A NOTE ON A THEOREM OF AX

PIOTR KOWALSKI^{∗}

1. Introduction

In [1], James Ax proved the following theorem:

Theorem 1.1. Let (K, ∂) be a differential field of characteristic 0 with the field of constants C. For x_{1}, …, x_{n} ∈ K, y_{1}, …, y_{n} ∈ K^{∗}, if

∂x_{1} = \frac{∂y_{1}}{y_{1}} , …, ∂x_{n} = \frac{∂y_{n}}{y_{n}}

and ∂x_{1}, …, ∂x_{n} are ℚ-linearly independent, then trdeg_{C}(x, y) ⩾ n+1.

The above theorem has some implications in mathematical logic. It was used by Boris Zilber [20] to prove Weak CIT, a weak version of Conjecture on Intersection with Tori (CIT) stated in [20]. CIT is a finiteness statement about intersections of subtori of a given torus with certain subvarieties of this torus. Weak CIT was crucial in the bad field construction [3].

During the Model Theory and Applications to Algebra and Analysis Semester in the Newton Institute in Cambridge (spring 2005), Boris Zilber suggested to look at possible generalizations of 1.1 to the positive characteristic case. Unfortunately, we have not fully succeeded as yet, therefore this note is concerned mostly with characteristic 0 generalizations of 1.1 and only a small discussion about the positive characteristic case is given.

Ax’s theorem above may be thought of as a certain statement about a torus and a vector group. Ax himself proved theorems going way beyond the torus case in [2], but they were not phrased in the differential terms. Brownawell and Kubota [7] proved a version of 1.1 in the case of an elliptic curve and Jonathan Kirby in his thesis [12] proved a generalization of 1.1 to the case of a semi-abelian variety and a vector group. Finally, Bertrand [4] proved a further generalization to the case of an algebraic group with no vectorial quotients and a vector group.

∗Supported by the Polish grant MEN no. N201 032 32/2231.

There is a common feature in all of these statements – there is no clear way how to transfer them to positive characteristic. In our statement, the main character is not an algebraic group, but a formal map (which may be though of as an analytic map) between algebraic groups. In this interpretation Kirby’s and Bertrand’s statements are about the exponential map on the Lie algebra of an algebraic group. The reason their statements do not transfer to positive characteristic becomes clear – there are no exponential maps in positive characteristic! The statement we get includes all the known statements of this form and is meaningful in the positive characteristic too. It is stated in terms of differential equations related to a formal map between (the formalizations of) algebraic groups. We do not claim the proofs here are very original. The main point – a usage of differential forms is almost the same as in [2] or in [12]. However, as the referee has pointed out, Theorem 6.12 is the first result in this area to depend on finer information than the algebraic/transcendental dichotomy. The theorem considers raising to a power α of degree over the field of constants greater than some n, and this degree is used essentially in the proof.

It is worth mentioning that Daniel Bertrand and Anand Pillay [5] have proved a generalization of 1.1 going into a different direction – they do not assume that a given algebraic group is defined over constants. We do not go into this direction here. It should be also mentioned that in [1] and [12], they deal with several commuting derivations. We do not think it would be difficult to transfer the main results of this paper to the multi-derivation case, but we stick with one derivation for simplicity of the presentation.

The paper is organized as follows. In Section 2, we recall several facts about differential forms. In Section 3, we relate logarithmic derivatives with invariant differential forms. In Section 4, we define the differential equation of a formal map. In Section 5, we define nowhere algebraic formal maps and prove an Ax-like statement about them (5.5), which is the main result of this paper. In Section 6, we show how 5.5 specializes to the previously known statements. In Section 7, we discuss an alternative formulation of 5.5 in terms of linear dependence. In Section 8, we give a brief discussion of the positive characteristic case.

I would like to thank the referee for the very helpful report.

2. Invariant Differential Forms

All the rings are of characteristic 0 unless we explicitly state that the characteristic is positive. Throughout the paper C is an algebraically closed field, C ⊆ K is a field extension, R is a C-algebra and V is an algebraic variety over C.

We will briefly recall the definition of Kähler differentials (see [9] and [10]). The module of Kähler differentials of R over C is the R-module Ω_{R/C} given together with the derivation d : R → Ω_{R/C} having the following universal property – for any R-module M and any C-derivation ∂ : R → M, there is a unique R-module map ∂^{∗} : Ω_{R/C} → M such that ∂ = ∂^{∗} ∘ d (see [9]). We will denote Ω_{R/C} just by Ω_{R}. If V is affine, Ω_{V} is defined as Ω_{C[V]}. If V is arbitrary, Ω_{V} is defined using an affine open cover of V (see [10]). Note that Ω_{•} is a functor, which is covariant on C-algebras and contravariant on C-varieties. For a subvariety W ⊆ V and ω ∈ Ω_{V}, ω|_{W} denotes the image of ω by the map Ω_{V} → Ω_{W} induced by the inclusion map W ⊆ V. On any commutative algebraic group A, we have Ω_{A}^{inv}, the C-subspace of Ω_{A} consisting of A-invariant forms. All results of this section about invariant forms generalize to the case of a non-commutative algebraic group, where invariant means left-and-right invariant.

If V is affine and x ∈ V (K) we have the following commutative diagram:

\xymatrix{C[V] \ar[r]^{x} \ar[d]_{d} & K \ar[d]^{d} \\ Ω_{V} \ar[r]^{Ω_{x}} & Ω_{K}}

For any ω ∈ Ω_{V}, we define:

ω_{K} : V (K) → Ω_{K}, ω_{K}(x) = Ω_{x}(ω).

If V is arbitrary, we define the map ω_{K} : V (K) → Ω_{K} using an open affine cover of V (see [18] and [13]).

Example 2.1. Consider

ω := dX ∈ Ω_{C[X]} = Ω_{𝔸^{1}} .

We clearly have ω_{K}(x) = dx. Thus ω_{K}^{−1}(0) = C. Hence ω_{K}^{−1}(0) is not even a constructible subset of K.

Lemma 2.2. For any ω ∈ Ω_{V}, the set ω_{K}^{−1}(0) is Aut(K/C)-invariant.

Proof. Take x ∈ ω_{K}^{−1}(0). We can assume V is affine and R = C[V]. Then x corresponds to a C-algebra map x : R → K. Take any σ ∈ G(K/C). Then σ(x) is just the composition of σ with x. Since Ω_{σ(x)} = Ω_{σ} ∘ Ω_{x}, the result follows. □

Question 2.3. What is the definable nature of ω_{K}^{−1}(0)? In particular: (1) Is ω_{K}^{−1}(0) definable in the structure (K, +, ·, C)?

(2) Is ω_{K}^{−1}(0) of the form f^{−1}(W (C)) for a certain defined over C morphism f : V → W?

We will define one more map using ω ∈ Ω_{V}. Let T V denote the tangent bundle of V. There is a natural bijection:

T V (K) ⟷ V (K[X]/(X^{2})).

We will denote the ring K[X]/(X^{2}) by K[ε], it is called the ring of dual numbers. Assume that V is affine and R = C[V]. Any point x ∈ T V (K) corresponds to a C-algebra morphism x : R → K[ε]. So, x is of the form x_{1} + x_{2}ε, where x_{1}, x_{2} are functions from R to K. It is easy to see that x_{1} is a ring homomorphism and x_{2} is a derivation from R to K, where the R-algebra structure on K is given by x_{1}. Therefore, x induces a map x^{∗} : Ω_{R} → K. We define an algebraic map

ω_{K}^{∗} : T V (K) → K, ω_{K}^{∗}(x) = x^{∗}(ω).

