Jet spaces of varieties over differential and difference fields - CiteSeerX

Jet spaces of varieties over differential and difference fields Anand Pillay∗ University of Illinois at Urbana-Champaign Humboldt University of Berlin Martin Ziegler Universit¨at Freiburg Humboldt Universit¨at zu Berlin May 27, 2003

Abstract We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed.

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Introduction

We study here mostly the categories of finite-dimensional differential algebraic varieties, and of finite-dimensional difference algebraic varieties. The first category belongs to Kolchin’s “differential algebraic geometry”. The second is the analogue when the derivation is replaced by an automorphism. ∗

Partially supported by an NSF grant and a Humboldt Foundation Research Award

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In each of these categories there are certain objects which can be seen as belonging entirely to algebraic geometry. For example in the differential case, these will be differential algebraic varieties of the form X(CK ) where X is an algebraic variety defined over the field of constants CK of an ambient differentially closed field (K, ∂). We may call a finite-dimensional differential algebraic variety “algebraic” if it is “differentally birational” to something of form X(CK ). The situation is somewhat analogous to the category of compact complex spaces. The Moishezon or “algebraic” compact complex spaces are precisely those which are bimeromorphic to some projective algebraic variety. The aim of the present paper is to state and prove a criterion for a finite-dimensional differential (difference) algebraic variety to be “algebraic”. Roughly speaking, if Z is a finite-dimensional differential algebraic variety, which parametrizes the family (Ya : a ∈ Z) of differential algebraic subvarieties of a finite-dimensional differential algebraic variety X which all pass through a common point a ∈ X, THEN Z will be “algebraic”. Our proofs are rather elementary and use a generalized Gauss jet map, together with the theory of linear differential (difference) equations. We were heavily influenced by work of F. Campana [5] on cycle spaces in compact complex spaces. The present paper may be of additional interest because our results subsume the structural results on definable sets in differentially closed fields and existentially closed difference fields which Hrushovski used to give proofs of the Mordell-Lang conjecture for function fields and the Manin-Mumford conjecture (all in characteristic zero). More details on the connection with these diophantine-geometric results will be given in section 7. However we should make it clear that our results do not simplify Hrushovski’s proof of MordellLang for function fields in the positive characteristic case, for which no other proof is currently known. But we will point out what our methods do yield in positive characteristic. We will be making use of model-theoretic terminology. One reason is that the general model-theoretic context provides a common and efficient language for the differential and difference-algebraic objects we consider. Another is that there is as yet no transparent and commonly accepted algebraicgeometric language for dealing with these objects. Of course there are Kolchin’s books for the differential case, but the language and machinery there is also somewhat obscure. So the present paper should be seen as one 2

on the model theory of differential and difference fields, although we will try to explain the “geometric” meaning of the terminology. For the differential case, we work with the theory DCF0 of differentially closed fields of characteristic zero, for which we refer to [13] for background. The language is that of rings (+, −, ·, 0, 1) together with a symbol ∂ for the derivation. DCF0 is complete, with quantifier elimination, and is moreover ω-stable. We work in a model (U, +, −, ·, 0, 1, ∂) which is ω1 -saturated. An affine differential algebraic set X is by definition the common solution set in U n of a finite system P1 (x1 , .., xn ) = 0, ..., Ps (x1 , .., xn ) = 0 of differential polynomial equations over U. It is rather important to know that such a set X has a smallest differential field of definition. Namely there is a smallest differential subfield k of U, such that X can be defined by differential polynomials with coefficients from k. k will be finitely generated as a differential field, and if d is a finite tuple of generators, we call d a canonical parameter for X. Note that by quantifier-elimination, any definable subset of U n is a finite Boolean combination of differential algebraic sets. If X ⊆ U n is definable (over a countable differential subfield k say), then X is said to be finite-dimensional if there is a finite bound on tr.deg(k < a > /k) for a ∈ X. Here k < a > denotes the differential field generated by k and a, and is precisely k(a, ∂(a), ∂ 2 (a), ...). C denotes the field of constants of U. Tuples a, b from U are said to be independent over k if k < a > is algebraically disjoint from k < b > over k. tp(c/k) is said to be internal to C if there is some b independent from a over k and a tuple d from C such that c ∈ k < b, d >. The notion of “canonical base” is fundamental in general stability theory. In the present situation it has the following meaning: Suppose k is a countable algebraically closed differential subfield of U. Let a be a n-tuple from U. Let X ⊆ U n be the locus of a over k, namely the common zero set of all differential polynomials over k which vanish at a. Then by Cb(tp(a/K)), the canonical base of the type of a over k, we mean a canonical parameter for X, as described earlier. Our main result in the differential case is: Theorem 1.1 Let X be a finite-dimensional definable set, defined over the countable algebraically closed differential subfield k of U. Let a ∈ X. Let k1 be a countable algebraically closed differential subfield containing k. Let c = Cb(tp(a/k1 )). Then tp(c/k < a >) is internal to C. 3

Explanation and translation. We may assume that X is the locus of a over k. Let Y be the locus of a over k1 . As c is a canonical parameter for Y , let us write Y as Yc . Let Z be the locus of c over k. We can view Z as an irreducible component of the “differential Hilbert scheme” of the differential algebraic variety X. Namely Z parametrizes the family (Yc0 : c0 ∈ Z) of differential algebraic subsets of X. Let Z0 ⊂ Z be the locus of c over k < a >. Z0 consists essentially of those c0 ∈ Z, such that a ∈ Yc0 . The conclusion of the theorem says that Z0 is “algebraic”. We are unaware of any systematic development of machinery and language (such as “differential Hilbert spaces”) in differential algebraic geometry which is adequate for the geometric translation above. This is among the reasons why we will stick with the language of model theory in our proofs below. The issue of algebraizing the content and proofs is a serious one which will be considered in future papers. In the difference field case, we have an essentially identical result to Theorem 1.1, but with the fixed field replacing the constants. A difference field is simply a field equipped with an (abstract) automorphism. The language for difference fields is the language of rings together with a sumbol σ for the automorphism. The relevant first order theory is ACF A, the theory of existentially closed difference fields, which was developed in great detail in [6]. A transparent translation between model theory and difference algebra is rather more problematic here, for various reasons. One reason is that ACF A (or rather its completions) does not have quantifier-elimination. Another is that ACF A is not ω-stable, in fact not even stable. The theory is simple, and the generalization of the machinery of stability theory to the broader class of simple theories has been an ongoing project since 1995. We will discuss ACF A in more detail in section 4, and for now just state the main result. Theorem 1.2 Let (U, +, −, ·, 0, 1, σ) be a model of ACF A of characteristic zero. Let X be a finite-dimensional definable set, defined over an algebraically closed difference subfield k of U. Let a ∈ X, let k1 be an algebraically closed difference subfield of U containing k. Let c = Cb(tp(a/k1 )). Then tp(c/k < a >) is almost internal to the fixed field F ix(σ). We will complete this introduction with a couple of additional comments. The first is for model-theorists. Theorems 1.1 and 1.2 imply directly the 4

