A, and some subgroup Γ of A. Then we would like to find an automorphism σ of some large model L of ACFA containing Γ, and a modular definable subgroup G of A(L) containing Γ. The result then would follow by (4.2). This is however too simple to work. There are two problems:

— In characteristic 0, every proper definable subgroup of 𝔾_{a}(L) is non-orthogonal to the fixed field (see (5.9)), and is therefore never modular. One gets around this difficulty by reducing to the case of a semi-abelian variety, using model theory.

— In order to get explicit bounds, one needs an explicit description of G (and not only its mere existence). When Γ is the subgroup T or_{p′}(A) of prime-to-p-torsion, for p a prime with good properties, then Weil’s result on abelian varieties defined over finite fields gives an equation of bounded complexity for σ a lifting of the Frobenius. However, this doesn’t say anything on the points of order a power of p. A further trick is needed, involving some model theory and ugly computations.

We indicate below some of the ingredients involved in the proof of Hrushovski. This chapter is organised as follows. We first introduce some tools and definitions from algebraic geometry, and state the main results used in Hrushovski’s proof. Of particular interest in my opinion is his description of definable subgroups of abelian varieties and of their definable endomorphisms. And of course, his criterion for modularity is absolutely fundamental in the proof, see (5.6). We then show in (5.13) how to obtain the qualitative result, and reduce the problem of finding an explicit bound for the number of cosets to the case where the group variety A is a semi-abelian variety.

We give a fairly detailed exposition on how to get the explicit bound. This part is essentially self-contained if one accepts the results stated earlier together with (5.12). We start with the “easy case” of the p′-torsion subgroup in (5.14). We then proceed slowly towards a proof of the full result, given in (5.17).

(5.1) Degrees of varieties. We embed our algebraic group A in some projective space ℙ_{n}. By the degree of a subset of ℙ_{n} we mean the degree of its Zariski closure. It is convenient to define a degree on algebraic subsets of cartesian powers of ℙ_{n}, in such a way that it satisfies the following:

(1) Let V_{1}, …, V_{r} be algebraic subsets of (ℙ_{n})^{ℓ}, and let Z_{1}, …, Z_{s} be the irreducible components of V_{1} ∩ ⋯ ∩ V_{r}. Then

∑_{i=1}^{s} deg(Z_{i}) ≤ ∏_{j=1}^{r} deg(V_{j}).

(2) Let V be an algebraic subset of (ℙ_{n})^{ℓ} × (ℙ_{n})^{k}, and consider the projection ρ : (ℙ_{n})^{ℓ} × (ℙ_{n})^{k} → (ℙ_{n})^{ℓ}. Then deg(ρ(V)) ≤ deg(V).

(3) Let V be an algebraic subset of (ℙ_{n})^{ℓ} × (ℙ_{n})^{k}, and let ρ be defined as above. For a ∈ (ℙ_{n})^{ℓ} define V (a) = V ∩ ρ^{−1}(a). Suppose that dim(V (a)) = r for generic a ∈ ρ(V). Then the Zariski closure V^{∗} of the set {a ∈ (ℙ_{n})^{ℓ} | dim(V (a)) > r} has degree ≤ deg(V).

For the definition of this degree and further properties, see [F] (Example 8.4.4) or [H95].

(5.2) Let A be an algebraic group. Then A has a unique maximal connected linear subgroup H, and A/H is an abelian variety, i.e., a commutative projective group variety. If H is commutative then the simple factors of H are isomorphic to either the multiplicative group 𝔾_{m} or the additive group 𝔾_{a}.

A semi-abelian variety is a commutative algebraic group A with no simple factor isomorphic to 𝔾_{a}. If A is an abelian variety, then there is an isogeny (epimorphism with finite central kernel) from some product A_{1} ×⋯×A_{n} onto A, with the A_{i}’s simple abelian subvarieties of A. A good reference for facts on abelian varieties is Lang’s book on abelian varieties [L83].

(5.3) Let us first show how to get from an effective description of G to an effective bound M. The group G will be described as {g ∈ A(K) | (g, σ(g), …, σ^{ℓ}(g)) ∈ S} for some algebraic subgroup S of A × σ(A) × ⋯ × σ^{ℓ}(A). We view A as embedded in ℙ_{n}, and define deg(S) and deg(X) with respect to this embedding. Let V = S ∩(X ×σ(X)×⋯×σ^{ℓ}(X)). Then dim(V) ≤ e = min{dim(S), (ℓ + 1)dim(X)} and deg(V) ≤ deg(X)^{ℓ+1}deg(S). Thus an effective bound for the number of components of the Zariski closure of G ∩ X is given by the following: Lemma. Let V ⊆ A^{ℓ+1} be an algebraic set, and let Ṽ = {g ∈ A(K) | (g, σ(g), …, σ^{ℓ}(g)) ∈ V }. Then the Zariski closure of Ṽ has degree at most deg(V)^{2^{dim(V )}}. If V is defined over L(c), where σ(L) = L, then Z is defined over L(σ^{−dim(V)}(c), …, c, σ(c), …, σ^{dim(V)}(c)).

The proof of this result uses the properties of degrees of varieties stated in (5.1). We’ll make a simple observation on how the irreducible components of Z are obtained. Let π_{0} denote the projection on the first copy of ℙ_{n}, π_{1} the projection on the first ℓ copies of ℙ_{n}, and π_{2} the projection on the last ℓ. The irreducible components of Z are images by π_{0} of algebraic subsets W of (ℙ_{n})^{ℓ+1} satisfying σπ_{1}(W) = π_{2}(W). Thus the procedure for getting the irreducible components of Z is as follows: start with some irreducible component W of V. If σπ_{1}(W) = π_{2}(W), π_{0}(W) will be an irreducible component of Z. If not, then consider W ∩ π_{2}^{−1}σπ_{1}(W), look at its irreducible components and repeat the procedure. This procedure stops after dim(V) + 1 steps.

