On algebraic σ-groups.

Piotr Kowalski^{∗} Wroclaw University University of Illinois

Anand Pillay^{†} University of Illinois University of Leeds

March 31, 2006

Abstract

We introduce the categories of algebraic σ-varieties and σ-groups over a difference field (K, σ). Under a “linearly σ-closed” assumption on (K, σ) we prove an isotriviality theorem for σ-groups. This theorem yields immediately the key lemma in a proof of the Manin-Mumford conjecture. The present paper uses crucially ideas from [10] but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski’s work [5] and are also closely related to papers of Pink and Roessler ([11] and [12]).

1 Introduction

The notion of an algebraic σ-variety over a difference field (K, σ) generalizes the notion of an algebraic variety equipped with a self-map. So, if K is an algebraically closed field, and σ an automorphism of K by an algebraic σvariety (over (K, σ)) we mean an algebraic variety X over K together with a morphism ϕ from X to X^{σ}, sometimes assumed to be dominant. The category of algebraic σ-varieties over a difference field (K, σ) belongs entirely to algebraic geometry, but captures some of the geometry of difference equations. A trivial σ-variety is one of the form (X, id) where X is defined over the fixed field of σ. The main result of this paper (Theorem 3.7) is an “isotriviality” theorem for algebraic σ-groups over a “linearly σ-closed” difference field (K, σ): Assume (G, ϕ) to be a separable algebraic σ-group, and X a σ-subvariety of G which generates G (all defined over K), then there is a normal algebraic σ-subgroup N < Stab(X) of G such that (G/N, ϕ/N) is isomorphic to a trivial algebraic σ-group, again with N and the trivializing isomorphism defined over K. The result thus ties up with both issues of descent (to the fixed field of σ) and “periodicity” of ϕ.

∗Supported by funds from NSF Focused Research Grant DMS 01-00979, and by the Polish KBN grant 2 P03A 018 24

†Supported by NSF grants

The “linearly σ-closed” assumption on (K, σ) is that K is algebraically closed and that linear difference equations over (K, σ) have “enough solutions” in K. The proof of the isotriviality theorem (Theorem 3.7) is elementary, and makes use of a higher Gauss map. It is an adaptation of the more model-theoretic proofs in [10], but the construction also appears in [1].

In section 4 we point out how Theorem 3.7 yields an elementary proof of the Manin-Mumford conjecture concerning the intersection of a subvariety X of a semi-abelian variety A with the torsion subgroup of A. In that section we will go into more details regarding the connection with other work and the benefits or even superiority of our methods.

Both authors would like to thank the organizers of the Arizona Winter School in Logic and Number Theory (March 2003) where some of the work presented here was done. Thanks also to the referee for his/her helpful comments on the organization and emphasis of the paper.

2 Algebraic σ-varieties and their basic properties.

Let K be an algebraically closed field, which we often assume to have uncountable transcendence degree. Let Fr denote the Frobenius map Fr(x) = x^{p} on K in case K has characteristic p > 0.

We identify an algebraic variety X over K with its set of K-rational points.

Let us fix an automorphism σ of K (so (K, σ) is a difference field.) Let C denote the fixed field {x ∈ K : σ(x) = x} of σ. For X a variety over K, X^{σ} is the variety over K obtained from X by applying σ to the coefficients of the defining data of X. Likewise if f : X → Y is a rational map defined over K, we obtain f^{σ} : X^{σ} → Y^{σ}. Note that if X is a quasiprojective variety over K, then X^{σ} = σ(X).

By an algebraic σ-variety we mean a pair (X, ϕ) where X is a variety over K and ϕ : X → X^{σ} is a morphism defined over K.

We will be interested in several classes of σ-varieties (all maps mentioned below are assumed to be defined over K): (X, ϕ) is said to be dominant if ϕ is dominant, namely ϕ(X) is Zariski-dense in X^{σ}. (X, ϕ) is separable if ϕ is dominant and separable. (X, ϕ) is trivial if X is defined over C and ϕ is the identity on X. A σ-morphism between (X, ϕ) and (Y, ψ) is a morphism f : X → Y such that the following diagram is commutative

X ⟶^{f} Y
ϕ↓  ↓^{ψ}
σ(X) ⟶_{f^{σ}} σ(Y) .

Note that the composition of σ-morphisms is also a σ-morphism, so that the family of σ-varieties becomes a category. A σ-variety (X, ϕ) will be called σ-isotrivial if it is σ-isomorphic to a trivial σ-variety. A σ-rational map from (X, ϕ) to (Y, ψ) is a rational map f : X → Y such that f^{σ} ∘ ϕ = ψ ∘ f holds on a Zariski-dense Zariski open subset of X. If (X, ϕ) is a σ-variety, by a (closed) σ-subvariety of (X, ϕ) we mean a (closed) subvariety Y of X such that ϕ(Y) ⊆ Y^{σ}, namely such that (Y, ϕ|Y) is a σ-variety itself. We will call a σ-subvariety Y of (X, ϕ) dominant (separable) if (Y, ϕ|Y) is dominant (separable). We will say that (X, ϕ) is an irreducible σ-variety if X is irreducible as an algebraic variety.