Zilber dichotomy for types of SU -rank 1 in DCF0 and ACF A0 : (*) Any type of SU -rank 1 is either modular or nonorthogonal to the constants (fixed field). On the other hand, Theorems 1.1 and 1.2 are substantially stronger than (*). For example, Theorem 1.1 rather directly implies that if G is a finitedimensional differential algebraic group, and X is a differential algebraic subset of G with trivial stabilizer, then X is “algebraic”. So the “socle argument” from [9] is included in Theorem 1.1. The second remark is that the jets or jet spaces used in the proofs below should not be confused with those in Buium’s paper [3]. We will be considering spaces of the form (M/Mn )∗ where M is the maximal ideal of the local ring at a point on a variety, together with additional structure in the presence of a derivation. Buium’s prolongations on the other hand correspond to arc spaces of varieties. Our main results (Theorems 1.1, 1.2) should be seen as analogies in the categories of finite-dimensional differential (difference) algebraic varieties, of results proved independently by Campana [5] and Fujiki [7] on the algebraicity of certain spaces of cycles in compact complex spaces. After reading Campana’s [5], the first author saw the possibility and implications of adapting the ideas. What was needed was the definition and properties of the differential jet of a finite-dimensional differential algebraic variety at a point. This was worked out together with the second author in a very enjoyable collaboration while both were visiting the Humboldt University of Berlin. Thanks to Andreas Baudisch for his hospitality. We would like to thank Elisabeth Bouscaren, Zoe Chatzidakis, Piotr Kowalski, Dave Marker, and Tom Scanlon, for comments on a first (Oct. 2001) draft of this paper. Thanks are also due to the referees for their comments on the draft (Dec. 2001) that was submitted for publication, and which led to a substantial rewriting of the paper.

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Jets in algebraic geometry

In this section, we recall the classical jets from algebraic geometry. The material is elementary and should be considered well-known, although we could not find a concise reference. 5

Let K be an algebraically closed field of any characteristic. Let X ⊆ K n be an irreducible affine variety, with ideal IX ⊂ K[x1 , .., xn ] and coordinate ring K[X] = K[x1 , .., xn ]/IX . Let a ∈ X. MX,a is by definition {f ∈ K[X] : f (a) = 0}. For each m ≥ 2, MX,a /Mm X,a is a finite-dimensional K-vector space, and we define jm−1 (X)a , the (m − 1)st jet of X at a, to be its dual space. For m = 2, we obtain the tangent space to X at a. Lemma 2.1 (With above notation.) Let a ∈ X. Then

T

m

Mm X,a = (0).

Proof. Corollary 10.18 of [2] says that if R is a Noetherian domain and I a T proper ideal of R then n I n = (0). In the special case where X = K n (affine n-space), let Ma denote MX,a , and jm,a the corresponding jet. So for arbitrary X we have canonical linear embeddings of jm (X)a in jm,a for all m. We will identify jm (X)a with its image. Lemma 2.2 Suppose X, Y are irreducible subvarieties of K n , and a ∈ X∩Y . Suppose jm (X)a = jm (Y )a for all m. Then X = Y . is annihilated by jm (X)a for all m and thus Proof. If f ∈ IX then f /Mm+1 a by jm (Y )a for all m. It follows that for all m, f /IY ∈ Mm Y,a for all m. By m Fact 2.1 ∩m MY,a = (0). Thus f ∈ IY . The following gives explicit equations for m-jets of affine varieties. Lemma 2.3 Let X be a subvariety of K n . Fix m ≥ 1. Let D be the set of differential operators of the form ∂s ∂xsi11 ∂xsi22 ...∂xsirr where 0 < s ≤ m, 1 ≤ i1 < i2 < .. < ir ≤ n, s1 + ... + sr = s, and 0 < si . Let a ∈ X. Let d = |D|. Then jm (X)a can be identified with the subspace P Va = {(uD )D∈D : D∈D DP (a)uD = 0, P ∈ IX } of K d . (Moreover, if X is defined over k < K, then we can restrict the polynomials P to those in IX ∩ k[X1 , .., Xn ].)

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Proof. Let Va act on MX,a as follows: if u¯ = (uD )D∈D ∈ Va , and Q ∈ MX,a , then P (¯ u) · Q = D∈D DQ(a)uD . This gives the isomorphism between Va and MX,a /Mm+1 X,a . Remark 2.4 The equations in the previous lemma give rise to the mth jet bundle jm (X) of X: {(a, u¯) : a ∈ X, u¯ ∈ Va }. We will be interested in the situation where X, Y are subvarieties of the affine variety Z, passing through a common point a ∈ Z (which will be generic on each of Z, X, Y ). The inclusions of X and Y in Z, give inclusions jm (X)a ⊆ jm (Z)a , and jm (Y )a ⊆ jm (Z)a and from 2.2. we obtain: Corollary 2.5 X = Y iff jm (X)a = jm (Y )a as subspaces of jm (Z)a , for all m.

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Differential fields of characteristic zero

In this section we prove Theorem 1.1. We refer the reader to [4] for background on “differential algebraic geometry” and to [13] for background on the model-theoretic approach to this subject. We will be working in a (saturated) differentially closed field (U, ∂) with field of constants C. Let us start with an informal description of the proof. ∂ We will define the differential m-jet jm (X)a to a finite-dimensional differential algebraic variety X at a point a ∈ X. This differential m-jet will be a finitedimensional vector space over C which can be identified with C d . If Y is a ∂ differential algebraic subvariety of X also passing through a, then jm (Y )a ∂ will be a C-subspace of jm (X)a . For sufficient large m, Y will be determined ∂ (over a) by jm (Y )a , a subspace of C d . ∂ We will obtain jm (X)a by embedding X in a certain canonically associated 0 algebraic variety X . The m-jet of X 0 at a will be endowed with a linear ∂ differential equation, and jm (X)a will be the solution space. In fact, in the actual proof below it will be convenient to work directly with the ∂-module structure on the m-jet of the associated algebraic variety X 0 . The “differential” jet will not appear explicitly in the proof, although we will give its definition.