This result needs to be refined to give more information when X varies in a family of varieties (we are now thinking of X + a for various a’s in the p-torsion subgroup).

(5.4) Notation. Let ℙ = (ℙ_{n})^{k} × (ℙ_{n})^{m}, and ρ : ℙ → (ℙ_{n})^{k} the projection. If Z is a subvariety of ℙ and a ∈ (ℙ_{n})^{k} we define Z(a) = {b ∈ (ℙ_{n})^{m} | (a, b) ∈ Z}, and if r = dim(Z(a)) for generic a ∈ ρ(Z), we set Z^{∗} = {a ∈ ρ(Z) | dim(Z(a)) > r}.

Proposition. Let (K, σ) ⊧ ACFA, and let V be a closed subset of ℙ^{ℓ} defined over K. There exist irreducible subvarieties Z_{i} of ℙ such that:

(1) If a ∈ Ṽ =_{def} {x ∈ ℙ(K) | (x, σ(x), …, σ^{ℓ−1}(x)) ∈ V }, then there is an i such that a ∈ Z_{i}, a ∉ ρ^{−1}(Z_{i}^{∗}).

(2) Z_{i}(K) ∩ Ṽ is dense in Z_{i} for every i.

(3) ∑_{i} deg(Z_{i}) ≤ ∑_{j=0}^{dim(V)} deg(V)^{2^{j}} ≤ 2deg(V)^{2^{dim(V )}}.

(4) If i ≠ j and Z_{i} is a proper closed subset of Z_{j}, then ρ(Z_{i}) ⊆ Z_{j}^{∗}.

(5.5) Definable endomorphisms and definable subgroups of an abelian variety. Let A be an abelian variety defined over a model (K, σ) of ACFA, and let End(A) denote the ring of algebraic endomorphisms of A, End_{σ}(A) the ring of definable endomorphisms of A. Denote by E(A) and E_{σ}(A) the rings ℚ ⊗_{ℤ} End(A) and ℚ ⊗_{ℤ} E_{σ}(A) respectively. Then E(A) and E_{σ}(A) have a description in terms of matrix rings over E(A_{i}) and E_{σ}(A_{i}) for some simple abelian subvarieties of A. The result is well-known for E(A) because of Poincaré’s reducibility theorem, and we will describe what happens for E_{σ}(A): Proposition. Let A be an abelian variety defined over K.

(1) Let A_{1}, …, A_{n} be abelian subvarieties of A such that A and A_{1} × ⋯ × A_{n} are isogenous. Let I ⊆ {1, …, n} be maximal such that if i ≠ j are in I and k ∈ ℕ, then A_{i} and σ^{k}(A_{j}) are not isogenous. For each i ∈ I let m(i) be the number of indices j ≤ n such that A_{j} and σ^{k}(A_{i}) are isogenous for some k ∈ ℕ. Then E_{σ}(A) ≃ ∏_{i∈I} M_{m(i)}(E_{σ}(A_{i})).

(2) Let k ≥ 1 and let B be a definable subgroup of A^{k}(K). Then B is commensurable with a finite intersection H of kernels of definable homomorphisms A^{k}(K) → A(K) (We say that B and H are commensurable if B ∩H has finite index in both B and H). If k = 1, a single homomorphism suffices.

Thus the study of definable subgroups of A(K) reduces to the study of the rings E_{σ}(A_{i}), i ∈ I. Of particular interest are the c-minimal subgroups of A(K), i.e., definable subgroups which are minimal up to commensurability, because of the following result:

Lemma. Let B be a definable subgroup of A(K). Then B is modular if and only if, for every f ∈ End_{σ}(A), every c-minimal subgroup of f(B) is modular. If B = Ker(fg) for f, g ∈ End_{σ}(A) with infinite kernels, then B is modular if and only if Ker(f) and Ker(g) are modular.

The result below gives a complete description.

(5.6) Theorem. Let A be a simple abelian variety defined over K ⊧ ACFA.

(1) Assume that for every n ∈ ℕ, A and σ^{n}(A) are not isogenous. Then E_{σ}(A) = E(A), and every definable proper subgroup of A(K) is finite.

(2) Assume that n ≥ 0 is minimal such that there is an isogeny h : A → σ^{n}(A), and let h′ : σ^{n}(A) → A and m ∈ ℕ^{>0} be such that h′h = [m]. Define ψ = σ^{−n}h and ψ′ = h′σ^{n}. Then E_{σ}(A) is isomorphic to the twisted Laurent polynomial ring E(A)[ψ, ψ′]. Note that ψ′ψ = [m].

From now on we assume that the hypotheses of (2) hold.

(3) Let f ≠ 0 be an element of End_{σ}(A). Then f is onto and Ker(f) has finite rank.

(4) A definable subgroup B of A(K) is c-minimal if and only if it is commensurable with Ker(f), for some f ∈ End_{σ}(A) which is irreducible in E_{σ}(A). Thus, if f ∈ End_{σ}(A) is non-zero, then Ker(f) is modular if and only if Ker(g) is modular for every irreducible divisor g of f.

(5) Let B be a c-minimal subgroup of A(K). If B is not modular, then there is an abelian variety A′ defined over F ix(τ) for some τ = σ^{m}ϕ_{p}^{−n}, and an algebraic isomorphism φ : A → A′ such that φ(B) ⊆ A′(F ix(τ^{ℓ})) for some ℓ. Similar results hold for the multiplicative group 𝔾_{m} (with E_{σ}(𝔾_{m}) ≃ ℚ[σ, σ^{−1}]), and

putting everything together, one obtains:

Theorem. Let A be a semi-abelian variety defined over F ix(σ), and let f(T) ∈ ℤ[T]. Assume that f(T) is relatively prime to all cyclotomic polynomials. Then Ker(f(σ)) is orthogonal to the formula σ(x) = x, and therefore modular if the characteristic is 0.