This gives us also a K-definable morphism ω^{∗} : V → T^{∗}V, which is the well-known section of the cotangent bundle often used to define differential forms. Hence ω induces two maps on V (K) which are quite different and should not be confused.

It will be useful to emphasize the relation between Ω_{V} and T V in functorial terms. The argument above implies the existence of a natural bijection

Hom_{C}(R, K[ε]) ⟷ Hom_{C}(SΩ_{R}, K),

where SΩ_{R} is the symmetric algebra on Ω_{R}. Therefore the functor of the ring of dual numbers is right-adjoint to the functor of the symmetric algebra on differential forms and for an affine algebraic variety V, we have C[T V] = SΩ_{V}. In this interpretation Ω_{V} (being a subspace of SΩ_{V}) corresponds to certain regular functions on V – those functions which are C-linear on fibers (similarly as C[X] = S(KX)). For an algebraic group A, the space Ω_{A}^{inv} corresponds to those fiber-wise linear functions which can be obtained by translations from C-linear functionals on T_{0}A. Hence Ω_{A}^{inv} corresponds exactly to T_{0}^{∗}A – the dual space to the Lie algebra of A. Let us quote a result of Rosenlicht [18] (generalized later by Kolchin [13]).

Proposition 2.4. If ω ∈ Ω_{A}^{inv}, then ω_{K} : A(K) → Ω_{K} is a group homomorphism.

The next proposition is crucial for an application of the theory of invariant forms to the proof of our main result. It is related to 5.4 from [12]. This is also the main obstacle for a positive characteristic generalization of the results here.

Proposition 2.5. Suppose ω ∈ Ω_{A}^{inv} and x ∈ ker(ω_{K}). Then there is a connected algebraic subgroup H ⩽ A defined over C such that x ∈ H(K) + A(C) and ω|_{H} = 0.

Proof. For any field extension K ⊆ M, the induced map Ω_{K} → Ω_{M} is an embedding – it follows e.g. from the description of differentials given in 16.4 of [9]. For any C-algebra map y : R → K (a “K-point”) and the corresponding map ỹ: R → M (the corresponding “M-point”), we have the following commutative diagram:

\xymatrix{Ω_{R} \ar[r]^{Ω_{y}} \ar[dr]_{Ωỹ} & Ω_{K} \ar[d] \\  & Ω_{M}}

Therefore, we can always replace K with M in this proof. Thus without loss K is algebraically closed and |C|^{+}-saturated.

Let O_{x} denote the orbit of x under Aut(K/C) and V be the locus of x over C. Take c ∈ V (C), which exists since C is algebraically closed. By a theorem of Chevalley (see Chapter II Section 7 in [8]), the group generated by V (K) − c is of the form H(K), where H is a connected algebraic subgroup of A defined over C. By the |C|^{+}-saturation of K, O_{x} is Zariski dense in V. Therefore, O_{x} − c generates H(K) as well. By 2.2, ω_{K} vanishes on O_{x}. By 2.4, ω_{K} vanishes on H(K). Clearly, x ∈ H(K) + A(C). We need the following:

Claim Assume V is smooth and irreducible, ω ∈ Ω_{V} and ω_{K} = 0. Then ω = 0.

Proof of Claim. We can assume again that V is affine and R = C[V]. Since R embeds in K over C (saturation again), we can also assume that R ⊂ K and it is enough to show that the induced map Ω_{R} → Ω_{K} is an embedding. Since we know it for extensions of fields, it is enough to show that Ω_{R} → Ω_{R_{0}} is an embedding, where R_{0} is the fraction field of R. By 16.9 in [9], Ω_{R_{0}} ≅ R_{0} ⊗_{R} Ω_{R}. Hence it is enough to show that Ω_{R} → R_{0} ⊗_{R} Ω_{R} is an embedding. It exactly means that Ω_{R} should be torsion-free. Take a non-zero ω ∈ Ω_{R} and I = {f ∈ R|fω = 0}. Since ω ≠ 0, I is proper. Take a maximal ideal P ◁ R containing I. By 16.22 in [9], Ω_{R} is locally free (here we finally use the smoothness assumption). In particular, the R_{P} -module Ω_{R} ⊗_{R} R_{P} is free. But in R_{P} we localize only by elements which do not annihilate ω, hence ω̃, the image of ω in Ω_{R} ⊗_{R} R_{P}, is non-zero. Let Ĩ = {f ∈ R|fω̃ = 0}. Clearly I ⊆ Ĩ. Since Ω_{R} ⊗_{R} R_{P} is free, it is torsion-free, thus Ĩ = 0. Hence I = 0 and Ω_{R} is torsion-free as well. □

Since H is smooth and connected, using Claim 2 we get ω|_{H} = 0. □

Corollary 2.6. For each ω ∈ Ω_{A}^{inv}, there is an algebraic subgroup H ⩽ A defined over C such that ker(ω_{K}) = H(K) + A(C) and ω|_{H} = 0.

Proof. For each x ∈ ker(ω_{K}), take H_{x} given by 2.5. Since each H_{x} is connected, the group H generated by all the H_{x}’s is algebraic (by a theorem of Chevalley as in the proof of 2.5) and gives the result. □

Remark 2.7. By the above corollary, we can see that Question 2.3(2) has an affirmative answer for an invariant form – one should take the projection morphism f : A → A/H.

3. Logarithmic Derivative

From now on (K, ∂) is a differential field and C is its field of constants. Most of the facts concerning logarithmic derivatives presented in this section can be found in [15] or [16]. The derivation ∂ induces a ring homomorphism:

K ∋ x ↦ x + ∂(x)ε ∈ K[ε],

which we will also denote by ∂ (recall from Section 2 that K[ε] = K[X]/(X^{2})). Therefore ∂ also induces a map

∂_{V} : V (K) → V (K[ε]) = T V (K).

In the next lemma we will show that this map is a natural transformation. For a morphism ϕ : V → W between algebraic varieties let ϕ′ : T V → T W denote the induced morphism on the tangent spaces. On the level of K-points, ϕ′ is just ϕ applied to V (K[ε]).

Lemma 3.1. For any morphism ϕ : V → W between algebraic varieties over C, the following diagram is commutative:

\xymatrix{T V (K) \ar[r]^{ϕ′} & T W (K) \\ V (K) \ar[u]^{∂_{V}} \ar[r]^{ϕ} & W (K) \ar[u]^{∂_{W}}}

Proof. It is enough to notice that ϕ is a natural transformation between the functors of rational points, so the following diagram is commutative:

\xymatrix{V (K[ε]) \ar[r]^{ϕ_{K[ε]}} & W (K[ε]) \\ V (K) \ar[u]^{∂_{V}} \ar[r]^{ϕ_{K}} & B(K) \ar[u]^{∂_{W}}}

□

The map ∂_{V} is usually not algebraic except the important case of the 0-section 0_{V} : V (K) → T V (K) coming from the 0-derivation on K. If V ⊆ 𝔸^{n} is affine, then T V ⊆ 𝔸^{2n} is affine as well, ∂_{𝔸^{n}} = (id, ∂^{×n}) and ∂_{V} = ∂_{𝔸^{n}} |_{V}. Hence the map ∂_{V} may be thought of as an application of ∂ to the K-points of V.