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We begin with recalling some facts about ∂-modules. We work in characteristic zero. A differential field (K, ∂) is a field K equipped with a derivation ∂ : K → K. By a ∂-module over the differential field (K, ∂) we mean a finitedimensional vector space V over K together with an additive endomorphism DV : V → V such that for any c ∈ K and v ∈ V , DV (c·v) = ∂(c)·v+c·DV (v). A ∂-module over (K, ∂) is often called a K-module equipped with a “connection”. Lemma 3.1 Let (V, DV ) be a ∂-module over (K, ∂). Let CK = {c ∈ K : ∂(c) = 0} be the field of constants of K. Let V ∂ = {v ∈ V : DV = 0}. Then (i) V ∂ is a CK -vector space, (ii) If (K, ∂) is differentially closed, then dimCK (V ∂ ) = dimK (V ) and there are v1 , ..., vn ∈ V ∂ which form a K-basis of V , and so also a CK -basis of V ∂ . Proof. (i) is clear. (ii) is well-known: after choosing a basis for V over K, V identifies with K n and the equation DV = 0 on V becomes a linear differential equation ∂(y) = Ay on K n , where y is a n × 1 vector of unknowns and A some n × n matrix over K. It is easy to see that a set of solutions of this equation is linearly independent over CK iff it is linearly independent over K. A fundamental matrix of solutions (that is a nonsingular n by n matrix whose columns are solutions) can be found in some differential extension (K1 , ∂1 ) of (K, ∂). As (K, ∂) is differentially closed such a fundamental matrix can already be found over K. Lemma 3.2 Let (K, ∂) be a differentially closed field. Let V be a ∂-submodule of (K n , ∂). Then V is defined over CK . Proof. Note that V ∂ is precisely V ∩(CK )n . By 3.1, we can find v1 , .., vr ∈ V ∂ which form a basis for V over K. So V is isomorphic to K r over {v1 , .., vr }. As each vi ∈ (CK )n , V is already defined over CK . Remark 3.3 Let (V, DV ) be a ∂-module over (K, ∂). Let V ∗ be the dual space to V and define DV ∗ : V ∗ → V ∗ by (DV ∗ (λ))(v) = ∂(λ(v)) − λ(DV (v)). Then (V ∗ , DV ∗ ) is a ∂-module over (K, ∂). Note that (V ∗ )∂ is precisely the set of K-linear ∂-module maps from (V, DV ) to (K, ∂). 8

We will now exhibit a canonical form for definable sets (types) of finite Morley rank in differentially closed fields. As mentioned above we work in a saturated differentially closed field U, ∂) and use the conventions discussed in the introduction. We first recall the “algebraic D-varieties” of Buium [4] For our purposes we need only consider the affine case. Definition 3.4 An (affine) algebraic D-variety is an irreducible affine variety X ⊆ U n together with an extension of ∂ to a derivation ∂ 0 of the coordinate ring U[X] of X. If P (x1 , .., xn ) is a polynomial over U, then by P ∂ we mean the polynomial obtained from P by hitting its coefficients with ∂. The next lemma is elementary and the proof is left to the reader. Lemma 3.5 Let (X, ∂ 0 ) be an algebraic D-variety, with X ⊆ U n . Let x1 , .., xn denote the coordinate functions. Let si = ∂ 0 (xi ) and s = (s1 , .., sn ). Then P (*) for every P (x1 , .., xn ) ∈ IX , ni (∂P/∂xi )(x)si (x) + P ∂ (x) is in IX . Conversely, if X ⊆ U n is an irreducible variety and s : X → U n is a polynomial map satisfying (*), then defining ∂ 0 (xi ) = si makes (X, ∂ 0 ) into an algebraic D-variety. By virtue of the above lemma, we will also write an algebraic D-variety as (X, s) where s satisfies (*). In fact it is clearly enough for (*) to hold for P ranging over generators of the ideal IX . Definition 3.6 Let (X, s) be an algebraic D-variety. Then (X, s)] (or just X ] if s is understood) is {x ∈ X : ∂(x) = s(x)}. Lemma 3.7 (i) Let (X, s) be an algebraic D-variety. Then (X, s)] is Zariskidense in X. (ii) Let k < U be a countable algebraically closed differential subfield. Let a be a finite tuple from U, such that tr.deg(k < a > /k) is finite. Then up to interdefinability over k, a is a k-generic point of some (X, s)] for (X, s) some algebraic D-variety defined over k. Namely tp(a/k) is determined by: a is a generic point of X over k (in the algebraic-geometric sense), and ∂(a) = s(a) (iii) Let (X, s) = (X, ∂ 0 ) be an algebraic D-variety, and a ∈ (X, s)] . Let M be the maximal ideal of U[X] at a. Then for each m ≥ 0, Mm is a differential ideal of (U[X], ∂ 0 ), namely Mm is closed under ∂ 0 . 9

Proof. (i) is by [16]. (ii) By replacing a by some (a, ∂(a), ..∂ r (a)), and using the properties of a derivation, we may assume that ∂(a) ∈ k(a). Adjoining to a a finite part of k(a) we may assume hat ∂(a) ∈ k[a] and so ∂(a) = s(a) for some polynomial function s : U n → U n . Let X be the irreducible affine algebraic variety defined over k whose k-generic point is a. Then (X, s) is an algebraic Dvariety. P (iii) Note that ∂ 0 acts on U[X] as (∂ 0 (f ))(x) = ni=1 (∂f /∂xi )si (x) + f ∂ (x). For a ∈ (X, s)] we see that (∂ 0 f )(a) = ∂(f (a)). So if also f (a) = 0 then (∂ 0 f )(a) = 0. By virtue of Lemma 3.6 (iii), if (X, s) is an algebraic D-variety, a ∈ (X, s)] , M is the maximal ideal of U[X] at a, and m ≥ 1, THEN M/Mm+1 becomes a ∂-module over (U, ∂) under ∂ 0 . By Remark 3.3, the dual space jm (X)a is equipped with the canonical dual ∂-module structure (over (U, ∂)). With this notation: Lemma 3.8 Suppose that (Y, s|Y ) is an algebraic D-subvariety of (X, s) (namely Y is a subvariety of X and (Y, s|Y ) is an algebraic D-variety in its own right), and a ∈ Y ] . Then for any m, jm (Y )a is a ∂-submodule of jm (X)a . m+1 Proof. The canonical U-linear surjection from MX,a /Mm+1 X,a to MY,a /MY,a is a ∂-module map. Hence by the definition (in 3.3) of the dual connection, the corresponding embedding of jm (Y )a in jm (X)a is also one of ∂-modules.