(5.7) Remark. In characteristic p, one obtains a similar criterion for semi-abelian varieties defined over F ix(σ) or over F ix(τ) for some τ = σ^{−m}ϕ_{p}^{n}.

Also, observe that if A is a simple abelian variety defined over F ix(σ) and if A(K) has a definable subgroup B of finite rank non-orthogonal to the formula σ^{m}(x) = x^{p^{n}} for some m > 0, n ∈ ℤ with n ≠ 0, then A must be isomorphic to a variety A′ defined over F ix(τ), where τ = σ^{−m}ϕ_{p}^{n}. This implies that the field of definition of A is contained in a finite algebraic extension of F ix(τ), and therefore is finite, since F ix(σ) ∩ F ix(τ^{ℓ}) = F ix(σ, ϕ_{p}^{nℓ}) ⊆ 𝔽_{p^{nℓ}}.

Let us assume that A is defined over the finite field 𝔽_{q} fixed by σ. Let f(T) ∈ ℤ[T], and consider the subgroup B of A(K) defined by the equation f(σ)(x) = 0. Let α_{1}, …, α_{r} be the roots of f(T) = 0 (in ℂ). Going to some power of τ, we may assume that σ^{m} and ϕ_{p}^{n} commute with all elements of End(A) (and fix the field of definition of A). To finish the discussion we need the following result of Weil, which will also be used in the proof: Theorem. (Weil. See [We], or [Mu], p. 203 and 205) Let A be an abelian variety defined over a finite field 𝔽_{q}, and consider the endomorphism ϕ_{q} : x ↦ x^{q} of A. If ω_{1}, …, ω_{2d} are the roots in ℂ of the characteristic polynomial of ϕ_{q} on A(𝔽_{q}^{alg}), then d = dim(A), the ω_{i}’s are algebraic integers of modulus |ω_{i}| = q^{1/2}, and q/ω_{i} is among the ω_{j}’s.

Hence, the endomorphism τ satisfies a functional equation g(T) = 0 on B, where the roots of g(T) are of the form α_{i}^{−m}ω_{j}^{ℓ}, where ℓ is such that q^{ℓ} = p^{n}. Thus, B is orthogonal to F ix(τ) if and only if no α_{i}^{−m}ω_{j}^{ℓ} is a root of unity.

Thus we obtain: B is modular if and only for every i and j, m ≠ 0 and ℓ, α_{i}^{−m}ω_{j}^{ℓ} ≠ 1.

(5.8) Before going on with Hrushovski’s proof, let us mention an easy corollary of his characterisation of modular subgroups. Proposition. Let A be a semi-abelian variety, and X a subvariety of A. Assume that m is an integer > 1 and prime to the characteristic of the field of definition of A, such that [m]X = X. Then X = a + C for some group subvariety C of A and element a ∈ A[m − 1]. Proof. Let k be an algebraically closed field over which X and A are defined, and embed k in a model (L, σ) of ACFA, with σ being the identity on k. By assumption, if u is a generic of X, then so is [m]u, and they have the same type (in the language of fields) over k. Hence, in L there is a generic u of X such that σ(u) = [m]u. Consider the subgroup B of A defined by the equation σ(x) = [m]x. Since m > 1 is prime to the characteristic of k, B is modular. Hence, B ∩ X is a finite union of cosets of definable subgroups of B. On the other hand, B ∩ X contains a generic point of X, which implies that one of these cosets is Zariski dense in X. This shows that X = a + C for some algebraic subgroup C of A. We also have: [m]X = [m]a + C = X = a + C, which implies that [m − 1]a ∈ C. Since C is divisible, we may choose a ∈ A[m − 1].

(5.9) Definable subgroups of 𝔾_{a}(K). The ring of endomorphisms of 𝔾_{a} definable in the model (K, σ) of ACFA, contains the twisted ring End_{K}(𝔾_{a})[σ, σ^{−1}], with the appropriate action of σ on End_{K}(𝔾_{a}). If the characteristic is 0, then End_{K}(𝔾_{a}) is canonically isomorphic to K. If the characteristic is p > 0, then End_{K}(𝔾_{a}) is generated over K by the Frobenius ϕ_{p} : x ↦ x^{p}.

In characteristic 0, a definable subgroup of 𝔾_{a}(K) will be commensurable to a subgroup defined by an equation ∑_{i=0}^{n} a_{i}σ^{i}(x) = 0 for some n and a_{0}, …, a_{n} ∈ K. One checks easily that a polynomial ∑_{i=0}^{n} a_{i}σ^{i} with a_{n} = 1 can be written as a product of linear terms of the form σ − a. Furthermore, if a ≠ 0 then the solution set of σ(x) = ax is non-orthogonal to the fixed field: if b_{1} and b_{2} are two solutions then σ(b_{1}/b_{2}) = b_{1}/b_{2}.

(5.10) Hypotheses and some notations. Let A be a commutative algebraic group defined over a number field K. Choose a sequence (0) = D_{0} ⊂ D_{1} ⊂ ⋯ ⊂ D_{s} = A of algebraic subgroups of A such that each factor D_{i+1}/D_{i} is a K-irreducible abelian variety or torus for i ≥ 1. Let {E_{j}} be a set of representatives of the K-isogeny classes of the factors D_{i+1}/D_{i} and define d = ∑ dim(E_{j}). Note that this number d does not change when we take powers of A.