The following summarizes some straightforward facts about the category of σ-varieties.

Lemma 2.1 Let (X, ϕ) and (X′, ϕ′) be σ-varieties and f : X → X′ a σmorphism.

(i) If (X, ϕ) is σ-trivial, and Y is a subvariety of X then Y is a σ-subvariety if and only if Y is defined over C. In particular a σ-subvariety of a trivial σ-variety is itself trivial.

(ii) The Zariski-closure of f(X) is a σ-subvariety of (X′, ϕ′). (iii) If Y′ is a σ-subvariety of (Y′, ϕ′) then f^{−1}(Y′) is a σ-subvariety of (X, ϕ). (iv) {(x, y) ∈ X × X : f(x) = f(y)} is a σ-subvariety of (X × X, ϕ × ϕ).

(v) If Y_{1}, Y_{2} are σ-subvarieties of (X, ϕ) then so are Y_{1} ∪ Y_{2} and Y_{1} ∩ Y_{2}. (vi) Suppose f is dominant, (X′′, ϕ′′) is a σ-variety, g : X′ → X′′ is a morphism (of varieties), and g ∘ f is a σ-morphism between (X, ϕ) and (X′′, ϕ′′) then g is also a σ-morphism from (X′, ϕ′) to (X′′, ϕ′′). (vii) The “σ-rational map” analogues of (ii), (iii), (iv) and (vi) hold.

Proof. (i) So our assumption is that X is defined over C and ϕ = id. Right to left is clear. Left to right: Assume Y is a σ-subvariety of (X, id). So Y^{σ} = Y. Let c be a finite tuple from K generating the field of definition of Y. So σ(c) = c, whereby the tuple c is from C and Y is defined over C. (ii) Let Z be the Zariski-closure of f(X). Then Z^{σ} is the Zariski-closure of f^{σ}(X^{σ}). For x ∈ X,

ϕ′(f(x)) = f^{σ}(ϕ(x)) ∈ f^{σ}(X^{σ}) ⊆ Z^{σ}.

So ϕ′(z) ∈ Z^{σ} for all z ∈ Z, as f(X) is Zariski-dense in Z. (iii) If x ∈ f^{−1}(Y), then

f^{σ}(ϕ(x)) = ϕ′(f(x)) ∈ Y^{σ}.

Hence

ϕ(x) ∈ (f^{σ})^{−1}(Y^{σ}) = (f^{−1}(Y))^{σ}.

(iv) follows from (iii), since the diagonal is a σ-subvariety of (X′×X′, ϕ′×ϕ′).

(v) This is immediate. (vi) Let x ∈ X and x′ = f(x).Then ϕ′(x′) = f^{σ}(ϕ(x)). Hence

g^{σ}(ϕ′(x′)) = g^{σ} ∘ f^{σ}(ϕ(x)) = (g ∘ f)^{σ}(ϕ(x)) = ϕ′′(g(f(x)) = ϕ′′(g(x′)).

As f(X) is Zariski-dense in X′, we see that for all x′ ∈ X′, g^{σ}(ϕ′(x′)) = ϕ′′(g(x′)) and g is a σ-morphism. (vii) The same proofs work.

By an algebraic σ-group, we mean an algebraic group G (over K) together with a homomorphism of algebraic groups, ϕ : G → G^{σ} (also defined over K). Equivalently, an algebraic σ-group is a group object in the category of algebraic σ-varieties, namely an algebraic σ-variety (G, ϕ) such that G is an algebraic group, and the group operation is a σ-morphism from (G×G, ϕ×ϕ) to (G, ϕ). By a σ-homomorphism of algebraic σ-groups we mean the obvious thing, a homomorphism of algebraic groups which is also a σ-morphism. Likewise for a σ-subgroup of an algebraic σ-group. For an algebraic group G, and subvariety X, Stab(X) denotes as usual {g ∈ G : gX = X}, an algebraic subgroup of G.

We begin by compiling some easy facts about σ-groups.

Lemma 2.2 Let (G, ϕ) be an algebraic σ-group, H a σ-subgroup of (G, ϕ), and X an irreducible dominant σ-subvariety of (G, ϕ). Then:

(i) Stab(X) is a σ-subgroup of (G, ϕ). (ii) The principal homogeneous space G/H has the structure of a σ-variety such that the action of G on G/H is a σ-morphism. If moreover H is normal, then G/H with its σ-variety structure is an algebraic σ-group. (iii) If ϕ is dominant (so surjective) then Z(G), the center of G is a σsubgroup. (iv) The connected component H^{0} of H is a σ-subgroup of (G, ϕ).

Proof. (i) Let g ∈ Stab(X) and x ∈ X. Then ϕ(g)ϕ(x) = ϕ(gx) ∈ X^{σ}. As ϕ(X) is Zariski-dense in X^{σ}, ϕ(g) ∈ Stab(X^{σ}) = (Stab(X))^{σ}. (ii) Note that (G/H)^{σ} = G^{σ}/H^{σ}. Define

ϕ/H : G/H → (G/H)^{σ}, ϕ/H(gH) := ϕ(g)H^{σ}.