Remark 3.9 The differential m-jet of the differential algebraic variety (X, s)] at a, will be (jm (X)a )∂ , the solution space of the associated linear differential equation on jm (X)a . We can now give: Proof of Theorem 1.1. The reader should bear in mind the explanation and translation following the statement of the theorem in the introduction. We are given countable algebraically closed differential subfields k < k1 of U, and a finite tuple a from U. We may replace a by anything interdefinable with it over k. So by Lemma 3.7 (ii) we may assume that there is an algebraic 10

D-variety (X, s) defined over k (that is, both X and s are defined over k) such that a is a generic point of (X, s)] over k (in the sense of 3.7 (ii)). Let now Y be the (algebraic-geometric) locus of a over k1 . Then (Y, s|Y ) is an algebraic D-variety, and a is a (differential) generic point over k1 of Y ] . The canonical base of tp(a/k1 ) is precisely a canonical parameter for the differential algebraic variety Y ] . As Y ] is defined by “x ∈ Y and ∂(x) = s(x)”, and s is defined over k, Cb(tp(a/k1 )) is interdefinable over k with the tuple c which generates the smallest field of definition of the variety Y . So it suffices to prove that tp(c/k, a) is internal to the field C of constants. For each m, let (Vm , DVm ) be jm (X)a equipped with its canonical ∂module structure. By Lemma 3.8, for each m, jm (Y )a is a DVm -submodule of Vm . By Lemma 3.1 let b be a basis for (Vm )∂ over C which is simultaneously a basis for Vm over U. b may be chosen to be independent from c over k < a > (in the differential sense). The basis yields an isomorphism (of ∂-modules) between (Vm , DVM ) and (U rm , ∂), which therefore takes jm (Y )a to a ∂-submodule Wm of (U rm , ∂). By Lemma 3.2, Wm is defined over C. Let em ∈ C be a tuple over which Wm is defined. By Corollary 2.5, Y is determined by the sequence of subspaces jm (Y )a of jm (X)a , so by jm (Y )a for sufficiently large m. As jm (X)a is defined over k(a) and jm (Y )a is isomorphic over b to Wm , it follows that for sufficiently large m, c ∈ k(a, b, em ). This concludes the proof of Theorem 1.1. We will give a couple of consequences of Theorem 1.1 for the model theory of differentially closed fields. In [11] the “Zilber dichotomy” was proved for strongly minimal sets in differentially closed fields. The proof used the deep and difficult theorem on Zariski geometries from [12]. Theorem 1.1 yields immediately this Zilber dichotomy. First some words of explanation. By a strongly minimal set in (U, ∂) we mean a definable set X of Morley rank and degree 1. This means that X is infinite, but every definable subset of X is finite or cofinite. For X strongly minimal, model-theoretic algebraic closure on X yields a pregeometry or matroid. X is said to be modular, if working over a sufficiently large set of parameters, we have dim(A) + dim(B) − dim(A ∩ B) = dim(A ∪ B), for A, B algebraically closed subsets of X. Modularity has attractive consequences, especially for definable groups. Now C is a strongly minimal set in (U, ∂) which is decidedly nonmodular. X is said to be nonorthogonal to C is there a finite-to finite definable relation 11

between X and C. Corollary 3.10 Let X be a strongly minimal set in (U, ∂). Then either X is modular or X is nonorthogonal to C. Proof. Assume X to be defined without parameters. If X is nonmodular there are tuples a, b from X such that c = Cb(tp(a/b)) is not contained in acl(a). So tp(c/a) is nonalgebraic and by Theorem 1.1, internal to C. As c is essentially a tuple from X this yields some nontrivial relation between X and C giving nonorthogonality. The next consequence is for finite-dimensional differential algebraic groups (equivalently groups of finite Morley rank definable in (U, ∂)). Certain weaker statements appear in [9], which go under the name of the “socle lemma” or “socle argument”. It should be said that the next result does not formally follow from 3.10 Corollary 3.11 Let G be a connected group of finite Morley rank definable in (U, ∂) over K. Let a ∈ G with p = tp(a/K) stationary. Let H < G be the left-stabilizer of p. Then tp(H · a/K) is internal to C. Proof. Let c ∈ G be generic over a. By Lemma 2.6 of [17], the right coset H ·a is interdefinable over K and c with the canonical base d of tp(c/K ∪ {a · c}). By Theorem 1.1, tp(d/K ∪ {c}) is internal to C. Hence tp(H · a/K ∪ {c}) is internal to C. As H · a is independent from c over K, tp(H · a/K) is internal to C.

4

Difference fields of characteristic 0

In this section we prove Theorem 1.2. We will be working in the context of existentially closed difference fields of characteristic zero. These are precisely the models of a first order theory, called ACF A0 (in the language of rings together with a symbol forthe automorphism). This theory was studied in detail in [6], which we refer the reader to for background. The Zilber dichotomy (types of SU -rank 1 are modular or nonorthogonal to the fixed field) was proved there, using somewhat involved and lengthy arguments. First will need another fact about jet spaces of varieties. 12

Fact 4.1 Suppose V1 , V2 , and W ⊆ V1 × V2 are irreducible varieties defined over a field K of characteristic zero. Suppose that the projections πi : W → Vi are dominant and generically finite-to-one for i = 1, 2. Let (a, b) be a generic point of W over K. Then jm (W )(a,b) is the graph of an isomorphism between jm (V1 )a and jm (V2 )b . Let us now give the analogue of Lemma 3.1 for difference modules. Suppose that (K, σ) is a difference field. By a σ-module over (K, σ) we mean a finite-dimensional K-vector space V together with an additive automorphism Σ of V such that Σ(cv) = σ(c)Σ(v) for c ∈ K and v ∈ V . Lemma 4.2 Let (V, Σ) be a σ-module over (K, σ). Let (V, Σ)] be {v ∈ V : Σ(v) = v}. Then (i) (V, Σ)] is a vector space over F ix(σ) of dimension at most dimK (V ). (ii) If (K, σ) is existentially closed, then there are v1 , .., vs ∈ (V, Σ)] which form a K-basis of V . Proof. Let k denote F ix(σ) < K. (i) Clearly (V, Σ)] is a k-vector space. Induction on n yields: if v1 , .., vn ∈ (V, Σ)] are linearly dependent over K, then they are linearly dependent over k. (ii) After choosing a basis for V over K, the σ module (V, Σ) is transformed into a σ-module (K d , Aσ) where A is the matrix of Σ with respect to the basis. It suffices to prove the desired result for (K d , Aσ). Note that (K d , Aσ)] = {v ∈ K d : σ(v) = A−1 v}. Matrix multiplication by A−1 is a surjective map from the set of n × n matrices over K to itself. As (K, σ) is existentially closed, there is a nonsingular n×n matrix U over K such that σ(U ) = A−1 U . The columns of U are elements of (K d , Aσ)] which form a basis of K d over K. We now work in saturated model (U, σ) of ACF A0 . K denotes an (algebraically closed) small difference subfield of U. For X a variety defined over K, X σ denotes the image of X under σ (also defined over K). Lemma 4.3 Let X, W be irreducible affine varieties defined over K such that W ⊆ X × X σ , and the projections π1 , π2 from W to X, X σ are dominant and generically finite-to-one. Let (a, σ(a)) be a generic point of W over K. Fix m ≥ 1 and let f : jm (X)a → jm (X σ )σ(a) be the isomorphism given by 13