We fix a prime p of good reduction, by which we mean: if 𝔽_{q} is the residue field of K modulo p and D̄_{i} denotes the algebraic set obtained by reducing modulo p, then each D̄_{i} is a reduced connected algebraic group. Moreover, for each i ≥ 1, D̄_{i+1}/D̄_{i} and D_{i+1}/D_{i} have the same dimension and are of the same type (i.e., an abelian variety or a torus). We also request that D̄_{1} be a vector group.

We denote by T or_{p′}(A) the subgroup of torsion elements of A of order prime to p, and by T or_{p}(A) the subgroup of torsion elements of A of order a power of p. Then T or(A) = T or_{p′}(A) ⊕ T or_{p}(A).

(5.11) Proposition. With notation as above, there is σ ∈ Aut(ℚ^{alg}) and an integral polynomial F (T) with no roots of unity among its roots, such that F (σ) vanishes on T or_{p′}(A). Furthermore, the degree of F is at most 2d and the sum of the absolute values of its coefficients is bounded by (1 + q^{1/2})^{2d}. Proof. Consider the Frobenius map ϕ_{q} : x ↦ x^{q} defined on 𝔽_{p}^{alg}, and let σ ∈ Gal(ℚ^{alg}/K) be a lifting of ϕ_{q}. Note first that our assumptions on p imply that reduction modulo p induces an isomorphism T or_{p′}(A) → T or_{p′}(Ā). Thus, for F (T) ∈ ℤ[T], if F (ϕ_{q}) vanishes on T or_{p′}(Ā), then F (σ) will automatically vanish on T or_{p′}(A).

Note also that if f(T), g(T) ∈ ℂ[T], then the sum of the absolute values of the coefficients of fg(T) is no greater than the product of the sums of the absolute values of the coefficients of f(T) and of g(T). It therefore suffices to show the assertion for each of the factors D̄_{i+1}/D̄_{i} and for D̄_{1}. Since D̄_{1} has no points of order prime to p, we may take the constant polynomial 1. If D̄_{i+1}/D̄_{i} is a K-simple abelian variety, then Weil’s result (5.7) gives us a monic polynomial of degree 2dim(D̄_{i+1}/D̄_{i}), with roots of modulus q^{1/2}. Hence the sum of the absolute values of the coefficients of this polynomial is ≤ (1 + q^{1/2})^{2dim(D_{i+1}/D_{i})}. Assume that D̄_{i+1}/D̄_{i} is a torus, isomorphic to 𝔾_{m}^{n} via an algebraic map φ defined over the finite field 𝔽_{q^{ℓ}}. The Frobenius map on D̄_{i+1}/D̄_{i} induces an automorphism ψ of 𝔾_{m}^{n}, which is the composition of θ = φ ∘ ϕ_{q}(φ)^{−1} with raising to the q-th power in 𝔾_{m}^{n}. Since θ ∈ End(𝔾_{m}^{n}) ≃ GL_{n}(ℤ), it is left fixed by ϕ_{p}. Hence,

θ^{ℓ} = θ ∘ ϕ_{q}(θ) ∘ ⋯ ∘ ϕ_{q}^{ℓ−1}(θ) = id,

which implies that the roots of the characteristic polynomial of θ in GL_{n}(ℤ) are roots of unity. Thus, the characteristic polynomial of ψ has degree n and its roots have absolute value q. Going back to D̄_{i+1}/D̄_{i} we get the result.

Since we may choose the same polynomial within a K-isogeny class, we get the correct bounds.

(5.12) Now comes the time to take care of the vector subgroup of G (a vector group is an algebraic group isomorphic to a product of copies of the additive group 𝔾_{a}). For that, we need two results, which we state below. The proof of the Proposition uses model theory and the full strength of Theorem (4.1). In positive characteristic the proof seems to work for quantifier-free definable subsets of G. The proof of the lemma is purely algebraic.

Definition. Let A be a commutative algebraic group, and let V be the maximal vector subgroup of A. A definable set X ⊆ A(K) is special if it is of the form Y + C where Y is a definable subset of V and C is a coset of a definable subgroup of A(K). Similarly, an algebraic subset of A is special if it is of the form Y + C where Y is an algebraic subset of V and C is a coset of an algebraic subgroup of A. Proposition (char. 0). Let (K, σ) ⊧ ACFA and let A be a commutative algebraic group defined over F ix(σ). Let F (T) ∈ ℤ[T] be a polynomial with no root of unity among its roots, and let G = {g ∈ A(K) | F (σ)(g) = 0}. Then every definable subset of G is a finite Boolean combination of special subsets of G. If X is a subvariety of A, then X ∩ G is a finite union of special subvarieties of A.

Lemma. Let A be a commutative algebraic group defined over ℚ^{alg}, and T the group of torsion points of A (or of prime-to-p torsion points of A). Let X be a subvariety of A and assume that X ∩ T ⊆ ⋃_{i=1}^{M} D_{i}, where each D_{i} is a special subvariety of X. Then the Zariski closure of X ∩ T is the union of at most M cosets of group subvarieties of A. More precisely, for every i, D_{i} ∩ T is either empty or its Zariski closure is the coset of a group subvariety of A.

(5.13) The qualitative result and reduction to the semi-abelian case. Let A be a commutative algebraic group defined over the number field K, let V be the maximal vector subgroup of A and B = A/V. We want to find σ′ ∈ Gal(ℚ^{alg}/K) and G(T) ∈ ℤ[T] such that G(σ′) vanishes on T or(A). Since the reduction map A → B is injective on T or(A), it suffices to find σ′ and G(T) such that G(σ′) vanishes on T or(B).

By (5.11) applied to two primes p and ℓ of good reduction for B, there are σ ∈ Gal(K(T or_{p′}(B))/K) and τ ∈ Gal(K(T or_{p}(B))/K), and polynomials F_{p}(T), F_{ℓ}(T) ∈ ℤ[T], with no roots of unity among their roots, and such that F_{p}(σ) vanishes on T or_{p′}(B) and F_{ℓ}(τ) vanishes on T or_{p}(B).