Let A be a commutative algebraic group defined over C. Then T A is a commutative algebraic group as well and the projection map T A → A is a group homomorphism. Hence T_{0}A, the fiber over 0, is a commutative algebraic group too and we have an exact sequence

0 → T_{0}A → T A → A → 0.

Since the 0-section is also an algebraic group homomorphism, this sequence splits and we have another projection map p_{A} : T A → T_{0}A.

Definition 3.2. The logarithmic derivative on A is defined as follows:

l∂_{A} : A(K) → T_{0}A(K), l∂_{A} = p_{A} ∘ ∂_{A}.

Example 3.3. The tangent spaces at identity are naturally identified with K below.

(1) l∂_{𝔾_{a}}(x) = ∂x.

(2) l∂_{𝔾_{m}}(x) = \frac{∂x}{x}.

(3) l∂_{E}(x, y) = \frac{∂x}{y}, where E is the elliptic curve given by y^{2} = f(x), for a cubic polynomial f.

Proposition 3.4. Let A, B be commutative algebraic groups over C and ϕ : A → B an algebraic group homomorphism. Then:

(1) ker(l∂_{A}) = A(C).

(2) The following diagram is commutative (naturality of the logarithmic derivative):

\xymatrix{T_{0}A(K) \ar[r]^{ϕ′} & T_{0}B(K) \\ A(K) \ar[u]^{l∂_{A}} \ar[r]^{ϕ} & B(K) \ar[u]^{l∂_{B}}}

Proof. (1) is exactly 2.2(iii) from [16]. For (2) it is enough to use 3.1 and notice that the following diagram is commutative:

\xymatrix{T_{0}A(K) \ar[r]^{ϕ′} & T_{0}B(K) \\ T A(K) \ar[u]^{p_{A}} \ar[r]^{ϕ′} & T B(K) \ar[u]^{p_{B}}}

which is easy to see after decomposing T A into T_{0}A × A. □

We add now differential forms into the picture. Let us fix ω ∈ Ω_{V}. In a presence of the derivation ∂, the two maps ω_{K} and ω_{K}^{∗} are related to each other by the diagram below.

Lemma 3.5. The following diagram is commutative

\xymatrix{V (K) \ar[r]^{∂_{V}} \ar[d]_{ω_{K}} & T V (K) \ar[d]^{ω_{K}^{∗}} \\ Ω_{K} \ar[r]^{∂^{∗}} & K}

Proof. For this proof one needs only to unravel the definitions and use the adjointness between functors R ↦ R[ε] and R ↦ SΩ_{R} (see Section 2). Let us assume that V is affine and R = C[V]. To simplify the diagrams we assume the following notation: for C-algebras R, S, T let [R, S] denote Hom_{C}(R, S) and for f ∈ [S, T] and r ∈ R, let f_{•} : [R, S] → [R, T] be the map obtained by the composition and ev_{r} : [R, S] → S the map obtained by the evaluation. We have to show that the following diagram is commutative († denotes the adjointness map):

\xymatrix{[R, K] \ar[r]^{∂_{•}} \ar[d]_{Ω} & [R, K[ε]] \ar[d]^{†} \\ [Ω_{R}, Ω_{K}] \ar[r]^{∂_{•}^{∗}} \ar[d]_{_{ev_{ω}}} & [Ω_{R}, K] \ar[d]^{_{ev_{ω}}} \\ Ω_{K} \ar[r]^{∂^{∗}} & K}

The lower diagram is clearly commutative. The upper one is commutative by the naturality of † and the observation that ∂^{†} = ∂^{∗}. □

As we have already noticed, we can identify Ω_{A}^{inv} with the dual space to T_{0}A. Using this identification, we give a description of the logarithmic derivative in terms of invariant forms (see also [15]). We denote by the same symbol the map T A(K) → K induced by a differential form and the restriction of this map to T_{0}A(K).

Lemma 3.6. Let A be a commutative algebraic group over C. Choose ω̄, a basis of Ω_{A}^{inv}. The following diagram is commutative:

\xymatrix{A(K) \ar[r]^{l∂_{A}} \ar[d]_{ω̄_{K}} & T_{0}A(K) \ar[d]^{ω̄_{K}^{∗}} \\ Ω_{K}^{×n} \ar[r]^{(∂^{∗})^{×n}} & K^{n}}

Proof. This time we need to show that the following diagram is commutative:

\xymatrix{A(K) \ar[r]^{∂_{A}} \ar[d]_{ω̄_{K}} & T A(K) \ar[d]^{ω̄_{K}^{∗}} \ar[r]^{p_{A}} & T_{0}A(K) \ar[dl]^{ω̄_{K}^{∗}} \\ Ω_{K}^{×n} \ar[r]^{(∂^{∗})^{×n}} & K^{n}}

The left-hand side diagram is commutative by 3.5. Commutativity of the right-hand side diagram is given by the invariance of ω̄ – the function ω̄_{K}^{∗} is determined by its values on T_{0}A(K). □

4. The differential equation of a formal map

Let us assume first that C = ℂ. For each algebraic group G over ℂ, the set of its ℂ-rational points G(ℂ) is a complex Lie group. Any local analytic homomorphism ϕ : A(ℂ) → B(ℂ) induces a linear map ϕ′ : T_{0}A(ℂ) → T_{0}B(ℂ). By tensoring, we also get a linear map (denoted by the same symbol) ϕ′ : T_{0}A(K) → T_{0}B(K).

Definition 4.1. The differential equation of ϕ is

l∂_{B}(y) = ϕ′(l∂_{A}(x)).

Example 4.2. The tangent spaces in (1) and (2) below are naturally identified with K.

(1) For exp : 𝔾_{a}(ℂ) → 𝔾_{m}(ℂ) its differential equation is

\frac{∂y}{y} = ∂x,

since exp induces the identity map on the tangent spaces. It is the same differential equation as the one in 1.1.

(2) Let α ∈ ℂ. Consider the following local analytic map

𝔾_{m}(ℂ) ∋ x ↦ x^{α} ∈ 𝔾_{m}(ℂ).

Its differential equation is:

\frac{∂y}{y} = α\frac{∂x}{x}.

(3) If ϕ is an algebraic map, we get from 3.4(2) that (x, y) is a solution of the differential equation of ϕ if and only if y−f(x) ∈ B(C). Therefore, the solution set of the differential equation of ϕ is the graph of ϕ up to constants.

By (3) above, we can intuitively think about the solution set of the differential equation of a local analytic homomorphism ϕ as (up to constants) the graph of a possibly non-existing function ϕ : A(K) → B(K).

If C ≠ ℂ, we need to replace the notion of local analytic with the notion of formal.

Definition 4.3 (Bochner [6]). An n-dimensional formal group (law) over C is a tuple of power series F ∈ C⟦X, Y ⟧^{×n} (|X| = |Y | = |Z| = n) satisfying:

• F (0, X) = F (X, 0) = X,

• F (X, F (Y, Z)) = F (F (X, Y), Z).

A formal group is commutative if it moreover satisfies:

• F (X, Y) = F (Y, X). A morphism from an n-dimensional formal group G into an m-dimensional formal group F is a tuple of power series f ∈ C⟦X⟧^{×m} such that:

• F (f(X), f(Y)) = f(G(X, Y)).