Fact 3.1. Then (i) (jm (X)a , f −1 σ) is a σ-module (over (U, σ)). (ii) if K1 is an algebraically closed difference field containing K and X1 is the variety over K1 whose generic point is a, then jm (X1 )a is a σ-submodule of (jm (X)a , f −1 σ). Proof. (i). It is clear that jm (X σ )σ(a) is precisely (jm (X)a )σ (namely the image of jm (X)a under σ). Hence f −1 σ is an additive automorphism of jm (X)a . As f is a linear isomorphism, we see that (jm (X)a , f −1 σ) is a σmodule. (ii) Let W1 be the irreducible variety over K1 whose generic point is (a, σ(a)). By Fact 3.1, jm (W1 )(a,σ(a)) is the graph of an isomorphism f1 between jm (X1 )a and jm (X1σ )σ(a) . Clearly f1 is the restriction of f to the subspace jm (X1 )a of jm (X)a , which yields (ii). We should now discuss a little more of the model theory of ACF A, especially canonical bases. ACF A the theory of existentially closed difference fields is not complete, even after fixing the characteristic. The completions of ACF A are all unstable, and do not have quantifier-elimination. On the other hand, these completions do have some “positive” properties. They admit elimination of imaginaries and have the technical property simplicity. Simplicity of a theory gives rise to a well-behaved notion of independence (nonforking) which is easy to describe in the ACF A case. If a and b are tuples from U and K is a difference subfield, then a is independent from b over K iff the difference fields k1 and k2 generated over k by a, b respectively, are algebraically disjoint over k. This can be discerned at the quantifier-free level. The “correct” notion of canonical base introduced in [8], is for so-called Lascar strong types and depends on the notion of an “amalgamation base”. In the ACFA context, any complete type p(x) over an algebraically closed difference field will be a Lascar strong type and we write Cb(p(x)) for its canonical base. This will be a finite tuple from K, unique up to interdefinability. On the other hand there is a somewhat more transparent notion of canonical base depending only on the quantifier-free type: if K is algebraically closed and a is some n-tuple, then the family of difference algebraic subvarieties of U n which contain a and are defined over K, has a smallest element X say, and we define Cb(qf tp(a/K)) to be a canonical parameter for X. For those familiar with “local stability theory”, Cb(qf tp(a/K)) is a tuple of generators for the 14

smallest difference subfield K0 of K such that a is independent from K over K0 and qf tp(a/K0 ) is stationary. With this notation we have: Fact 4.4 Cb(qf tp(a/K)) ⊆ dcl(Cb(tp(a/K))), and Cb(tp(a/K)) ⊆ acl(Cb(qf tp(a/K))). As in the differential case we will be reducing these canonical bases to data of an algebraic-geometric nature. aclf (−) denotes field-theoretic algebraic closure. Lemma 4.5 Suppose that σ(a) ∈ aclf (K, a). Let K1 ⊃ K be algebraically closed (in ACF A). Let V1 be the irreducible variety over K1 with a as a K1 -generic point. Then the field of definition c of V1 is contained in Cb(qf tp(a/K1 )) and the latter is contained in the algebraic closure of K and c. Proof. It is clear that the field of definition of V1 is contained in Cb(qf tp(a/K1 )). For the second part, note that a is independent from K1 over K, c in the sense of algebraically closed fields. Our hypotheses imply that (σ i (a) : i ∈ Z) is contained in aclf (K, a), whereby (σ i (a) : i ∈ Z) is independent from K1 over K(c) in the sense of algebraically closed fields. We will say that a type p(x) over algebraically closed K is almost internal to the ∅-definable set X, if for some A ⊃ K and a realizing p such that a is independent from A over K, a ∈ acl(A, X). The definable set X ⊆ U n is said to be finite-dimensional if there is a finite bound on tr.deg(K < a > /K) for a ∈ X, where K is some difference field over which X is defined, and K < a > denotes the difference field generated by K and a. Proof of Theorem 1.2. Replacing a by some (a, σ(a), .., σ r (a)) we may assume that aclf (K, a) = aclf (K, σ(a)). Let X be the variety over K whose generic point is a, and W the variety over K whose generic point is (a, σ(a)). Then X, W and a satisfy the hypotheses of Lemma 4.3. Let Y be the variety over K1 whose generic point is a. By 4.5, Cb(tp(a/K1 ) is interalgebraic over K with a tuple c which generates the smallest field of definition of Y . Y is determined by jm (Y )a ⊂ jm (X)a for large enough m. Let f be as in Lemma 4.3. So 15

jm (Y )a is a σ-submodule of the σ-module (jm (X)a , f −1 σ). Let b be a basis of jm (X)a over U as in 4.2(ii), which is moreover independent from c over K < a >. With respect to this basis (jm (X)a , f −1 σ) becomes isomorphic (as a σ-module) to some (K d , σ), and this isomorphism takes jm (Y )a to some σsubmodule of K d , namely to a vector subspace Lm of K d which is σ-invariant, hence defined over F ix(σ). Let em be a finite tuple from F ix(σ) over which Lm is defined. Then c ∈ K(a, b, em ), completing the proof. As in Corollary 3.10, we deduce from Theorem 1.2 the “Zilber dichotomy” for types of SU -rank 1 in completions of ACF A0 : any type of SU -rank 1 is modular or nonorthogonal to the fixed field of σ. This was the culminating result of [6]. A suitable difference version of Corollary 3.11 also follows.

5

Further remarks on jet bundles and prolongations

In Remark 3.9 we introduced the differential m-jet of (X, s)] at a point a ∈ (X, s)] , as a subset of jm (X)a . Let us denote this differential m-jet ∂ ∂ by jm ((X, s)] )a . These fit together to form jm ((X, s)] ), the differential m] jet bundle of (X, s) , another differential algebraic variety. In 2.3 and 2.4 we gave explicit equations for the m-jet bundle jm (X) of X. Here, for the benefit of the interested reader, we will give explicit differential equations ∂ for jm ((X, s)] ) as a subset of jm (X). This actually constituted our original approach before we noticed the rather simpler account using ∂-modules. We work again in the saturated differentially closed field (U, +, ., ∂). X is an irreducible subvariety of U n defined over K. The first (Buium) prolongation τ (X) (implicit in our treatment above) is the subvariety of U 2n defined by the equations P (x1 , .., xn ) = 0 and P ∂ i ((∂P/∂xi )(x))yi + P (x) = 0 for P ∈ IX . τ (X) projects canonically onto X. Let s : X → τ (X) be a regular section, defined over K, and X ] = {a ∈ X : (a, ∂(a)) = s(a)}. jm (X) is the mth jet bundle as in Remark 2.4. Let j = jm for some fixed m ≥ 1. The “functoriality” of j yields j(s) : j(X) → j(τ (X)). We will exhibit a canonical morphism h : j(τ (X)) → τ (j(X)) such that for a ∈ X ] ,