Using a result of Serre [Se], one can show that the field L = K(T or_{p′}(B))∩K(T or_{p}(B)) is a finite Galois extension of K, over which K(T or_{p′}(B)) and K(T or_{p}(B)) are linearly disjoint. Hence for m = [L : K], there is σ′ ∈ Gal(ℚ^{alg}/L) which extends σ^{m} on K(T or_{p′}(B)) and τ^{m} on K(T or_{p}(B)). Let α_{1}, …, α_{2d} and β_{1}, …, β_{2d} be the roots of F_{p}(T) and F_{ℓ}(T) respectively, and define G(T) = ∏_{i=1}^{2d}(T −α_{i}^{m})(T −β_{i}^{m}). Then G(σ′) vanishes on T or(B), and Ker(G(σ′)) defines a modular subgroup of B in any model of ACFA extending (ℚ^{alg}, σ′).

This gives us directly using the two results of (5.12), that the Zariski closure of X ∩ T or(A) is the union of finitely many cosets of group subvarieties of A. However, since we don’t know [L : K], we cannot expect to get an explicit bound on the number of cosets. To get the explicit bound we reduce to the semi-abelian case via the following observation:

Let Y be the image of X in B. Then the map A → B, which is injective on T or(A), establishes a bijection between the irreducible components of the Zariski closure of X ∩ T or(A) and the irreducible components of the Zariski closure Z of Y ∩ T or(B). Thus the Zariski closure of X ∩ T or(A) is the union of at most deg(Z) cosets of algebraic subgroups of A. So, we have

Theorem. Let A be a commutative algebraic group defined over the number field K, let X be a subvariety of A. Then X ∩ T or(A) = ⋃_{i=1}^{M} a_{i} + T or(A_{i}) where each A_{i} is an algebraic subgroup of A. Let V be the maximal vector subgroup of A, and Y the image of X in B = A/V. The number M is bounded by the number of irreducible components of the Zariski closure of Y ∩ T or(B).

(5.14) The bound on M in the case of the p′-torsion subgroup.

Theorem. Let A be a commutative algebraic group over a number field K, let X be a subvariety of A, and fix a prime p such that A has good reduction at p. Let q be the size of the residue field of K at p. Then

X ∩ T or_{p′}(A) = ⋃_{i=1}^{M} a_{i} + T or_{p′}(A_{i}),

where each A_{i} is an algebraic subgroup of A. If d ≤ dim(A) is defined as in (5.10), and if d_{+} is the degree of the graph of addition in A^{3} then:

M ≤ (deg(X)^{2d+1}d_{+}^{2d^{2}(2d+1)(log_{2}(1+q^{1/2})+1)^{2}})^{2^{d dim(X)}}.

Proof. Choose σ and F (T) = ∑_{i=0}^{2d} m_{i}T^{i} as in (5.11), and work in a model of ACFA extending (ℚ̃, σ). Let S̃ = KerF (σ) and consider the subgroup S of A^{2d+1} defined by {(a_{0}, …, a_{2d}) ∈ A^{2d+1} | ∑_{i=0}^{2d}[m_{i}]a_{i} = 0}. Using the fact that ∑_{i} |m_{i}| ≤ (1 + q^{1/2})^{2d}, and that multiplication by a number M ≥ 2 can be achieved with log_{2}(M)(log_{2}(M) + 1)/2 additions, one obtains deg(S) ≤ d_{+}^{2d^{2}(2d+1)(log_{2}(1+q^{1/2})+1)^{2}}. Let Z be the Zariski closure of S̃ ∩ X. Then (5.3) gives

deg(Z) ≤ (deg(X)^{2d+1}deg(S))^{2^{2d dim(X)}}.

Furthermore, by modularity of S̃, Z consists of cosets of algebraic subgroups of A. Each of these cosets intersects T or_{p′}(A) in either the empty set or a Zariski dense set. This gives the result.

(5.15) The whole torsion subgroup. Finding the bound on M in the case of all torsion is rather involved. By the qualitative result (5.13) we may assume that A is semi-abelian. Fix two primes p and ℓ of good reduction for A, and let F_{p}(T), F_{ℓ}(T) ∈ ℤ[T], σ, τ ∈ Aut(ℚ^{alg}) be as in (5.13). Choose also some (K_{p}, σ) and (K_{ℓ}, τ) models of ACFA and extending (ℚ^{alg}, σ) and (ℚ^{alg}, τ) respectively. We may, and will, identify the fields K_{p} and K_{ℓ}. That is, we are working in a large algebraically closed field K_{p} = K_{ℓ}, with two distinguished automorphisms σ and τ. Write F_{p}(T) = ∑_{i=0}^{2d} m_{i}T^{i} and F_{ℓ}(T) = ∑_{i=0}^{2d} n_{i}T^{i}. Define also

S_{q} = {(a_{0}, …, a_{2d}) | ∑_{i}[m_{i}]a_{i} = 0}, and

S̃_{q} = {a ∈ A(K_{p}) | (a, σ(a), …, σ^{2d}(a)) ∈ S_{q}},

and similarly the sets S_{ℓ} and S̃_{ℓ}. The groups S̃_{q} and S̃_{ℓ} are modular in the structures (K_{p}, σ) and (K_{ℓ}, τ) respectively.

We also set

ω_{1} = 2d dim(X) (≤ 2d dim(A)), ω_{2} = 2ω_{1} + 1, ω_{3} = 2dω_{2}dim(A).