There is a well-known formalization functor (see pages 5 and 13 in [14]) from the category of algebraic groups to the category of formal groups. Briefly, for an algebraic group G, let R := O_{G,1} be the local ring at identity. The group multiplication from G induces a comultiplication on R, i.e. a map

μ : R → R ⊗_{C} R.

Then μ induces a comultiplication on the completion of R, i.e. a map

μ̂ : R̂ → R̂ ˆ⊗_{C} R̂.

Since G is smooth, R̂ ≅ C⟦X⟧ (|X| = dim G), so R̂ ˆ⊗_{C} R̂ ≅ C⟦X_{1}, X_{2}⟧. The formalization of G, denoted Ĝ, is defined as μ̂(X) ∈ C⟦X_{1}, X_{2}⟧. One can check that any algebraic group homomorphism f : A → B functorially induces a formal map f̂ : Â → B̂. But there are formal maps between Â and B̂ which are not algebraic (as the exponential map), i.e. not being the formalization of an algebraic homomorphism.

Abusing the language a bit, we will sometimes say for a formal map ϕ : Â → B̂ that ϕ is a formal map between algebraic groups A and B.

Example 4.4. The formalization of 𝔾_{a} is X +Y and the formalization of 𝔾_{m} is X+Y +XY. Note that the axioms of formal groups correspond to the fact that 0 is the neutral element, so XY is not a formal group.

It is easy to see that formal maps between algebraic groups still induce linear maps on their tangent spaces. If C = ℂ, then the category of formal groups is equivalent to the category of local complex Lie groups and this equivalence commutes with the linear maps induced by formal maps on tangent spaces. Hence the following definition generalizes 4.1:

Definition 4.5. Let A, B be commutative algebraic groups over C and ϕ a formal map between A and B. The differential equation of ϕ is

l∂_{B}(y) = ϕ′(l∂_{A}(x)).

By 4.2(3) again, we can think of the set of solutions of the differential equation of ϕ as (up to constants) the graph of a possibly non-existing function ϕ : A(K) → B(K). Note that the category of formal groups is not concrete, so morphisms can not be understood as functions (unless K is a metric field and a series may converge). However, the solution set of the differential equation of ϕ may be thought of as the “blurred graph” of ϕ (blurred by constants). There may be a relation between this “blurred graph” and the blurred (pseudo-)exponentiation from Section 8 of [12]. Since for two commutative algebraic groups A, B we have the following bijection:

Hom_{Formal}(A, B) ∋ f ↦ f′ ∈ Hom_{Linear}(T_{0}A, T_{0}B),

we could have forgotten about formal or even analytic maps and just talk about the induced linear maps on tangent spaces. However we find the language of formal maps more natural as e.g. the original theorem of Ax should say something about the exponential map and not about the identification (a trivial one) of the tangent spaces it induces. This formalism is also useful for finding positive characteristic analogues of Ax’s theorem.

5. Ax theorem for nowhere algebraic formal maps

Recall the statement of Ax’s theorem from the introduction:

Theorem 5.1 (Ax). If (∂x_{i} = \frac{∂y_{i}}{y_{i}})_{1⩽i⩽n} and (∂x_{i})_{1⩽i⩽n} are ℚ-linearly independent, then trdeg_{C}(x, y) ⩾ n + 1.

Note the following scheme in the above statement:

A good differential equation + linear independence ⟹ large trdeg.

The differential equation there is actually the differential equation of the exponential map (see 4.2(1)). In Theorem 5.7 from [12] and Proposition 1.b from [4], the differential equations were those of the map exp : Lie(A) → A for A semi-abelian (Kirby) or without 𝔾_{a}-quotients (Bertrand). Thus our statement should identify those formal maps which yield differential equations giving large transcendence degree. Note that algebraic maps are not good – if ϕ is algebraic, then any (x, ϕ(x)) is a solution to the differential equation of ϕ (see 4.2(3)) and the transcendence degree is always smaller that the dimension of the domain. However even certain non-algebraic formal maps are not good as the example below shows.

Example 5.2. For the differential equation \frac{∂z}{z} = \sqrt{2}\frac{∂y}{y} take:

• y_{1}, y_{2} such that \frac{∂y_{1}}{y_{1}} = 1, \frac{∂y_{2}}{y_{2}} = \sqrt{2} + 1,
• z_{1} = y_{1}^{−1}y_{2}, z_{2} = y_{1}y_{2}.

Then \frac{∂y_{1}}{y_{1}} , \frac{∂y_{2}}{y_{2}} are ℚ-linearly independent, trdeg_{C}(y_{1}, y_{2}, z_{1}, z_{2}) ⩽ 2 and:

• \frac{∂z_{1}}{z_{1}} = \sqrt{2} + 1 − 1 = \sqrt{2}\frac{∂y_{1}}{y_{1}} , \frac{∂z_{2}}{z_{2}} = \sqrt{2}\frac{∂y_{2}}{y_{2}} .

We see that our map should be “very far” from being algebraic. Below is a definition which works (to some extent) for our purposes, but it is perhaps not very beautiful.

Definition 5.3. A formal map ϕ between algebraic groups A, B is nowhere algebraic if there are no:

• algebraic subgroups A_{0} ⩽ A, B_{0} < B,

• algebraic epimorphism ψ : A_{0} ↠ B/B_{0},

• v ∈ T_{0}A_{0}(C) such that ψ′(v) = ϕ̄′(v) ≠ 0, where ϕ̄ is the following composition:

\xymatrix{A_{0} \ar[r] & A \ar[r]^{ϕ} & B \ar[r] & B/B_{0}.}

At first, it seemed to me it would be more natural just to demand in the definition above that ϕ and ψ should not coincide on A_{0}, so “nowhere” would refer to the algebraic structure (rather than formal) then. But the formal map X^{\sqrt{2}} satisfies this condition and still does not yield an Ax-like statement as was shown in 5.2. Therefore a “good” formal map should not coincide with an algebraic one even in the formal sense (i.e. on the tangent bundle) as in the definition above. Before the statement of the main result, we need to phrase the linear dependence condition in more geometric terms.

Lemma 5.4. For a_{1}, …, a_{n} ∈ K^{∗}, \frac{∂a_{1}}{a_{1}} , …, \frac{∂a_{n}}{a_{n}} are linearly dependent over ℚ if and only if (a_{1}, …, a_{n}) belongs to a constant coset of a proper subtorus of 𝔾_{m}^{n}.

Proof. Note that \frac{∂x}{x} = l∂_{𝔾_{m}}(x). Let a = (a_{1}, …, a_{n}). ⟹ Take (k_{1}, …, k_{n}) ∈ ℤ^{n}\{0} such that ∑ k_{i}l∂_{𝔾_{m}}(a_{i}) = 0 and define

f : 𝔾_{m}^{n} → 𝔾_{m}, f(x_{1}, …, x_{n}) = ∏_{i=1}^{n} x_{i}^{k_{i}}.

Since l∂_{𝔾_{m}}(f(a)) = 0, f(a) ∈ C. Let T := ker(f) and take c ∈ 𝔾_{m}^{n}(C) such that f(c) = f(a). Then f(c^{−1}a) = 1, so a ∈ cT (K). ⟸ If a ∈ T (K) + 𝔾_{m}^{n}(C) for a proper subtorus T < 𝔾_{m}^{n}, then (by 3.4(2)) l∂_{𝔾_{m}^{n}}(a) ∈ T_{0}T (K). Since T_{0}T (K) is a ℚ-definable proper linear subspace of T_{0}𝔾_{m}^{n}(K), the tuple l∂_{𝔾_{m}^{n}}(a) is ℚ-linearly dependent. □

We are ready now to state and prove the main result of this paper.