Then it is easily checked that (G/H, ϕ/H) is a PHS for (G, ϕ) in the category of algebraic σ-varieties. (iii) As ϕ : G → G^{σ} is surjective, ϕ(Z(G)) ⊆ Z(G^{σ}) = (Z(G))^{σ}. (iv) Clearly ϕ(H^{0}) is a connected algebraic subgroup of H^{σ}, hence contained in (H^{σ})^{0} = (H^{0})^{σ}.

The next lemma requires a little more work.

Lemma 2.3 Suppose that (G, ϕ) is a σ-group, (Y, id) is a trivial σ-variety and h : Y → G is a dominant σ-rational map. Then (G, ϕ) is σ-isotrivial.

Proof. Note that the connected component of G is a σ-subgroup of (G, ϕ). Likewise every irreducible component of Y is (trivially) a σ-subvariety of (Y, id). Hence we may and will assume that Y and G are irreducible. Let W be the Zariski-closure of {(x, y) ∈ Y × Y : h(x) = h(y)}. By Lemma 2.1 (iv), W is a σ-subvariety of (Y × Y, id) so by Lemma 2.1 (i) is defined over C. Let k_{0} be a countable subfield of C over which Y and W are defined. Let K_{0} be a countable field containing k_{0} over which (G, ϕ) and h are defined. Note that for x, y generic points of Y over K_{0}, (x, y) ∈ W if and only if h(x) = h(y). It follows that the restriction of W to generic points of Y over k_{0} is an equivalence relation. Hence there is an irreducible variety Z defined over k_{0} and a dominant rational map h′ : Y → Z defined over k_{0} such that for x, y ∈ Y, each generic over K_{0}, h′(x) = h′(y) if and only if h(x) = h(y). Note that h is a σ-rational dominant map between the trivial σ-varieties (Y, id) and (Z, id). Clearly we obtain a σ-birational isomorphism h′′ between (Z, id) and (G, ϕ). The pullback of the group operation on G to Z then gives a generically associative and invertible σ-rational map ∗ : Z × Z → Z. Again by 2.1, ∗ is defined over C. A classical theorem of Weil [15] then yields a connected algebraic group H defined over C and a birational isomorphism (defined over C) between Z and H which takes ∗ to the group operation of H. Putting everything together we obtain a σ-birational isomomorphism between (G, ϕ) and (H, id) which takes the group operation of G to that of H. This clearly extends to a σ-isomorphism of σ-groups between (G, ϕ) and (H, id) which completes the proof.

We will need one more σ-isotriviality result, for which the following wellknown fact about algebraic groups is needed.

Lemma 2.4 Let G be a connected algebraic group, and U the unipotent radical of the maximal normal linear subgroup L of G. Let n > 0, and let G_{n} be the connected algebraic subgroup of G generated by {x^{n} : x ∈ G}. Then G_{n} ∪ U generates G.

Proof. In the characteristic zero case G_{n} already equals G, but we give a proof valid in all characteristics.

Note that G_{n} projects onto the abelian variety G/L. Hence we may assume that G = L is linear. Now G/U is reductive, hence by [6], Exercise 12, is generated by its semisimple elements. For any semisimple s ∈ G/U there is semisimple s′ ∈ G such that s′/U = s. By [6], 15.3, s′ is contained in a maximal algebraic torus T of G. But T^{n} = T. Hence G^{n} projects onto G/U, and we are finished.

The next lemma is obvious.

Lemma 2.5 Let τ be an automomorphism of K which commutes with σ. Then for any σ-variety (X, ϕ), (X^{τ}, ϕ^{τ}) is also a σ-variety. Moreover (X, ϕ) is σ-isotrivial if and only if (X^{τ}, ϕ^{τ}) is σ-isotrivial.

Proposition 2.6 Let (G, ϕ) be a connected σ-group, and f : (G, ϕ) → (H, id) a surjective σ-homomorphism with finite kernel. Then (G, ϕ) is σisotrivial.

Proof. To begin with, let us note that as f is an n-to 1 homomorphism (some n), and f^{σ} : G^{σ} → H is likewise, and f^{σ} = f ∘ϕ it follows that ϕ is a bijective homomorphism G → G^{σ}. (In fact as the referee noted, a degree counting argument shows that ϕ is an isomorphism of algebraic groups.)