We know that if b is any element of A, then (X − b) ∩ S̃_{q} is of the form C_{b}(K_{p}) ∩ S̃_{q}, with C_{b}(K_{p}) ∩ S̃_{q} Zariski dense in C_{b}, and where C_{b} is a finite union of cosets of algebraic subgroups of A; moreover we know that C_{b} is defined over L(σ^{−ω_{1}}(b), …, σ^{ω_{1}}(b)), where L = σ(L) is a finite Galois extension of K over which X is defined.

Let us first define the various components of C_{b} uniformly in b. For that we need to look at the Zariski closure of the set (σ^{−ω_{1}}(b), …, σ^{ω_{1}}(b), a) when b ranges over A(K_{p}), a ∈ S̃_{q} and a + b ∈ X, and more precisely at the algebraic set which defines it, i.e., at the algebraic subset S of (A^{ω_{2}} × A)^{2d+1} defined by:

(1) (x_{0}, …, x_{2d}) ∈ S_{q};

(2) (x_{i} + y_{0,i}) ∈ σ^{i}(X), for 0 ≤ i ≤ 2d;

(3) y_{j,i+1} = y_{j+1,i} for 0 ≤ i ≤ 2d − 1, −ω_{1} ≤ j ≤ ω_{1}.

A word about the indices: x_{i} corresponds to σ^{i}(a), and y_{j,i} to σ^{i}(σ^{j}(b)). One verifies that

dim(S) = 2d dim(A) + (2d + 1)dim(X) + 2ω_{1}dim(A) ≤ (4d + ω_{2})dim(A)
deg(S) ≤ deg(A)^{2ω_{1}}deg(S_{q})(deg(X)d_{+})^{2d+1}.

We now apply Lemma (5.4) to S and obtain a set of irreducible subvarieties W_{i} of A^{ω_{2}} × A, such that, if ρ : A^{ω_{2}} × A → A^{ω_{2}} is the projection then:

(i) If (b, a) ∈ S̃ =_{def} {(b, a) | ((b, a), …, σ^{2d}(b, a)) ∈ S}, then for some i we have (b, a) ∈ W_{i}, b ∉ W_{i}^{∗}.

(ii) The set W̃_{i} = W_{i}(K_{p}) ∩ S̃ is Zariski dense in W_{i}. (iii) ∑_{i} deg(W_{i}) ≤ ∑_{i=0}^{dim(S)} deg(S)^{2^{i}} ≤ 2deg(S)^{2^{dim(S)}}.

By (ii) we may choose (b, a) ∈ S̃ which is a generic of W_{i} (in the sense of the Zariski topology). Since W_{i} is irreducible, we know that the irreducible components of ρ^{−1}(b)∩W_{i} are conjugate over L(b). Since S̃_{q} is modular we also know that these components are cosets of some algebraic subgroups of A. Let A_{i} be the algebraic subgroup of A such that the component of ρ^{−1}(b) ∩ W_{i} containing (b, a) is a coset of A_{i}. If c is a generic of A_{i}, then (b, a + c) is a generic of ρ^{−1}(b) ∩ W_{i} and therefore (b, a + c) is a generic of W_{i}. Since W_{i} is closed this shows that W_{i} = W_{i} + ((0) × A_{i}), and therefore that for every y ∈ ρ(W_{i}), ρ^{−1}(y) ∩ W_{i} is a union of cosets of A_{i}. Furthermore these cosets are finite in number if y ∉ W_{i}^{∗}.

(5.16) Working on W_{i}. Fix i, let W_{i}^{0} = W_{i} \ ρ^{−1}(W_{i}^{∗}), B_{i} = A/A_{i} and let θ_{i} : A → B_{i} be the natural projection. For j ∈ ℤ let τ^{j}(B_{i}) = A/τ^{j}(A_{i}) and τ^{j}(θ_{i}) = τ^{j}θ_{i}τ^{−j} : A → τ^{j}(B_{i}). Define also B′_{i} = ∏_{j=0}^{ω_{3}} τ_{j}(B_{i}), C = A^{ω_{2}}.

We are interested in the set Θ_{i} = {(b, θ_{i}(a)) | (b, a) ∈ W̃_{i}, b ∉ W_{i}^{∗}, b ∈ S̃_{ℓ}^{ω_{2}}}. Note that if (b, c) ∈ Θ_{i} then c ∈ L(b)^{alg}. From deg_{τ} (b) ≤ 2dω_{2}dim(A) = ω_{3} we deduce that deg_{τ} (c) ≤ ω_{3}.

Let R ⊆ C^{ω_{3}+1} be defined by

R = {(y_{0}, …, y_{ω_{3}}) ∈ C^{ω_{3}+1} | ∑_{j=0}^{2d}[n_{j}]y_{i+j} = 0 for 0 ≤ i ≤ ω_{3} − 2d}.

Then dim(R) = ω_{3} and deg(R) ≤ deg(S_{ℓ})^{ω_{2}(ω_{3}−2d+1)}. Consider now the closed set U_{i} ⊆ (C ×B_{i})×(C ×τ(B_{i}))×⋯×(C ×τ^{ω_{3}}(B_{i})) which is the Zariski closure of the set of tuples ((y_{0}, z_{0}), …, (y_{ω_{3}}, z_{ω_{3}})) satisfying:

— (y_{0}, …, y_{ω_{3}}) ∈ R,

— For every 0 ≤ j ≤ ω_{3}, and x_{j} such that τ^{j}(θ)(x_{j}) = z_{j}, (y_{j}, x_{j}) ∈ τ^{j}(W_{i}^{0}).

We also let V_{i} be the image of U_{i} in B′_{i} under the natural projection (C × A)^{ω_{3}+1} → A^{ω_{3}+1} → ∏_{j=0}^{ω_{3}} τ^{j}(B_{i}). Then deg(V_{i}) ≤ deg(U_{i}) ≤ deg(R)deg(W_{i})^{ω_{3}+1}, and dim(U_{i}) = dim(V_{i}) ≤ dim(R) ≤ ω_{3}.