Theorem 5.5. Let A, B be commutative algebraic groups over C of dimension n and ϕ : Â → B̂ be a nowhere algebraic formal map. If (a, b) ∈ (A × B)(K) satisfies the differential equation of ϕ and trdeg_{C}(a, b) ⩽ n, then there is a proper algebraic subgroup B_{0} < B defined over C such that b ∈ B_{0}(K) + B(C).

Proof. From the description of invariant differential forms given in Section 2, we know that the choice of a basis of Ω_{A}^{inv} corresponds exactly to a choice of a linear isomorphism between T_{0}A and 𝔾_{a}^{n}. Take ω̄ = (ω_{1}, …, ω_{n}), an arbitrary basis of Ω_{A}^{inv} and the basis η̄ = (η_{1}, …, η_{n}) of Ω_{B}^{inv} such that the following diagram is commutative:

\xymatrix{T_{0}A(K) \ar[rr]^{ϕ′} \ar[dr]_{ω̄_{K}^{∗}} &  & T_{0}B(K) \ar[dl]^{η̄_{K}^{∗}} \\  & K^{n}}

From 3.6 we get a commutative diagram:

\xymatrix{T_{0}A(K) \ar[rr]^{ϕ′} \ar[dr]_{ω̄_{K}^{∗}} &  & T_{0}B(K) \ar[dl]^{η̄_{K}^{∗}} \\  & 𝔾_{a}^{n}(K) \\  & Ω_{K}^{×n} \ar[u]_{(∂^{∗})^{×n}} \\ A(K) \ar[uuu]^{l∂_{A}} \ar[ur]^{ω̄_{K}} &  & B(K) \ar[uuu]_{l∂_{B}} \ar@{-->}[ul]_{η̄_{K}}}

Chasing this diagram, we get:

∂^{∗}(ω_{1}(a) − η_{1}(b)) = 0, …, ∂^{∗}(ω_{n}(a) − η_{n}(b)) = 0.

Let g := (a, b), G := A × B and π_{A} : G → A, π_{B} : G → B be the projection maps. For i ⩽ n let us define:

ξ_{i} := π_{A}^{∗}(ω_{i}) − π_{B}^{∗}(η_{i}) ∈ Ω_{G}^{inv}.

We clearly have:

∂^{∗}(ξ_{1}(g)) = 0, …, ∂^{∗}(ξ_{n}(g)) = 0.

In this paragraph we proceed exactly as in [12], so we are going to be brief. Since trdeg_{C}(g) ⩽ n, we have dim_{C(g)} Ω_{C(g)} ⩽ n. Since

ξ_{1}(g), …, ξ_{n}(g) ∈ ker(∂^{∗} : Ω_{K} → K) ∩ Ω_{C(g)}

(Ω_{C(g)} is naturally embedded into Ω_{K} here), we get that ξ_{1}(g), …, ξ_{n}(g) are linearly dependent over K. Since each ξ_{i} is invariant on the commutative group G, it is closed. Therefore each ξ_{i}(g) is closed, hence ξ_{1}(g), …, ξ_{n}(g) are linearly dependent over C by a Lie derivative argument (see [2] or [12]). Take a non-zero tuple (c_{1}, …, c_{n}) ∈ C^{n} such that c_{1}ξ_{1}(g) + … + c_{n}ξ_{n}(g) = 0 and define:

ξ := c_{1}ξ_{1} + … + c_{n}ξ_{n} ∈ Ω_{G}^{inv}.

We have:

ξ|_{A} ≠ 0, ξ|_{B} ≠ 0, ξ(g) = 0.

By 2.5, there is an algebraic subgroup H < G over C such that g ∈ H(K) + G(C) and ξ|_{H} = 0.

Let us define H^{A} := π_{A}(H), H_{A} := H ∩ (A × 0) and similarly for H^{B}, H_{B}. Clearly:

g ∈ (H^{A} × H^{B})(K) + G(C).

Thus we are done, if H^{B} is proper. Since ξ|_{B} ≠ 0, we have H_{B} ≠ H. Hence we can also assume that b ∉ H_{B}(K) + B(C). We will show that H^{B} = B contradicts the assumption that ϕ is nowhere algebraic.

Assume H^{B} = B. The algebraic subgroup H < G induces an algebraic epimorphism ψ : H^{A} → B̄ := B/H_{B} such that ψ(a) = b̄:= b + H_{B}. By 3.4(2), we have:

ψ′(l∂_{A}(a)) = l∂B̄(b̄), ϕ̄′(l∂_{A}(a)) = l∂B̄(b̄),

where ϕ̄ is the composition of ϕ with the formalization of the projection map B → B̄. Hence ϕ̄′ coincides with ψ′ on l∂_{A}(a). Since b ∉ H_{B}(K)+ B(C), by 3.4 we get l_{∂}B̄(b̄) ≠ 0, which contradicts the assumption that ϕ is nowhere algebraic. □

The statement of the above theorem is non-symmetric, since the notion of a nowhere algebraic formal map is non-symmetric. But in many cases (including all the previously known ones), we get a symmetric statement, i.e proper algebraic subgroups of both A and B. It is discussed in details in the next section.

6. Specializations of the main theorem and a torus case

In this section we will show how Theorem 5.5 generalizes some of the known Ax-like statements. We will put these statements into more general contexts and still show how 5.5 specializes to these contexts. We will also discuss another Ax-like statement regarding certain formal automorphisms of tori which does not fit to the scope of 5.5.

Definition 6.1. We call two algebraic groups G and H essentially different if any algebraic subgroup T < G × H is isogenous to a group of the form G_{0}×H_{0} for G_{0} (resp. H_{0}) an algebraic subgroup of G (resp. H).

Example 6.2. Essentially different algebraic groups:

(1) G = 𝔾_{a}^{n}, H is a semi-abelian variety;

(2) G = 𝔾_{m}^{n} × 𝔾_{a}^{n}, H is an abelian variety;

(3) (generalization of (2) by Poincaré’s Reducibility Theorem) G = G_{1} × ⋯ × G_{n}, H = H_{1} × ⋯ × H_{m}, where each G_{i}, H_{j} is a simple algebraic group, and G_{i} is not isogenous to H_{j};

(4) G = 𝔾_{m}^{n}, H is an extension of an abelian variety by a vector group.

Proposition 6.3. If A, B are essentially different and ϕ : Â → B̂ is an isomorphism, then ϕ and ϕ^{−1} are nowhere algebraic.

Proof. By symmetry, it is enough to deal with ϕ. Assume ϕ is not nowhere algebraic and take algebraic subgroups A_{1} ⩽ A, B_{1} < B with infinite index [B : B_{1}] and an algebraic epimorphism ψ : A_{0} ↠ B/B_{1}. Then the preimage in A × B of the graph of ψ is not isogenous to a group of the form A_{0} × B_{0}. □

Theorem 6.4. Assume A, B are essentially different commutative algebraic groups of dimension n and ϕ is a formal isomorphism between them. Let (a, b) ∈ (A × B)(K) satisfy the differential equation of ϕ and trdeg_{C}(a, b) ⩽ n. Then there are proper algebraic subgroups A_{0} < A, B_{0} < B such that a ∈ A_{0}(K) + A(C) and b ∈ B_{0}(K) + B(C).