We first prove: Claim 1. The proposition is true if G is unipotent. Proof. So assume G to be unipotent and connected and (G, ϕ) to be a σgroup. We show by induction on dim(G) that (G, ϕ) is σ-isotrivial. By Lemma 2.2 (iii) and (iv) Z(G)^{0} is a normal σ-subgroup of (G, ϕ), and as G is unipotent dim(Z(G)^{0}) > 0. Now let L be the subgroup of Z(G)^{0} consisting of elements of order p. Then L is an infinite connected normal σ-subgroup of (G, ϕ). We will first show that (L, ϕ|L) is σ-isotrivial. By Proposition 11, Chapter VII of [14], L is a vector group, namely isomorphic (over K) to K^{d} for some d. Hence replacing L by K^{d} we may assume that L^{σ} = L, hence ϕ|L is an (abstract) automorphism of the group L. Our assumptions give us a finite-to-one homomorphism f′ from (L, ϕ|L) onto a trivial σ-group. So ϕ fixes ker(f′) setwise and induces the identity on L/Ker(f′). It follows easily that F ix(ϕ|L) = {x ∈ L : ϕ(x) = x} is a subgroup of L of finite index. As L is connected, ϕ is the identity on L. We have shown that L is σ-isotrivial.

By induction hypothesis (G/L, ϕ/L) is σ-isotrivial. By Lemma 2, Chapter VII of [14], the projection π : G → G/L has a rational section s. As π is a σ-homomorphism, s is a σ-rational map. As G is generated by L and the image of s, it follows that (G, ϕ) is σ-isotrivial. We have proved Claim 1.

We now return to the general case. Let N = ker(f). Then N is central (as G is connected). Let n = |N|. Thus the n-th power map g ↦ g^{n} induces a function s : H → G (such that f(s(g)) = g^{n}) which is clearly constructible. Claim 2. s^{σ} = ϕ ∘ s. Proof. The graph of s is {(f(g), g^{n}) : g ∈ G}. As f is a σ-morphism and H is σ-trivial, we have that f(g) = f^{σ}(ϕ(g)) for all g ∈ G. Hence for g ∈ G, (f(g), ϕ(g)^{n}) is in the graph of s^{σ}, which suffices.

As s is constructible there is some k ≥ 0 and a rational map s′ : H → Fr^{k}(G) such that (generically) s coincides with the composition of s′ with Fr^{−k} : Fr^{k}(G) → G. By 2.4, (Fr^{k}(G), Fr^{k}(ϕ)) is a σ-group. It follows from Claim 2 that s′ is a σ-rational map (from (H, id) to (Fr^{k}(G), Fr^{k}(ϕ)). By Lemma 2.3, the algebraic subgroup of Fr^{k}(G) generated by im(s′) is a σ-isotrivial σ-subgroup. As im(s′) = Fr^{k}(im(s)) we conclude by 2.4 that: Claim 3. The algebraic subgroup of G generated by im(s) is a σ-isotrivial σ-subgroup.

Now let U be the unipotent radical of the linear part of G. Claim 4. U is a σ-isotrivial σ-subgroup of (G, ϕ). Proof. Clearly U^{σ} is the unipotent radical of the linear part of G^{σ}. Moreover the bijective homomorphism ϕ takes U to the unipotent radical of G^{σ}. This shows that U is a σ-subgroup of G. By Claim 1, (U, ϕ|U) is σ-isotrivial.

As im(s) = {g^{n} : g ∈ g}, it follows from Claims 3 and 4 together with Lemma 2.4 that (G, ϕ) is σ-isotrivial.

3 Linearly closed difference fields and an isotriviality theorem

In this section we will we define a genericity property (for (K, σ)) which allows us to prove an important isotriviality theorem for algebraic σ-groups. This isotriviality theorem has its origin in Proposition 4.3 of Hrushovski’s [4] which has come to be known as the “socle theorem”.

Definition 3.1 Let (K, σ) be a difference field.

(i) By a σ-module over (K, σ) we mean a finite-dimensional vector space V over K together with an additive automorphism Φ : V → V such that Φ(cv) = σ(c)Φ(v) for all v ∈ V and c ∈ K. (Equivalently, a σ-module over (K, σ) is a left module for for the noncommutative ring K[σ].) (ii) We say that (K, σ) is linearly closed if (K is algebraically closed and) for any σ-module (V, Φ) over (K, σ) we can find a basis for V over K consisting of vectors v ∈ V such that Φ(v) = v.

Remark 3.2 A σ-module over (K, σ) can be thought of as yielding the linear difference equation Φ(v) = v over (K, σ) and linear closedness of (K, σ) means that we can always find over K a “fundamental system of solutions” of such an equation.

Definition 3.3 A difference field (K, σ) is said to be existentially closed if any finite system of difference equations and inequations over K with a solution in some difference field extension of (K, σ) already has a solution in K.

Fact 3.4 The difference field (K, σ) will be linearly closed in either of the following cases:

(i) (K, σ) is existentially closed, (ii) K is an algebraically closed field of characteristic p > 0 and σ is some integer power of the Frobenius.

Proof. Let (V, Φ) be a σ-module over (K, σ), where dim_{K}(V) = n say. After choosing a basis v_{1}, .., v_{n} for V over K, this σ-module becomes (K^{n}, Aσ) where A is the (invertible) n × n matrix over K such that Φ((v_{1}, .., v_{n})^{t}) = A(v_{1}, .., v_{n})^{t}. So to prove that V has a basis consisting of solutions of Φ(v) = v it suffices to find a nonsingular n×n matrix B over K such that Aσ(B) = B. Equivalently, find B ∈ GL(n, K) such that A = B^{−1}σ(B). (*)

Case (i). We can easily solve A = X^{−1}σ(X), X ∈ GL(n, −) in some difference field extension of (K, σ). Namely simply extend σ to an automorphism of the function field of GL(n, −) over K by putting σ(X) = AX. So assuming (K, σ) to be existentially closed we find a solution in GL(n, K).