By (5.14) the Zariski closure Z_{i} of V_{i}∩T_{p′}(B′_{i}) is a finite union of cosets of definable subgroups of B′_{i}, and dim(Z_{i}) ≤ ω_{3}. Moreover, deg(Z_{i}) ≤ (deg(V_{i})^{2d+1}deg(S_{q})^{ω_{3}+1})^{2^{2d dim(R)}}.

Claim. Z̃_{i,τ} = {a ∈ B_{i} | (a, τ(a), …, τ^{ω_{3}}(a)) ∈ Z_{i}} is a finite union of cosets of τ-definable subgroups of B_{i} of finite SU-rank. Proof. Being a coset of a subgroup is a property preserved under homomorphisms and intersections. The first assertion follows since Z̃_{i,τ} is obtained from Z_{i} using projections, intersections, and the maps τ, τ^{−1}.

Let a be a generic of Z̃_{i,τ}. Since dim(Z_{i}) ≤ ω_{3}, we have tr.deg(a, τ(a), …, τ^{ω_{3}}(a)) ≤ ω_{3}, which implies that deg_{τ} (a) < ∞ and that the SU-rank of Z̃_{i,τ} in (K_{l}, τ) is finite.

Consider now the set

Ũ_{i,τ} = {(b, a) ∈ S̃_{ℓ}^{ω_{2}} × B_{i} | ((b, a), …, τ^{ω_{3}}(b, a)) ∈ U_{i}, (a, …, τ^{ω_{3}}(a)) ∈ Z_{i}}.

Since Z_{i} ⊆ B′_{i} and ((b, a), …, τ^{ω_{3}}(b, a)) ∈ U_{i} implies b ∈ S̃_{ℓ}^{ω_{2}}, Lemma (5.3) gives that the Zariski closure of Ũ_{i,τ} has degree ≤ (deg(U_{i})deg(Z_{i}))^{2^{dim(R)}}, and Ũ_{i,τ} ⊆ S̃_{ℓ}^{ω_{2}} × Z̃_{i,τ}. By the claim, Z̃_{i,τ} is a union of cosets of definable subgroups of B_{i} of finite SU-rank. We also know that Z̃_{i,τ} is modular (since every element in it is algebraic over a tuple from S̃_{ℓ}). Hence, every definable subset of S̃_{ℓ}^{ω_{2}} × Z̃_{i,τ} is a Boolean combination of cosets of definable subgroups of A^{ω_{2}} × B_{i}.

This implies that {(θ_{i}^{−1}(a) + b_{0}) | (b_{−ω_{1}}, …, b_{ω_{1}}, a) ∈ Ũ_{i,τ} } is the union of at most (deg(U_{i})deg(Z_{i}))^{2^{dim(R)}} cosets of definable subgroups of A.

(5.17) Theorem. Let A be a commutative algebraic group defined over a number field K and X a subvariety of A. Then the Zariski closure of X ∩ T or(A) consists of finitely many cosets of algebraic subgroups of A, and a bound on the number M of these cosets can be effectively computed (modulo the choice of the primes p and ℓ). Proof. The first assertion is proved in (5.13). It remains to show that the results of the previous paragraph give us the bound. For that we need to show:

If c ∈ T or(A) ∩ X, if c = a + b with a ∈ T or_{p′}(A) and b ∈ T or_{p}(A) and b^{∗} = (σ^{−ω_{1}}(b), …, σ^{ω_{1}}(b)), then (b^{∗}, θ_{i}(a)) ∈ Ũ_{i,τ} for some i.

By definition, a ∈ S̃_{q} and a + b ∈ X so that (b^{∗}, a) ∈ S̃ (see (5.15)). Choose i such that (b^{∗}, a) ∈ W_{i}^{0}. Then τ^{j}(b^{∗}, a) ∈ τ^{j}(W_{i}^{0}) and τ^{j}(b^{∗}) ∈ S̃_{ℓ}^{ω_{2}} for every j. Hence, ((b^{∗}, θ_{i}(a)), …, τ^{ω_{3}}(b^{∗}, θ_{i}(a)) ∈ U_{i}, so that (b^{∗}, θ_{i}(a)) ∈ Ũ_{i,τ}.

Note also that if (c_{−ω_{1}}, …, c_{ω_{1}}, d) ∈ Ũ_{i,τ}, then (c_{0}+d) ∈ X, so that the Zariski closure of the coset containing a + b is contained in X.

To conclude, we obtain the following bound on the number of cosets: M is bounded by the sum over i of the degrees of the Zariski closures of Ũ_{i,τ}. Unwinding back, we get

M ≤ ∑_{i} degŨ_{i,τ} ≤ ∑_{i}(deg(U_{i})deg(Z_{i}))^{2^{ω_{3}}} ,

∑_{i} deg(U_{i})deg(Z_{i}) ≤ deg(S_{q})^{(ω_{3}+1)2^{2dω_{3}}} ∑_{i} deg(V_{i})^{(2d+1)2^{2dω_{3}} +1},

∑_{i} deg(V_{i}) ≤ deg(S_{ℓ})^{ω_{2}(ω_{3}−2d+1)} ∑_{i} deg(W_{i})^{ω_{3}+1},

∑_{i} deg(W_{i}) ≤ 2deg(S)^{2^{dim(S)}} ≤ 2(deg(A)^{2ω_{1}}deg(S_{q})deg(X)^{2d+1}d_{+}^{2d+1})^{2^{(4d+ω_{2})dim(A)}},

So that

M ≤ 2^{M_{1}}deg(S_{q})^{M_{2}}deg(S_{ℓ})^{M_{3}}deg(A)^{M_{4}}(d_{+}deg(X))^{M_{5}},

where

\begin{matrix}M_{1} = & (ω_{3} + 1)((2d + 1)2^{2dω_{3}} + 1)2^{ω_{3}}, \\ M_{2} = & 2^{(4d+ω_{2})dim(A)}M_{1} + (ω_{3} + 1)2^{(2d+1)ω_{3}}, \\ M_{3} = & ω_{2}(ω_{3} − 2d + 1)((2d + 1)2^{2dω_{3}} + 1)2^{ω_{3}} \\ M_{4} = & 2ω_{1}2^{(4d+ω_{2})dim(A)}M_{1}, \\ M_{5} = & (2d + 1)2^{(4d+ω_{2})dim(A)}M_{1}. \end{matrix}

The order of magnitude of ω_{3} is 8d^{2}dim(A)dim(X) ≤ 8d^{2}dim(A)^{2}.