16

j ∂ (X ] )a = {u ∈ j(X)a : ∂(u) = h ◦ j(s)(u)}. Let D be the set of differential operators of order at most m in variables x1 , .., xn as in 2.3 with the natural identifications. Let D1 be the same thing, but with variables x1 , .., xn , y1 , .., yn . Note that D is canonically a subset of P D1 . j(X) = {(a, uD )D∈D : a ∈ X, D DP (a)uD = 0, P ∈ IX } and j(τ (X)) = P {(a, b, uD )D∈D1 : (a, b) ∈ τ (X), D∈D1 DQ(a, b)uD = 0, Q ∈ Iτ (X) }. For D ∈ D, let LD be the set of those operators in D1 which can be obtained from D by replacing exactly one occurrence of some ∂/∂xi by ∂/∂yi . So for example if D is ∂ 2 /∂x21 then LD = {∂ 2 /∂x1 ∂y1 }, and if D = ∂ 2 /∂x1 ∂x2 then LD = {∂ 2 /∂x1 ∂y2 , ∂ 2 /∂x2 ∂y1 }. Now for (a, b, uD )D∈D1 ∈ j(τ (X)), let h(a, b, uD )D∈D1 = (a, uD , b, vD )D∈D P where vD = D0 ∈LD uD0 . We leave the proof of the following to the interested reader: Lemma 5.1 (i) h : j(τ (X)) → τ (j(X)). Moreover if πi (i = 1, 2) are the natural projections from j(τ (X)), τ (j(X)) respectively to τ (X), then π2 .h = π1 . (ii) for a ∈ X ] , j ∂ (X ] )a = {u ∈ j(X)a : h(J(s)(a, u)) = (a, u, b, ∂u)}, where s(a) = (a, b) ∈ τ (X).

6

Remarks on the positive characteristic case

We mention here some problems and partial results concerning the generalization of the results above to the positive characteristic cases. The general problem here is that certain relevant finite morphisms need not be separable and so will not induce isomorphisms at the level of jets. Let us first consider ACF Ap . As the πi in Fact 4.1 need not be separable, we may not obtain a linear isomorphism between j(V1 )a and j(V2 )b . In fact this cannot even be expected, as there are many different “fixed fields”, F ix(σ n F rm ) for n, m ∈ Z. So the most one can hope for in Theorem 1.2 is almost internality to the union of all fixed fields. The obvious idea is as follows: suppose we are given finite-dimensional tp(a/K) where a is a generic point of V over K and (a, σ(a)) is a generic point of W over K where the projections from W to both V and V σ are finite-to-one. Replace σ by τ = σ n F rm (still yielding a model of ACF Ap ). (a, τ (a)) is now a generic point of some W 0 over K which again projects finite-to-one to V and 17

V τ . Even though these projections need still not be separable, they may induce an isomorphism between some nonzero subspace of j(V )a and one of j(V τ )τ (a) . One may be tempted to try to show that the union of all these subspaces of j(V )a (as τ varies) generates (or is Zariski-dense) in j(V )a . However it is not clear if this approach works even in some specific examples such as the following pointed out by Zoe Chatzidakis: Let V be 2-space and W = {(x1 , x2 , y1 , y2 ) : y1 = x2 , y2p + xp1 + y1 = 0}. Let us now consider the characteristic p differential case. By this we mean separably closed fields of finite (nonzero) Ershov invariant e. It is convenient for our purposes to consider such fields equipped with several Hasse derivations (mainly because the ∂-module theory extends smoothly). A theory was developed by Messmer and Wood [15], and an alternative approach was recently developed by the second author [19]. In the case e = 1 these approaches essentially coincide. For convenience we restrict our attention to this case (e = 1) although everything we say generalizes (using the theory in [19]). So the relevant theory is the theory SCHp,1 of separably closed fields K of characteristic p and Ershov invariant 1, equipped with a strict Hasse (or iterative) derivation. The Hasse derivation is by definition a sequence D = (D0 , D1 , ...) of additive maps from K to K such that D0 = id, P Dn (xy) = i+j=n Di (x)D (y) and ! j i+j Di ◦ Dj (x) = Di+j (x) for all n, i, j. i Strictness means that K p is the field of constants of D1 . The theory is complete with quantifier-elimination in the obvious language (the language of rings together with the Di ’s), as well as being stable. If {t} is a p-basis of K over K p , then by considering a suitable Wronskian, we see that for any x ∈ K, the sequence (Di (x) : i < pn − 1) is birational over {Di (tj ) : i, j < pn − 1} P with the sequence (ai : i < pn − 1) of pn th powers in K such that x = i ai ti . Work in a saturated model (U, +, ., Di )i of SCHp,1 . K will denote a relatively algebraically closed substructure (or even a model). C is the field of absolute constants of U, that is {a ∈ U : Di (a) = 0 for all i}. C coincides with n n ∩n U p . In fact more precisely U p is the common zero set of D1 , ..., Dpn−1 . Definition 6.1 (i) tp(a/K) is thin if trdeg(K(Di (a) : i < ω)/K) is finite. (ii) tp(a/K) is very thin if L = K(Di (a) : i < ω) is finitely separably generated over K, that is if there is a finite tuple b from L such that L is separably 18

algebraic over K(b). Remark 6.2 (i) This notion of thinness coincides with that in [9]. (ii) Any extension of a very thin type is very thin. Our partial result is: Proposition 6.3 Let tp(a/K) be very thin. Then for any b, tp(Cb(stp(a/K, b)/K, a) is internal to C. We sketch how to adapt the previous arguments. The main point is that for any m the mth jet at a of a suitable variety is (as a U-vector space) equipped with a D-module structure. The theory of iterative linear differential equations ([14]) allows the proof of Theorem 1.2 to go through. Let L = K(Di (a) : i < ω). By the properties of the Hasse derivation D and as tp(a/K) is assumed to be be very thin, we can find finite tuples a0 ⊆ a1 ⊆ a2 ... in L such that (i) a ⊆ a0 , (ii) L = K(a0 , a1 , ....), (iii) K(ai ) is closed under D0 , .., Dpi , and (iv) L is separably algebraic over K(a0 ). Let Di = {D0 , .., Dpi }. Let Xi be the (absolutely irreducible) variety over K whose generic point is ai . Let fi : Xi → Xi−1 be the surjective morphism induced by the inclusion ai−1 ⊆ ai . Let gi : Xi → X0 be likewise. Let Oi be the local ring of rational functions over U on Xi which are defined at ai . As K(ai ) is closed under the operators in Di , Oi is naturally equipped with a “truncated” Hasse ring structure, namely with an action of D0 , .., Dpi extending the action on U and satisfying the relevant properties. Clearly the maximal ideal Mi of Oi (that is, the functions which are 0 at ai ) is a Di subring, as are all powers of Mi . Let us fix m ≥ 1. So the finite-dimensional U-vector space Vi = Mi /Mm+1 is equipped with a Di -module structure. In i particular, Vi is a Dj -module for j < i. Now, as fi is etale, it induces an isomorphism fi∗ between Vi−1 and Vi , which is moreover an isomorphism of Di−1 modules. So, by virtue of the isomorphisms gi∗ , the U-vector space V0 is equipped with a D-module structure. Let jm = jm (X0 )a0 be the dual space to V0 (the mth jet to X0 at a0 ). As in Remark 3.3, jm is equipped with a D-module structure Djm say. By [14], the 0-set jm,] of Djm is a vector space over C with the same dimension as the dimension of the U-vector space jm . 19