Proof. By 6.3, ϕ and ϕ^{−1} are nowhere algebraic. By 5.5, we get B_{0} < B with the required properties. Clearly, (b, a) satisfies the differential equation of ϕ^{−1}. By 5.5 again, we get A_{0} < A with the required properties. □

Remark 6.5. If A is a semi-abelian variety, then A and 𝔾_{a}^{n} are essentially different. Hence 6.4 generalizes 5.7 from [12]. Note that the choice of an arbitrary basis of the space of invariant forms in [12] corresponds exactly to the choice of an arbitrary formal isomorphism here.

Definition 6.6. We say that two algebraic groups G and H have no common quotients if for any normal algebraic subgroups G_{0} ◁ G, H_{0} ◁ H; G/G_{0} ≅ H/H_{0} implies G/G_{0} is finite.

Fact 6.7. If G and H are essentially different, then they have no common subquotients, i.e. for any algebraic subgroups G_{1} ⩽ G, H_{1} ⩽ H, G_{1} and H_{1} have no common quotients.

Proof. Assume G_{1}, H_{1} have a common infinite algebraic quotient Q. Then there are algebraic epimorphisms ϕ : G_{1} → Q, ψ : H_{1} → Q. It is easy to see that the preimage of the diagonal by ϕ × ψ is not isogenous to a group of the form H_{0} × G_{0}. □

Example 6.8. (1) Let G = 𝔾_{a}^{n}, H = E(A), where A is a semiabelian variety and E(A) is its maximal vectorial non-split extension. Then G and H have no common quotients but are not essentially different.

(2) If A and B are abelian varieties, then by Poincaré’s Reducibility Theorem they are essentially different if and only if they have no common quotients.

Fact 6.9. Assume A, B are commutative algebraic groups and for any algebraic subgroup A_{1} ⩽ A, A_{1} and B have no common quotients. Let ϕ be a formal isomorphism between A and B. Then ϕ is nowhere algebraic.

Proof. Obvious from the definitions. □

Theorem 6.10. If A, B, ϕ are as in 6.9, (a, b) ∈ (A × B)(K) satisfies the differential equation of ϕ and trdeg_{C}(a, b) ⩽ dim(A), then there is a proper algebraic subgroup B_{0} < B such that b ∈ B_{0}(K) + B(C).

Proof. Clear by 6.9 and 5.5. □

Remark 6.11. Note that if A = 𝔾_{a}^{n} and we are in the situation of 6.10, then we also get a proper subgroup A_{0} < A defined over C such that a ∈ A_{0}(K) + A(C). This is because algebraic subgroups over C of 𝔾_{a}^{n} are in 1-to-1 correspondence (modulo constants) with linear subspaces defined over C of T_{0}𝔾_{a}^{n} and this correspondence is given by the logarithmic derivative map, which is here just the n-th cartesian power of ∂. Therefore, 6.10 includes Bertrand’s Proposition 1.b from [4] – an Ax-like statement for exp : Lie(A) → A, where A has no 𝔾_{a}-quotients.

Unfortunately, it seems like the argument given in the proof of 5.5 alone is not enough to prove an Ax-like statement for another class of formal maps – raising to irrational powers. The proof of the theorem below combines an argument from 5.5 with some arguments which seem to be specific to the case of torus – rigidity and decomposition into the product of 1-dimensional subgroups (see also 6.15). It is also easy to see that raising to an irrational power is still not nowhere algebraic.

Theorem 6.12. If α ∈ C, [ℚ(α) : ℚ] > n, y_{1}, z_{1}, …, y_{n}, z_{n} ∈ K^{∗} satisfy

\frac{∂z_{1}}{z_{1}} = α\frac{∂y_{1}}{y_{1}} , …, \frac{∂z_{n}}{z_{n}} = α\frac{∂y_{n}}{y_{n}}

and \frac{∂z_{1}}{z_{1}} , …, \frac{∂z_{n}}{z_{n}} are ℚ-linearly independent, then

trdeg_{C}(y_{1}, z_{1}, …, y_{n}, z_{n}) ⩾ n + 1.

Proof. Let g := (y_{1}, …, y_{n}, z_{1}, …, z_{n}) and assume trdeg_{C}(g) ⩽ n. If we proceed as in the proof of 5.5, we get (a_{1}, …, a_{n}) ∈ C^{n} \ {0} such that for

ξ := ∑_{i=1}^{n} a_{i} (α\frac{dX_{i}}{X_{i}} − \frac{dY_{i}}{Y_{i}} ) ,

we have ξ_{K}(g) = 0. Let T be a proper subtorus of 𝔾_{m}^{2n} such that ker(ξ_{K}) = T (K) + 𝔾_{m}^{2n}(C) (see 2.6). For l ⩽ 2n, let π_{l} : 𝔾_{m}^{2n} → 𝔾_{m}^{l} denote the projection on the first l coordinates. If π_{n}(T) ≠ 𝔾_{m}^{n} we are done as in the proof of 5.5. Assume π_{n}(T) = 𝔾_{m}^{n}. We will reach a contradiction. Let dim T = n + k, 0 ⩽ k < n. After permuting some coordinates, we may assume that π_{n+k}(T) = 𝔾_{m}^{n+k}. It means there is a matrix B = (b_{ij}) ∈ M_{n−k,n+k}(ℤ) and non-zero integers l_{1}, …, l_{n−k} such that T is given by the following equations:

(†)  X_{n+k+j}^{l_{j}} = ∏_{i=1}^{n+k} X_{i}^{b_{ij}}, 0 < j ⩽ n − k.

By 2.4, there is a matrix A ∈ M_{n−k,n+k}(ℚ) (we divide the i-th row of B by l_{i}) such that for any x = (x_{1}, …, x_{2n}) ∈ T (K) we have

(∗)  A (\frac{dx_{1}}{x_{1}} , …, \frac{dx_{n+k}}{x_{n+k}} ) = (\frac{dx_{n+k+1}}{x_{n+k+1}} , …, \frac{dx_{2n}}{x_{2n}} ) .

On the other hand (since ξ_{K}(x) = 0) we have:

(∗∗)  α (a_{1} \frac{dx_{1}}{x_{1}} + ⋯ + a_{n} \frac{dx_{n}}{x_{n}} ) = a_{1} \frac{dx_{n+1}}{x_{n+1}} + ⋯ + a_{n} \frac{dx_{2n}}{x_{2n}} .

Since we are not going to use the derivation ∂ anymore, we can assume as in the proof of 2.5 that K is sufficiently saturated. Hence we can and will take x generic over C in T (K). Since π_{n+k}(T) = 𝔾_{m}^{n+k}, the elements x_{1}, …, x_{n+k} are algebraically independent over C. Therefore

(∗ ∗ ∗)  dim_{C} (\frac{dx_{1}}{x_{1}} , …, \frac{dx_{n+k}}{x_{n+k}} ) = n + k.

By (∗), (∗∗) and (∗ ∗ ∗) we get (A^{T} is A transposed):

A^{T} (a_{k+1}, …, a_{n}) = (αa_{1}, …, αa_{n}, −a_{1}, …, −a_{k}).