6. Some other applications In this chapter we state without proofs some other applications of the results on

difference fields. We start with a result of Hrushovski, and conclude with two results by T. Scanlon.

(6.1) Reduction of a conjecture of Lang. Let A be a commutative algebraic group defined over a number field K, let Γ be the division group of A(K), i.e., the set of elements a ∈ A(K^{alg}) such that [m]a ∈ A(K) for some non-zero integer m. A conjecture of Lang states that if X is a subvariety of A containing no cosets of infinite algebraic subgroups of A, then X ∩ Γ is finite.

The techniques used in the previous paragraph give the following reduction of the conjecture (also proved by Raynaud, Hindry, McQuillan), with effective bounds: Theorem ([H95]). Let A be a commutative algebraic group defined over a number field, and let Γ be the division group of A(K). Suppose that X is a subvariety of A, containing no cosets of infinite algebraic subgroups of A. One can effectively find an integer M such that Γ ∩ X(K^{alg}) ⊆ \frac{1}{M} A(K). Moreover one can effectively find coset representatives r_{i} of A(K)/MA(K) such that Γ ∩ X(K^{alg}) ⊆ ⋃_{i} \frac{r_{i}}{M} + A(K).

Idea of the proof. Fix a prime p of good reduction for A and let Γ_{p′} denote the p′-division subgroup of A(K), i.e. we require that the integer m in the definition of division group be prime to p. Let 𝔽_{q} be the residue field of K at p, σ a lifting of the Frobenius ϕ_{q} : x ↦ x^{q} to K, and F_{p}(T) ∈ ℤ[T] the Weil polynomial.

One first shows that (σ − 1)F_{p}(σ) vanishes on Γ_{p′}. Using the fact that Ker(F_{p}(σ)) is orthogonal to the fixed field and the assumption on X, one then shows that that X(K^{alg})∩ Ker((σ − 1)F_{p}(σ)) is contained in finitely many cosets of Ker(σ − 1) = A(F ix(σ)). From this one deduces a number M_{p} such that M_{p}(X(K^{alg}) ∩ Γ_{p′}) ⊆ A(K).

Choosing another prime ℓ of good reduction for A one obtains a number M_{ℓ} such that M_{ℓ}(X(K^{alg}) ∩ Γ_{ℓ′}) ⊆ A(K). Then M_{p}M_{ℓ}(X(K^{alg}) ∩ Γ) ⊆ A(K). The bound M_{p}M_{ℓ} is effective, modulo the choice of the two primes p and ℓ.

(6.2) A conjecture of Tate-Voloch. Conjecture. Let G be a semi-abelian variety defined over ℂ_{p}, and let X be a subvariety of G. There is a constant N such that for any P ∈ T or(G), either P ∈ X or d(P, X) > N.

Here ℂ_{p} is the completion of the algebraic closure of ℚ_{p} (with respect to the p-adic valuation on ℚ_{p}^{alg}), and d(P, X) is a p-adic distance associated to the valuation. If X is a subvariety of an affine space, one defines d(P, X) = max{p^{−v(f(P))} | f ∈ I}, where I is the ideal of polynomials defining X. In the general case, one extends the definition by using a cover by affine sets.

When G is a torus, this is a theorem of Tate and Voloch [TV]. Hrushovski [H96e] proved the conjecture when G is over ℚ_{p}^{alg}, has good reduction, and for prime-to-p torsion points. Scanlon ([Sc1] [Sc2]) proves the conjecture when G is defined over ℚ_{p}^{alg}. He considers liftings σ of the Frobenius, a Weil polynomial F_{q}(T), and uses the fact that T or(G) ⊆ Ker((σ − 1)F_{q}(σ)).

(6.3) Drinfeld modules. Let K be an algebraically closed field of positive characteristic p and of positive transcendence degree. Consider the ring End_{K}(𝔾_{a}) of endomorphisms of 𝔾_{a} defined over K. Then End_{K}(𝔾_{a}) is isomorphic to the twisted polynomial ring K[ϕ_{p}]. Let A = 𝔽_{p}[T] and view it as a subring of K, by identifying T with some transcendental t ∈ K.

A Drinfeld module (over A) is given by a ring homomorphism φ : A → End_{K}(A) so that if φ(T) = ∑_{i=0}^{n} a_{i}ϕ_{p}^{i}, then a_{0} = t and a_{n} = 1.

Theorem ([Sc3]). Let φ be a Drinfeld module. Consider K^{N} as an A-module via φ. If X is a subvariety of K^{N} then the intersection of X with the A-torsion subgroup of K^{N} (= {x ∈ K^{N} | φ(a)(x) = 0 for some non-zero a ∈ A}) is a finite union of translates of A-torsion subgroups of algebraic subgroups of K^{N}.

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UFR de Mathématiques, Case 7012 Université Paris 7 2, place Jussieu 75251 Paris Cedex 05, FRANCE e-mail: zoe@logique.jussieu.fr