Case (ii). Suppose first that σ is a positive power of the Frobenius, say σ(x) = x^{q}. Then we can find B as in (*) by for example Proposition 3 of [14] which says that for a connected algebraic group G defined over F_{q} the map taking g ∈ G to g^{−1}σ(g) is surjective. Equivalently, use that fact that H^{1}(k, GL(n)) is trivial for k perfect.

The general case (where σ is a possibly negative power of the Frobenius) follows because any power of the Frobenius yields a bijection GL(n, K) → GL(n, K).

We now point out that a σ-module over (K, σ) is really the same thing as a “σ-vector group”. Recall that a vector group G over an algebraically closed field K is simply an algebraic group over K which is isomorphic (as an algebraic group) to some power of the additive group. As such G has the structure of a vector space over K. By a σ-vector group over (K, σ) we mean a vector group G over K together with an isomorphism (of vector groups, so K-linear), ϕ : G → G^{σ}. Note that if V is a vector group, then Gr(V), the set of linear subspaces of V has the structure of an algebraic variety, and GL(V) the group of linear transformations of V has the structure of an algebraic group. If (V, ϕ) is a σ-vector group, then Gr(V) is equipped with the structure of a σ-variety (Gr(V), Gr(ϕ)) where if W is a linear subspace of V, (Gr(ϕ))(W) = ϕ(W) a linear subspace of V^{σ}. Likewise, GL(V) acquires an algebraic σ-group structure (GL(V), GL(ϕ)) where for α ∈ GL(V)), and v ∈ V^{σ}, ((GL(ϕ))(α))(v) = ϕαϕ^{−1}(v).

Fact 3.5 Suppose (K, σ) is linearly closed, and (V, ϕ) is a σ-vector group. Then

(i) (V, ϕ) is isotrivial, in fact is linearly isomorphic to a trivial σ-vector group. (ii) Both (Gr(V), Gr(ϕ)) and (GL(V), GL(ϕ)) are also isotrivial.

Proof. (i) Note that (V, ϕ^{−1}σ) is a σ-module over (K, σ). By Fact 3.4, let v_{1}, .., v_{n} be a basis of V consisting of solutions of ϕ^{−1}σ(v) = v, namely of ϕ(v) = σ(v). This basis gives rise to an isomorphism f : V → K^{n}, which is a σ-isomorphism between (V, ϕ) and (K^{n}, id). (ii) follows directly.

Lemma 3.6 Let (G, ϕ) be a separable σ-group. Then (G/Z(G), ϕ/Z(G)) is σ-isotrivial.

Proof. By 2.2 (ii) and (iii), (G/Z(G), ϕ/Z(G)) is a σ-group and the quotient map from G to G/Z(G) is a σ-homomorphism.

Let V denote the Lie algebra (tangent space to identity) of G, and let Ad_{G} : G → GL(V) be the adjoint representation. Note that V^{σ} is the Lie algebra of G^{σ}. As ϕ : G → G^{σ} is separable, its differential ϕ′ : V → V^{σ} is an isomorphism (of K-vector spaces), whence (V, ϕ′) is a σ-vector group. By Fact 3.5, (GL(V), GL(ϕ′)) is an isotrivial algebraic σ-group. It is rather easy to see that Ad_{G} is actually a σ-homomorphism from (G, ϕ) to (GL(V), GL(ϕ′)). As ker(Ad_{G}) = Z(G), we obtain a σ-embedding

ψ : (G/Z(G), ϕ/Z(G)) → (GL(V), GL(ϕ′)).

By Lemma 2.1 and Fact 3.5(ii), the image of (G/Z(G), ϕ/Z(G)) under ψ is σ-isotrivial.

Here is our “isotriviality theorem” for σ-groups.

Theorem 3.7 Let (G, ϕ) be a separable algebraic σ-group, and X an irreducible σ-subvariety of G which contains the identity and generates G. Then there is a normal connected σ-subgroup N of (G, ϕ) such that N < Stab(X) and (G/N, ϕ/N) is isotrivial.

Proof. First we recall the generalized Gauss map (see [12] and [10]). Suppose H is an algebraic group, and W an irreducible subvariety of H. Let V be the r-jet of H at the identity, for some r ≥ 1. (If r = 1, V is the tangent space.) Let F_{W} : W → Gr(V) be defined by F_{W} (x) is the image of the r-jet at the identity of the variety x^{−1}W in V, a linear subspace of V. F_{W} is a rational map, and for r sufficiently large, after quotienting by Stab(W), F_{W} becomes a birational embedding. Note also that W/ Stab(W) is Zariskiclosed in G/ Stab(W).