If we decompose A into a block form A = [A_{1}A_{2}A_{3}], where A_{1} ∈ M_{n−k,k}(ℚ), A_{2} ∈ M_{n−k,n−k}(ℚ) and A_{3} ∈ M_{n−k,k}(ℚ), then we have:

A_{2}^{T} (a_{k+1}, …, a_{n}) = α(a_{k+1}, …, a_{n}).

If (a_{k+1}, …, a_{n}) ≠ 0, then α is an eigenvalue of A_{2}^{T}. Therefore, α is a zero of the characteristic polynomial of A_{2}^{T}, which has rational coefficients and degree n−k ⩽ n, a contradiction. Hence (a_{k+1}, …, a_{n}) = 0. Take v ∈ K \ C. Then (0, …, 0, v) ∈ ker(ξ_{K}). Therefore, there is (c_{1}, …, c_{2n}) ∈ 𝔾_{m}^{2n}(C) such that (c_{1}, …, c_{2n−1}, vc_{2n}) ∈ T (K). By the description of T given in (†), we see that v is algebraic over C, a contradiction, since C is algebraically closed. □

Note that Example 5.2 shows that the statement of 6.12 can not be improved.

Remark 6.13. A statement similar to 6.12 appears as 5.8 in [12]. The constant α ∈ C there is arbitrary, but after assuming trdeg_{C}(y, z) ⩽ n the conclusion there is weaker:

(y, z) ∈ T (K) + 𝔾_{m}^{n}(C) for a proper subtorus T < 𝔾_{m}^{2n}.

This condition is equivalent to ℚ-dependence of \frac{∂y_{1}}{y_{1}} , \frac{∂z_{1}}{z_{1}} , …, \frac{∂y_{n}}{y_{n}} , \frac{∂z_{n}}{z_{n}} (see 5.4).

Question 6.14. What is a general definition of a “good” formal map for which:

• one can prove 5.5,

• it includes the nowhere algebraic case and formal maps from

6.12? Is it the condition (much nicer and weaker than 5.3) that for ϕ, a formal map, ker(ϕ′ − ψ′) = 0 for any algebraic ψ? I am not able to prove 5.5 for such formal maps as yet. Formal maps from 6.12 are such. Note that the formal map from Example 5.2 is not like this: for

ψ : 𝔾_{m}^{2} → 𝔾_{m}^{2}, ψ(x, y) = (x^{−1}y, xy)

we have (1, \sqrt{2} + 1) ∈ ker(ϕ′ − ψ′).

Let E be an elliptic curve, α ∈ C and ϕ_{α} a formal automorphism of E such that ϕ′_{α} is the multiplication by α on T_{0}E. By similar arguments as in the proof of 6.12 one can show:

Theorem 6.15. Let (y, z) ∈ E^{2n}(K) satisfy the differential equation of ϕ_{α}^{×n} and trdeg_{C}(y, z) ⩾ n + 1. Then:

• If E has no complex multiplication and [ℚ(α) : ℚ] > n, then there are proper algebraic subgroups E_{1}, E_{2} < E^{n} such that y ∈ E_{1}(K) + E^{n}(C) and z ∈ E_{2}(K) + E^{n}(C).

• If E has a complex multiplication τ and [ℚ(α, τ) : ℚ(τ)] > n, then there are proper algebraic subgroups E_{1}, E_{2} < E^{n} such that y ∈ E_{1}(K) + E^{n}(C) and z ∈ E_{2}(K) + E^{n}(C).

One can guess from 6.12 and 6.15 that the field ℚ plays a different role for a torus and an elliptic curve without a complex multiplication, than for an elliptic curve with a complex multiplication (see also [7] or [11]). This relation will be made clearer in the next section.

Analyzing Poizat’s proof of Weak CIT (see 3.2 in [17]) one can see that there are three crucial ingredients in it: Ax’s theorem 1.1, rigidity of tori (countably many algebraic subgroups) and the following property:

(∗)  trdeg_{C}(a_{1}, …, a_{n}) ⩽ dim_{C}(∂a_{1}, …, ∂a_{n}).

Since Ax’s theorem also holds when torus is replaced by any semiabelian variety, Weak CIT holds in such a context as well, which was proved by Kirby, see 5.15 in [12]. Let us concentrate on the property (∗). Note that for a given k ∈ ℕ, the condition

dim_{C}(∂a_{1}, …, ∂a_{n}) ⩽ k

is definable in (a_{1}, …, a_{n}) and implies the non-definable condition

trdeg_{C}(a_{1}, …, a_{n}) ⩽ k.

For a commutative algebraic group A let us formulate the following condition:

(∗_{A})  ∀a ∈ A(K) trdeg_{C}(a) ⩽ dim_{C}(l∂_{A}(a)).

Then (∗) is just (∗_{𝔾_{a}^{n}}). One could wonder for which algebraic groups A, the property (∗_{A}) holds. E.g. if it held for a torus, it would give another proof of Weak CIT using 6.12. Similarly for elliptic curves using 6.15. Unfortunately (∗_{A}) does not hold neither for A being a torus nor for A being a power of an elliptic curve. Quite ironically, a counterexample is given by the corresponding Ax-like statement 6.12 or 6.15. Let us focus on the torus case. Take α ∈ C non-algebraic and a, b ∈ K^{∗} \ C^{∗} such that \frac{∂a}{a} = α\frac{∂b}{b}. Clearly dim_{C} l∂_{𝔾_{m}^{2}}(a, b) = 1. But by 6.12, trdeg_{C}(a, b) = 2. Therefore if for an algebraic group A the property (∗_{A}) holds, then A should not have “very non-algebraic” formal endomorphisms, since we get counterexamples to (∗_{A}) along the “blurred graph” (see comments at the end of Section 4) of such a formal map. Note that vector groups are exactly like that – any formal endomorphism of a vector group is algebraic. However we should have in mind we are in the characteristic 0 situation here (note the last sentence of this paper).

7. Schanuel Conditions

The original theorem of Ax (1.1) is stated in terms of Schanuel-like inequalities. By 5.4, the linear condition there is equivalent to a condition about subtori. The latter condition is easier to deal with if one goes from torus to more complicated algebraic groups. The problem is that in 1.1 the base field for linear dependence is clear (being just ℚ), but for algebraic groups other than tori it gets more complicated. Jonathan Kirby has proved results into this direction about products of elliptic curves (see Propositions 1. and 2. in [11]). In this section, we will give a general result in terms of Schanuel-like inequalities, which includes the Ax’s theorem and the statements from [11].

Definition 7.1. For an algebraic group G defined over K, we define its absolute field of definition:

ℚ_{G} := ℚ(⋃{field of definition of H|H, a connected algebraic subgroup of G}).

Example 7.2. Absolute fields of definitions:

(1) G = 𝔾_{a}^{n}, n > 1: ℚ_{G} = K;

(2) G = 𝔾_{m}^{n}: ℚ_{G} = ℚ;

(3) E is an elliptic curve over ℚ, G = E^{n}, n > 1:

ℚ_{G} = End_{ℚ} (E) := End(E) ⊗_{ℤ} ℚ = ℚ(τ),

where τ is a complex multiplication on E (if any);

(4) More generally, if A is a simple commutative algebraic group, then for n > 1:

ℚ_{A^{n}} = End_{ℚ} (E);

(5) G is a semi-abelian variety defined over ℚ: ℚ_{G} ⊆ ℚ^{acl}.

The next theorem is a version of 5.5 stated in terms of linear dependence.