Moreover there are v1 , .., vd ∈ jm,] which form a U-basis of jm . With respect to such a basis the D-module (jm , Djm ) becomes isomorphic to (U d , D), and note that, as in 3.2, any U-subspace of U d which is closed under D is defined over C. Now suppose that tp(a/K 0 ) is a stationary extension of tp(a/K). Let Y0 be the variety over K 0 whose generic point is a0 . Let c be generators of the smallest field of definition of Y0 . We may assume that the basis (v1 , .., vd ) of jm above is chosen to be independent from c over K, a (in the sense of the 0 ambient stable structure (U, D)). Let jm denote the the mth jet of Y0 at a0 . 0 d jm is then a D-submodule of (U , D) and hence is defined over some em in C. 0 As Y0 is determined by all these jm , it follows that c ∈ K(a, (v1 , .., vd ), (em )m ). Thus tp(Cb(tp(a/K 0 ))/K, a) is internal to C. The proof is complete. Remark 6.4 Suppose that A is an ordinary semi-abelian variety over definably closed K < U. Let A] = ∩n pn (A(U)) (a type-definable connected subgroup of A(U)). Let a be a generic point of A] over K. Then tp(a/K) is very thin. Proof. We will make use of the “Verschiebung” as described in [1] to which the reader is referred for more background and references. First, what is the meaning of “ordinary”? Let us work for now geometrically, that is inside an algebraically closed field. The semi-abelian variety A is by definition an extension of an abelian variety B by an algebraic torus T . Let b = dim(B). A is said to be ordinary if the group of p-torsion point of the abelian part B of A is precisely (Z/pZ)b (which implies that the group of pt -torsion points of B is (Z/pt Z)b for all t > 0). Now for the Verschiebung. We can hit the coefficients defining A with the Frobenius F r to obtain another semiabelian variety A(p) . Moreover, acting on coordinates, F r yields a bijective isogeny F r : A → A(p) . The dual isogeny from A(p) to A is called the Verschiebung V , and both V ◦ F r : A → A and F r ◦ V : A(p) → A(p) are just multiplication by p in the relevant groups (that is x → px in additive notation). Similarly the dual isogeny Vn to n F rn : A → A(p ) has the feature that composition with F rn is multiplication by pn . The main fact we use is: n (*) if A is ordinary then each Verschiebung map Vn : A(p ) → A is separable. Now let A, K, A] , a be as in the hypotheses of the proposition. We will show that Di (a) is separably algebraic over K(a) for all i. Fix n. Then by 20

definition of A] there is c ∈ A(U) such that a = pn c (in the group A written additively). As multiplication by pn in A is the same as Vn ◦ F rn , it follows n n that a = Vn (b) where b = F rn (c) = cp , and so a ∈ K(cp ). It follows that n n D0 (a), .., Dpn−1 (a) ∈ K(cp ). But by (*) b = cp is separably algebraic over K(a). The same is thus true of the Di (a). This completes the proof. Finally let us give an example of a thin but not very thin type (in the context of SCHp,1 ). Let K be an elementary submodel of U. In particular K contains n a p-basis {t} of U. We may assume that Dpn (tp ) = 1 for all n ≥ 0. Let c ∈ C \ K. By saturation of U we can find an element a ∈ U, transcendental −1 −n n over K(c), such that a − (ct + cp tp + .. + cp tp ) is a pn+1 th power in U −n for all n. Then Dpn (x) = cp for all n ≥ 0, and K(D0 (a), D1 (a), ...) is not finitely separably generated over K (but is of course of transcendence degree 2 over K). On the other hand tp(a/K) is “2-step analyzable” in C: first −1 −2 c ∈ dcl(K, a). Let K 0 = dcl(K, c) = K(c, cp , cp , ...). Then a ∈ / K 0 , but 0 Di (a) ∈ K for all i > 0, and so the difference of two realizations of tp(a/K 0 ) is in C. Clearly a type of U -rank 1 which is nonorthogonal to C is very thin. We do not know of any type of U -rank 1 which is thin but not very thin.

7

Relations with Mordell-Lang and ManinMumford

In [9], Hrushovski gave model-theoretic proofs of the Mordell-Lang conjecture for function fields in all characteristics. In [10] he gave a model-theoretic proof of the Manin-Mumford conjecture over number fields. In this section we will discuss the impact of our methods and results on these proofs. The general thrust is that our methods are able to eliminate the most delicate part of the model theory in these proofs, at least in the characteristic zero cases. The Mordell-Lang conjecture for function fields (or the relative MordellLang conjecture) is, in its most general form the following. Theorem 7.1 Let k < K be algebraically closed fields. Let A be a semiabelian variety over K, X an irreducible subvariety of A defined over K, and 21

Γ the group of prime-to-p division points of a finitely generated subgroup of A(K). Assume that X ∩Γ is Zariski-dense in X and that StabA (X) is trivial. THEN, after possibly translating X, there are a semiabelian subvariety A0 of A containing X, a semiabelian variety A1 defined over k, and an isomorphism f : A0 ∼ = A1 such that f (X) is defined over k. Let us begin with the characteristic zero case. Buium [3] essentially proved Theorem 7.1 in the case where A is an abelian variety. His proof had two steps. Step 1 was to embed the situation into Kolchin’s differential algebraic geometry. That is K is equipped with a derivation ∂ whose field of constants is k, and a finite-dimensional differential algebraic subgroup G of A is constructed such that that Γ < G. So now X ∩G is Zariski-dense in X. Step 2 is to use this additional differential algebraic structure to perform the required descent to k. Buium used various analytic arguments in Step 2, including Big Picard and a result of Hamm on Lie groups. Hrushovski’s proof of Theorem 7.1 in the characteristic zero case had the same overall structure. Step 1 was exactly as in Buium. But Step 2 used a model-theoretic analysis of sets of finite Morley rank definable in differentially closed fields. The deepest ingredient was the Zilber dichotomy: that a strongly minimal set is either modular or nonorthogonal to the field of constants. This in turn used the main theorem on Zariski geometries from [12] whose proof is extremely complicated. Another ingredient concerned a reduction to sets very close to strongly minimal sets (the “socle argument”). Our Corollary 3.11 subsumes both these ingredients. We will point out now how Corollary 3.11 plugs into the end of Hrushovski’s proof. This remains a proof in “model-theoretic language” and we use freely model-theoretic notions. With some additional work, operating in the category of “algebraic D-groups”, it is possible to transform the proof below into a purely algebraic one, and this has been done in [18]. Sketch of proof of Theorem 7.1 in characteristic zero. First, possibly enlarging K, equip K with a derivation ∂ such that (K, ∂) is a differentially closed field with field of constants k. We now work in the structure (K, +, ·, ∂). Identify A and X with their sets of K-rational points. As in Buium [3], let G be a connected definable subgroup of A(K) of finite Morley rank which contains Γ. Then X ∩ G is Zariski-dense in X. X ∩ G is a differential alge22