We now return to the context of the theorem. As ϕ is finite and separable, it induces a linear isomorphism ϕ′ from V to V^{σ} where V is the r-jet of G at the identity for sufficiently large r. So (Gr(V), Gr(ϕ′)) is a σ-variety. It is easy to check that Gr(ϕ′) ∘ F_{X} = F_{X^{σ}} ∘ (ϕ|X). Thus F_{X} is a σ-rational map from (X, ϕ|X) to (Gr(V), Gr(ϕ′)). By 2.2(i) and (ii), (G/ Stab(X), ϕ/ Stab(X)) is a σ-variety and the quotient map G → G/ Stab(X) is a σ-morphism. By 2.1(vii), the birational embedding X/ Stab(X) → Gr(V) is σ-rational. By 3.5 (Gr(V), Gr(ϕ′)) is isotrivial. By 2.1(i) and (ii) we conclude that X/ Stab(X) is σ-birational with a trivial σ-variety.

By 2.1(v) and 2.2(iii), Stab(X) ∩ Z(G) is a σ-subgroup of G which is clearly normal. Let N′ denote this subgroup. By 2.1(v) the natural embedding of G/N′ in G/ Stab(X) × G/Z(G) is a σ-embedding. By the previous paragraph X/ Stab(X) is σ-birational with some trivial (Y, id). So X/ Stab(X)×G/Z(G) is σ-birational with Y ×(G/Z(G)) which is σ-isotrivial. We clearly obtain a σ-birational isomorphism of X/N′ with a σ-subvariety of Y × (G/Z(G)), and so by 2.1, we see that X/N′ is σ-birational with a trivial σ-variety. But X/N′ generates G/N′, so we easily find some trivial σ-variety and a dominant σ-rational map from it to G/N′. We conclude by Lemma 2.3 that (G/N′, ϕ/N′) is σ-isotrivial. Let N be the connected component of N′. Then by Proposition 2.6, G/N is also σ-isotrivial.

4 On ϕ-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture

In this section we give some applications of the general formalism of algebraic σ-groups and in particular of Theorem 3.7. Our main result here is Proposition 4.1(i) and (ii), which concerns algebraic groups G equipped with an isogeny ϕ, and ϕ-invariant subvarieties X of G. This corresponds to Propositions 7.1 and 7.3 in [12]. However their 7.3 deals only with semiabelian varieties but part (ii) of our Proposition 4.1 below deals with arbitrary algebraic groups.

The importance of Proposition 4.1(i) is that it yields, by standard techniques, the Manin-Mumford conjecture. This was the route in Theorem 3.6 of [12]. So as to make the current paper self-contained and complete, we take the liberty of stating and proving the Manin-Mumford conjecture this way (Theorem 4.3 below). We also discuss a positive characteristic version of Manin-Mumford, and the role of Proposition 4.1 (i) and (ii).

Hrushovski’s proof [5] of the Manin-Mumford conjecture over number fields has a strong model-theoretic character, depending on a detailed study of definable sets in existentially closed difference fields and a crucial dichotomy theorem [2]. In [10] we gave another proof of the dichotomy theorem using higher Gauss maps (in characteristic zero). We wanted to apply our methods to obtain a direct proof of Manin-Mumford without the model-theoretic detour (as was done for Mordell-Lang in [9]). After seeing [11] and [12] the second author saw how to accomplish this and this is more or less the approach in the current paper. But [12] also contains a version of Manin-Mumford in positive characteristic, which essentially says that any counterexample is defined over a finite field. Answering a question of the second author, the first author saw how to prove a key lemma in [12] by simply applying the isotriviality theorem in the case where σ is a power of the Frobenius. In fact we generalize the key lemma from [12] from semi-abelian varieties to arbitrary algebraic groups. This is 4.1(ii) of the current paper. Another proof of the positive characteristic version of Manin-Mumford was given by Scanlon [13], using the “dichotomy theorem” in positive characteristic [3]. Our methods have so far been unable to yield this positive-characteristic dichotomy theorem in full generality.

In any case our treatment of the Manin-Mumford issues here is very closely related in mathematical content to [12], although we believe that our account of the key lemmas (our Proposition 4.1 (i) and (ii)) is more elementary and direct. The paper [11] (see also [8]) dealing just with the abelian variety case uses relatively simple arguments, but requiring more sophisticated background theories, such as a result of Ueno on stabilizer-free subvarieties of abelian varieties being of general type, as well as Matsumura’s theorem on the automorphism groups of varieties of general type.

Proposition 4.1 (i) Let K be an algebraically closed field. Let A be a semiabelian variety over K, ϕ : A → A a separable isogeny of A, and X an irreducible subvariety of A containing 0 which generates A and is ϕ-invariant (ϕ(X) ⊆ X). Assume also that Stab_{A}(X) is finite. Then ϕ is an automorphism of A of finite order. (ii) Let K be an algebraically closed field of characteristic p > 0, and let G be a connected algebraic group over K. Suppose ϕ : G → G is a surjective homomorphism (of algebraic groups). Let X be an irreducible subvariety of G which contains the identity, generates G, is ϕ-invariant, and has finite stabilizer. Assume that for some r, s > 0, ϕ^{s} ∘ Fr^{−r} : Fr^{r}(G) → G is separable. Then there is an algebraic group H defined over 𝔽_{p^{r}} and an isomorphism f of G with H such that f takes ϕ^{s} to Fr^{r} (so (G, ϕ^{s}) and (H, Fr^{r}) are isomorphic as algebraic id-groups).