Theorem 7.3. Let A, B, ϕ, a, b, n be as in 5.5. If dim_{ℚ_{B}∩C} l∂_{B}(b) = n then trdeg_{C}(a, b) ⩾ n + 1.

Proof. If trdeg_{C}(a, b) ⩽ n, we get by 5.5 a proper algebraic subgroup B_{0} < B defined over C, b_{0} ∈ B_{0}(K) and c ∈ B(C) such that b = b_{0} + c. Since B_{0} is also defined over ℚ_{B}, it is defined over ℚ_{B} ∩ C. By 3.4 we get:

dim_{ℚ_{B}∩C} l∂_{B}(b) = dim_{ℚ_{B}∩C} l∂_{B}(b_{0}) = dim_{ℚ_{B}∩C} l∂_{B_{0}}(b_{0}) ⩽ dim B_{0} < n.

□

Remark 7.4. We do not know if 5.5 can be obtained from 7.3. It is unclear if an analogue of 5.4 holds, i.e. for b ∈ B(K), if

dim_{ℚ_{B}∩C} l∂_{B}(b) < n

implies that b belongs to a constant coset of a proper algebraic subgroup of B. It is quite clear that 𝔾_{m} in 5.4 may be replaced by any simple commutative algebraic group. It is worth noticing that for B = 𝔾_{a}, we get the (∗) property considered at the end of the previous section.

Now we state a version of 6.4.

Theorem 7.5. Let A, B, ϕ, a, b, n be as in 6.4 and assume A, B, ϕ′ are defined over ℚ. If

max[dim_{ℚ_{A}∩C} l∂_{B}(b), dim_{ℚ_{B}∩C} l∂_{B}(b)] = n,

then trdeg_{C}(a, b) ⩾ n + 1.

Proof. Using 6.4 we get (as in the proof of 7.3):

dim_{ℚ_{A}∩C} l∂_{A}(a) < n, dim_{ℚ_{B}∩C} l∂_{B}(b) < n.

Since ϕ′ is defined over ℚ and l∂_{B}(b) = ϕ′(l∂_{A}(a)), we get

dim_{ℚ_{A}∩C} l∂_{B}(b) = dim_{ℚ_{A}∩C} l∂_{A}(a) < n.
□

Example 7.6. (1) If ϕ is a formal isomorphism between 𝔾_{a}^{n} and a semi-abelian variety A and (v, a) satisfies the differential equation of ϕ, then we get:

dim_{ℚ_{A}} (l∂_{A}(a)) = n ⟹ trdeg_{C}(v, a) ⩾ n + 1.

(2) Assume A, ϕ and (v, a) are as in (1).

• If A = E^{n} for an elliptic curve E, we get:

dim_{ℚ(τ)} (l∂_{A}(a)) = n ⟹ trdeg_{C}(v, a) ⩾ n + 1,

where τ is a complex multiplication in E (if any). This is Proposition 1. from [11].

• More generally, if E above is any simple semi-abelian variety, then we get:

dim_{End_{ℚ} (E)}(l∂_{A}(a)) = n ⟹ trdeg_{C}(v, a) ⩾ n + 1.

• Yet more generally, if A = A_{1}^{l_{1}}×⋯×A_{k}^{l_{k}}, where A_{i} are pairwise non-isogenous simple semi-abelian varieties we get a condition involving all End_{ℚ} (A_{i}) which is a bit cumbersome to write down. If A_{i} are 1-dimensional, we get Proposition 2. from [11].

(3) If ϕ is a formal isomorphism between 𝔾_{m}^{n} and an abelian variety A (or even an extension of an abelian variety by a vector group) and (a, b) satisfies the differential equation of ϕ, then we get:

dim_{ℚ} (l∂_{𝔾_{m}^{n}}(a)) = n ⟹ trdeg_{C}(a, b) ⩾ n + 1.

8. Positive characteristic

In this section we assume char(K) = p > 0. If ∂ is a derivation on K with the field of constants C, then K^{p} ⊆ C, so K is algebraic over C and any Ax-like statement is meaningless. To get a more reasonable field of constants we replace the derivation ∂ with a Hasse-Schmidt derivation D = (D_{i})_{i<ω}, i.e.: each D_{i} : K → K is additive, D_{0} is the identity map,

D_{i}(xy) = ∑_{n+m=i} D_{n}(x)D_{m}(y),

D_{i} ∘ D_{j} = \binom{i + j}{i}D_{i+j}.

Let C be the constant field of all D_{i}. K is usually not algebraic over C. To proceed as in Section 2, we need to:

• Replace the tangent bundle T V (K) = V (K[X]/X^{2}) with the bundle of arcs:

Arc V (K) = V (K⟦X⟧).

• For A, a commutative algebraic group, replace T_{0}A with U_{A}, which denotes the fiber of Arc A → A over 0.

• It is more difficult to find a good replacement of differential forms. It turns out that Vojta’s Hasse-Schmidt differential forms [19] work well. However 2.5 remains the main problem.

To proceed as in Sections 3 and 4 we need the following:

Definition 8.1. lD_{A} : A(K) → U_{A}(K) is the HS-logarithmic derivative – the composition of D_{A} : A(K) → Arc A(K) and the projection map Arc A(K) → U_{A}(K).

Example 8.2.

lD_{𝔾_{a}}(x) = (D_{i}(x))_{i⩾1}, lD_{𝔾_{m}}(x) = (\frac{D_{i}(x)}{x})_{i⩾1} .

Remark 8.3. Any formal map ϕ : A → B induces an algebraic homomorphism U_{ϕ} : U_{A} → U_{B}.

Definition 8.4. The HS-differential equation of ϕ is

U_{ϕ}(lD_{A}(x)) = lD_{B}(y).

One can now state and try to prove a version of 5.5 in this context. This is work in progress.

The main difference between the characteristic 0 case and the positive characteristic case is that for A, B commutative algebraic groups of the same dimension U_{A} need not be isomorphic to U_{B} if characteristic is positive. In characteristic 0 case, all commutative formal groups of dimension n are isomorphic to 𝔾̂_{a}^{n}. This is no longer true in positive characteristic even for 1-dimensional algebraic groups, e.g. pU_{𝔾_{a}} = 0 and pU_{𝔾_{m}} ≠ 0, so U_{𝔾_{a}} ≇ U_{𝔾_{m}} (note that A ≅ B implies U_{A} ≅ U_{B}). Actually, the situation in dimension 1 has a nice description: to each 1dimensional formal group F one associates its height ht(F) ∈ ℕ∪{∞}, which is the height of the p-th power map on F. It is well-known that F ≅ G if and only if ht(F) = ht(G). Let E_{o} be an ordinary elliptic curve and E_{s} a supersingular elliptic curve. Then we have:

ht(𝔾_{a}) = ∞, ht(𝔾_{m}) = 1, ht(E_{o}) = 1, ht(E_{s}) = 2.

And for each n ∈ ℕ, there is a formal group F such that ht(F) = n. Hence there are formal algebraic groups in dimension 1 which are not algebraic, so Riemann’s Existence Theorem does not hold in this context. But there are still formal isomorphisms between E_{o} and 𝔾_{m}. They are the main candidates for a positive characteristic version of 5.5. We finish with a remark that unlike in the characteristic 0 case, there are formal endomorphisms of 𝔾_{a} which are not algebraic e.g. ∑_{n=0}^{∞} X^{p^{n}} (thanks to Thomas Scanlon for pointing out this example to me).

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