braic subset of G, so has finitely many irreducible components in the Kolchin topology. So one of them, Y say, is Zariski-dense in X. As X has trivial stabilizer in G, Y has trivial stabilizer in G. Assume K1 is a countable differential subfield of K over which all the data so far is defined. Let p be the generic type of Y over K1 . By Corollary 3.11, p is internal to k. By translating, we may assume that Y (and so X) contains the identity. By Zilber’s indecomposability theorem, Y generates (in finitely many steps) a connected definable subgroup H of G. Then H is also internal to k, and thus definably isomorphic to a definable group of the form B(k) where B is a commutative algebraic group defined over k. Let f : B(k) ∼ = H be the definable isomorphism. By quantifier-elimination in DCF0 , f is the restriction to B(k) of some rational map (over K) and so extends to a rational homomorphism h from B into A. The kernel of h contains the unipotent radical of B, hence is defined over k. So if ker(h) is nontrivial then it meets B(k) nontrivially. But h|B(k) is injective, hence Ker(h) is trivial, and h is an isomorphism between B and a semiabelian subvariety A1 of A (which contains X). Now Y is a Zariski-dense subset of X, so h−1 (Y ) is a Zariski-dense subset of h−1 (X). As h−1 (Y ) is contained in B(k), it follows that g −1 (X) is defined over k. The proof is complete. Let us now turn to the positive characteristic case. Hrushovski’s proof of Theorem 7.1 is the only known one. On the other hand, our 6.3 and 6.4 yield as above a proof in the case where the semiabelian variety A is ordinary. W can reduce to the case where the data, A, X, Γ are defined over the separable closure of k(t). We call this separable closure K and equip it with an interative Hasse derivation such that k is the field of absolute constants. We may also assume the iterative Hasse field (K, D) to be saturated. Let G = p∞ (A(K)), a type-definable subgroup of A which is very thin by 6.4. Y = X ∩ G is Zariski-dense in X. Y has trivial stabilizer in G, so by 6.3 and the analogue of Corollary 3.11, the subgroup H of G generated by Y is internal to k. So H is definably isomorphic to B(k) for some algebraic group B defined over k, and as in the proof above, we obtain a rational isomorphism h between B and a semiabelian subvariety A1 of A such that h−1 (X) is alo defined over k. In [1], Abramovich and Voloch proved various cases of the Theorem 7.1 in postive characteristic, including this case where A is ordinary. Some kind of 23

Gauss jet map also plays a role in their proof. We do not as yet understand the complete picture of the relationship Abramovich and Voloch’s work and the work here. The crucial ingredient of Hrushovski’s proof of the full statement of Theorem 7.1 was again the Zariski geometries theorem in [12], or rather a version of it for “type-definable” Zariski geometries. We consider it important and urgent to find an algebraic proof. Finally let us discuss the Manin-Mumford conjecture. The result here is: Theorem 7.2 (characteristic zero) Let A be a semiabelian variety, and X a subvariety, all defined over a number field k. Let T or(A) denote the group of torsion elements of A. Then the Zariski closure of X ∩ T or(A) is a finite union of translates of semiabelian subvarieties of A. Many proofs of this theorem have been given. Hrushovski’s used definability in models of ACF A0 . The first step, purely algebraic, is to work in an existentially closed difference field (K, σ), such that k is contained in the fixed field of σ and such that T or(A) is contained in a very special finite rank difference algebraic subgroup G of A. (Specifically G = Ker(P (σ)) where P (T ) is a polynomial over Z with no complex roots of unity among its roots.) The second step is to show that G is “modular” which implies that all definable subsets are Boolean combinations of translates of subgroups. The deepest ingredient of this second step is the Zilber dichotomy for types of SU -rank 1 in ACF A0 (they are modular or nonorthoghonal to the fixed field), which was proved in [6], but also follow from our Theorem 1.2. So in this sense, our work again simplifies the proof. It is quite likely that our methods can be applied directly to give a new, algebraic proof of Manin-Mumford. This will be treated in a future paper.

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[3] A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Annals of Math. 136 (1992), 557-576. [4] A. Buium, Differential algebraic groups of finite dimension, Lecture Notes in Math. 1506, 1992. [5] F.Campana, Algebricit´e et compacite dans l’espace des cycles d’un espace analytique complex.Math.Annalen 251 (1980), 7-18. [6] Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, Transactions AMS 351 (2000), 2887-3071. [7] A. Fujiki, On the Douady space of a compact complex space in the category A, Nagoya Math. J.85 (1982), 189-211. [8] B. Hart, B. Kim and A.Pillay, Canonical bases and coordinatization in simple theories, JSL 2000. [9] E. Hrushovski, The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690. [10] E. Hrushovski, The Manin-Mumford conjecture and the model theory ofdifference fields, Annals of Pure and Applied Logic, 112 (2001), 43-115. [11] E. Hrushovski and Z. Sokolovic, Minimal subsets of differentially closed fields, preprint 1992. [12] E. Hrushovski and B. Zilber, Zariski geometries, Journal AMS 9(1996). [13] D. Marker, Model theory of differential fields, in Model theory of fields, ed. Marker, Messmer, Pillay, LectureNotesin Logic 5, Springer, 1996. [14] B. H. Matzat and M. van der Put, Iterative differential equations and the Abhyankar conjecture, preprint 2001. [15] M. Messmer and C. Wood, Separably closed fields with Hasse derivations I, Journal of Symbolic Logic 60 (1995), 898-910. [16] D. Pierce and A. Pillay, A note on the axioms for differentially cloed fields of characteristic zero, Journal of Algebra,204 (1998), 108-115.

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[17] A. Pillay, Remarks on a theorem of Campana and Fujiki, Fundamenta Mathematicae 174 (2002), 187-192. [18] A. Pillay, Mordell-Lang for function fields in characteristic zero, revisited, to appear in Compositio Math. [19] M. Ziegler, Separably closed fields with Hasse derivations, Journal of Symbolic Logic, 68 (2003), 311-318.

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