Proof. (i) Without loss of generality there is an automorphism σ of K such that (K, σ) is existentially closed and A, ϕ, X are defined over the fixed field C of σ. By 3.4 (K, σ) is weakly generic and Theorem 3.7 applies. So as Stab(X) is finite, (A, ϕ) is σ-isotrivial. Thus there is a σ-isomorphism f of (A, ϕ) with (B, id) for some semiabelian variety B defined over C. Now f is defined over a finite extension of C, so for some s > 0, σ^{s}(f) = f. But clearly σ^{s}(f)ϕ^{s} = f : A → B. It follows that ϕ^{s} is the identity. (ii) Let ψ = ϕ^{s} ∘ Fr^{r}. So (Fr^{r}(G), ψ) is a separable Fr^{−r} group. By 3.4(ii) and 3.7, (Fr^{r}(G), ψ) is Fr^{−r}-isotrivial, hence there is an algebraic group H defined over 𝔽_{p^{r}} and an isomomorphism f : Fr^{r}(G) → H such that

Fr^{−r}(f) ∘ ϕ^{s} ∘ Fr^{−r} = f : Fr^{r}(G) → H.

But also clearly

f ∘ Fr^{r} = Fr^{r} ∘ Fr^{−r}(f) : G → H.

So putting these together we see that

Fr^{−r} ∘ϕ^{s} = Fr^{r} ∘ Fr^{−r}(f) : G → H.

So Fr^{−r}(f) is an isomorphism between G and H which takes ϕ^{s} to Fr^{r}.

We will now show how the Manin-Mumford conjecture (for semiabelian varieties) follows from Proposition 4.1(i). The proof is an elementary consequence of Proposition 4.1 (i) together with the following nontrivial numbertheoretic fact:

Fact 4.2 Let A be a semiabelian variety defined over a number field k. Then there is an automorphism σ of k̄ over k, and a monic integral polynomial P (T) ∈ ℤ[T] such that P (σ) annihilates T (A), the torsion subgroup of A, and neither 0 nor any roots of unity are among the zeroes of P (T).

Explanation and remarks. Write A additively. If P (T) = T^{n} + a_{n−1}T^{n−1} + .. + a_{1}T + a_{0} and σ ∈ Gal(k̄/k) then P (σ) is the (abstract) endomorphism of A(k̄) given by P (σ)(x) = σ^{n}(x) + a_{n−1}σ^{n−1}(x) + .. + a_{0}x. Details about Fact 4.2 are given in sections 5 and 6.2 of [5]. Although the condition that P (0) ≠ 0 is not explicitly mentioned it is clearly true from the construction there. The abelian variety case is dealt with in section 3 of [11]. It should be said that a nontrivial result of Serre concerning the intersection of the fields generated over k by the p-torsion and prime-to-p torsion subgroups of A(k̄) is involved here. One can avoid recourse to Serre’s result at the expense of weakening the conclusion of Fact 4.2 to: for arbitrarily large primes there is suitable P_{p}(T) and σ such that P_{p}(σ) annihilates the prime-to-p torsion of A. All that is involved here is lifting the characteristic polynomial of the Frobenius acting on the Tate module of the reduction of A mod p. In any case, using two primes as in [5], the methods of the present paper still yield an elementary proof of Manin-Mumford.

Theorem 4.3 Let k be a number field, A a semiabelian variety defined over k, X an irreducible subvariety of A, also over k. Assume that T (A) ∩ X is Zariski-dense in X, where T (A) is the torsion subgroup of A. Then X is the translate of a semiabelian subvariety of A.

Proof. After translating by an element of T (A) ∩ X, we may assume that X contains 0. Let σ ∈ Gal(k̄/k) and P (T) = T^{n} + a_{n−1}T^{n−1} + .. + a_{1}T + a_{0} be as given by Fact 4.2. Let A^{n} = A × .. × A (n times), also a semiabelian variety. Let ϕ be the endomorphism of A^{n} defined as ϕ(x_{0}, .., x_{n−1}) = (x_{1}, x_{2}, .., x_{n−1}, −a_{0}x_{0} − a_{1}x_{1} − .. − a_{n−1}x_{n−1}). Claim. ϕ is an isogeny of A^{n}, and for each r, ϕ^{r} − id is an isogeny of A^{n}.

Proof. Note that P (ϕ) = 0. If ϕ were not an isogeny of A^{n} with itself then there would be a positive-dimensional semiabelian subvariety B of A^{n} such that ϕ|B = 0. Evaluating P (ϕ) at an element x ∈ B of infinite order yields that P (0) = 0 a contradiction. So ϕ is an isogeny of A^{n}. For each r ≥ 1 there is an integral polynomial P_{r}(T) without complex roots of unity among its zeroes such that P_{r}(ϕ^{r}) = 0. It follows as above, using the assumptions on P (T) that ϕ^{r} − id is an isogeny of A^{n} with itself.

Let π : A^{n} → A be projection on the first coordinate. Let ∇ : A(k̄) → A^{n}(k̄) be given by ∇(x) = (x, σ(x), .., σ^{n−1}(x)). Let Y be the Zariski-closure of ∇(X∩T (A)). As P (σ) vanishes on T (A) it follows that ϕ(∇(x)) = ∇(σ(a)) ∈ ∇(X ∩ T (A)) for x ∈ X ∩ T (A). Hence ϕ(Y) ⊆ Y. Also X is the Zariskiclosure of π(Y). Let Y′ be an irreducible component of Y such that X is the Zariski-closure of π(Y′). Then as ϕ “permutes” the components of Y, ϕ^{m}(Y′) ⊆ Y′ for some m ≥ 1. Let X′ be the translate of Y′ by y^{−1} for some y ∈ Y′ ∩ ∇(X ∩ T (A)). Then X′ contains the identity of A^{n}, is still ϕ^{m}-invariant, and some translate of X is the Zariski closure of π(X′).

CASE 1. X′ is a semiabelian subvariety of A^{n}. Then, as 0 ∈ X, π(X′) = X and X is a semiabelian subvariety of A, proving the theorem.

CASE 2. Otherwise. Let B be the semiabelian subvariety of A^{n} generated by X′. Then ϕ^{m} is an isogeny of B with itself. Let S = Stab_{B}(X′). Then S is ϕ^{m}-invariant, and X′/S is a positive-dimensional irreducible subvariety of the semiabelian variety B/S with trivial stabilizer. Moreover ϕ^{m} induces an isogeny ϕ^{m}/S of B/S with itself, X′/S is ϕ^{m}/S-invariant, and generates B/S. By Proposition 4.1(i), some positive power of ϕ^{m}/S is the identity on B/S. This contradicts the Claim above and so proves Theorem 4.3.

We complete this section with remarks on the positive characteristic analogue of Manin-Mumford. This is theorem 4.4 below. It was proved in [12] (Theorem 3.6 (b)). A proof was also given by Scanlon [13], using the “dichotomy theorem” for rank 1 types in existentially closed difference fields of positive characteristic from [3]. The Pink-Roessler proof has two ingredients. The first is a result stating that if A is a semiabelian variety, and ϕ : A → A an isogeny, then there are semiabelian varieties A_{1}, .., A_{n} and for each i an isogeny ϕ_{i} of A_{i} such that for each i, there is s > 0 and r ≥ 0 such that ϕ^{s} is a composition of Fr^{r} with a separable isogeny, and such that (A, ϕ) is isogenous to (∑ A_{i}, ∑ ϕ_{i}). The proof uses formal groups and Dieudonné modules. The second ingredient is Theorem 4.1 above for which we have given a somewhat elementary proof. The statement below follows relatively easily (as in the proof of 4.3 above) from these ingredients.

Theorem 4.4 Let A be a semiabelian variety over an algebraically closed field K of characteristic p > 0, and X an irreducible subvariety of A such that T (A) ∩ X is Zariski-dense in X. Let B be the identity component of Stab_{A}(X). Then there is a semiabelian variety A′ and an irreducible subvariety X′ of A′, both defined over a finite field, and a homomorphism h : A′ → A/B with finite kernel, such that X/B is a translate of h(X′).

5 Additional remarks

In [7], the authors studied algebraic D-varieties and D-groups and proved a “Chevalley-type theorem”: the image of a D-constructible subset of an algebraic D-group under a D-homomorphism is also D-constructible. Here D-constructible means Boolean combination of D-closed. This result is precisely “quantifier-elimination” for the many-sorted structure of algebraic Dgroups with predicates for algebraic D-subvarieties of Cartesian powers.

Our original aim was to do something similar for the category of separable algebraic σ-groups equipped with predicates for dominant subvarieties. However, among the problems is that the intersection of two dominant subvarieties may no longer be dominant as the following example shows.

Let the algebraic σ-group be (𝔾_{m}^{2}, ϕ) where ϕ(x_{1}, x_{2}) = (x_{1}^{2}, x_{2}^{2}). We have two dominant subvarieties: X given by x_{1} = x_{2}^{2} and Y given by x_{1} = x_{2}^{4}. X ∩ Y = {(1, 1), (1, −1)} which is not dominant.

However we can prove the following weak version of quantifier-elimination.

Theorem 5.1 Suppose that G and G_{0} are separable σ-groups and f : G → G_{0} is a σ-homomorphism. Let X be an irreducible dominant σ-subvariety of G. Then f(X) is a Boolean combination of dominant σ-varieties.

Proof. The proof goes as in the proof of the analogous Theorem 3.2 of [7]. The relevant lemmas were proved in section 2. One should only notice that the result holds for trivial σ-varieties because of 2.1 and the Chevalley’s theorem (quantifier-elimination) for algebraically closed fields.

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