MULTIPLES OF SUBVARIETIES IN ALGEBRAIC GROUPS OVER FINITE FIELDS

BJORN POONEN

Abstract. Let X be a subvariety of a commutative algebraic group G over \overline{𝔽}_{q} such that X generates G. Then ⋃_{ϕ∈End G} ϕ(X(\overline{𝔽}_{q})) = G(\overline{𝔽}_{q}). If G is semiabelian, this can be strengthened to ⋃_{n≥1} nX(\overline{𝔽}_{q}) = G(\overline{𝔽}_{q}), and there is a density-1 set of primes S such that X(\overline{𝔽}_{q}) projects surjectively onto the S-primary part of G(\overline{𝔽}_{q}). These results build on work of Bogomolov and Tschinkel.

1. Statements of results

This introductory section states our main results; the proofs are contained in later sections. Throughout this paper, 𝔽_{q} is a finite field of q elements, and \overline{𝔽}_{q} is an algebraic closure of 𝔽_{q}.

1.1. A result of Bogomolov and Tschinkel. Bogomolov and Tschinkel proved the following result and used it to deduce the existence of non-uniruled K3 surfaces over \overline{𝔽}_{q} whose rational curves cover all \overline{𝔽}_{q}-points.

Theorem 1.1 ([BT03]). Let X be a smooth projective integral curve of genus g ≥ 1 over \overline{𝔽}_{q}, embedded in its Jacobian J in the usual way, using a basepoint O. Then J(\overline{𝔽}_{q}) = ⋃_{ϕ∈End J} ϕ(X(\overline{𝔽}_{q})).

The first goal of this paper is to give a new proof requiring nothing deeper than the Weil conjectures for curves and the geometric class field theory fact that if X ↪ J is a curve embedded in its Jacobian in the usual way and ϕ: A → J is a separable isogeny, then ϕ^{−1}(X) is geometrically irreducible.

1.2. Generalizations. Actually this new proof works also for generalized Jacobians, so we prove the result in this generality, in Section 2.

In later sections, we generalize further to the case of a subvariety X of arbitrary dimension in an arbitrary commutative algebraic group G, subject to the obviously necessary condition that X generate G. (In this paper, an algebraic group over a field k is a connected reduced group scheme of finite type over k. See Section 3 for the definition of “generate”.)

Theorem 1.2. Let G be a commutative algebraic group over \overline{𝔽}_{q}. Let X ⊆ G be an irreducible (but not necessarily closed) subvariety that generates G. Then ⋃_{ϕ∈End G} ϕ(X(\overline{𝔽}_{q})) = G(\overline{𝔽}_{q}). Moreover, it suffices to take the union over ϕ ∈ ℤ[F], where F is a Frobenius endomorphism of G.

Date: November 30, 2004.

2000 Mathematics Subject Classification. Primary 14G15; Secondary 14K15.

Key words and phrases. Algebraic group, abelian variety, semiabelian variety, generalized Jacobian, finite field.

This research was supported by NSF grant DMS-0301280 and a Packard Fellowship.

Theorem 1.2 shows that each point of G is contained in the image of X under some endomorphism. In fact, given any finite collection of points of G, we can find a single endomorphism that works for all of them:

Corollary 1.3. Under the hypotheses of Theorem 1.2, if S ⊆ G(\overline{𝔽}_{q}) is finite, then S ⊆ ϕ(X(\overline{𝔽}_{q})) for some ϕ ∈ ℤ[F] ⊆ End G.

Proof. Apply Theorem 1.2 to X^{s} ⊆ G^{s} where s = #S. □

1.3. Semiabelian varieties. In the case where G is a semiabelian variety, we show below that instead of using all ϕ ∈ End G, it is enough to use the multiplication-by-n maps for n ∈ ℤ_{≥1}.

Theorem 1.4. Let G be a semiabelian variety over \overline{𝔽}_{q}. Let X ⊆ G be an irreducible subvariety that generates G. Then ⋃_{n≥1} nX(\overline{𝔽}_{q}) = G(\overline{𝔽}_{q}).

The crucial case where X is a curve was obtained independently by Bogomolov and Tschinkel [BT04]. (After discovering our proof, we learned that they had found their proof a few weeks earlier.)

We get a corollary analogous to Corollary 1.3, proved in the same way:

Corollary 1.5. Under the hypotheses of Theorem 1.4, if S ⊆ G(\overline{𝔽}_{q}) is finite, then S ⊆ n(X(\overline{𝔽}_{q})) for some n ≥ 1.

Remark 1.6. The semiabelian restriction in Theorem 1.4 is necessary, as the following example shows. Let G = 𝔾_{a} × 𝔾_{a} with coordinates x, y, and let p be the characteristic. Let X be the curve xy = 1 in G. By criterion (i) of Corollary 3.3, X generates G. But ⋃_{n≥1} nX(\overline{𝔽}_{q}) = ⋃_{n=1}^{p} nX(\overline{𝔽}_{q}), which is an algebraic subset of G of dimension 1. Thus ⋃_{n≥1} nX(\overline{𝔽}_{q}) ≠ G(\overline{𝔽}_{q}).

Remark 1.7. One faces similar counterexamples if one tries to generalize Theorem 1.4 to noncommutative groups, such as semisimple algebraic groups. (One interprets n as the n-th power map in the group.) For instance, for G = SL_{2} over \overline{𝔽}_{q} of characteristic p, if

a = (\begin{matrix}1 & α \\ 0 & 1 \end{matrix}) with α ∈ \overline{𝔽}_{q}, then a calculation shows that the only elements of SL_{2}(\overline{𝔽}_{q}) having

a power equal to a are those of the form ± (\begin{matrix}1 & mα \\ 0 & 1 \end{matrix}) with m ∈ 𝔽_{p}. If X is an irreducible

curve containing one such element for every α ∈ \overline{𝔽}_{q}, then X has infinite intersection with

either \{(\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) : x ∈ \overline{𝔽}_{q}\} or \{− (\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) : x ∈ \overline{𝔽}_{q}\}, so X is a dense open subset of one of

these curves. Thus there is no curve X in SL_{2} satisfying ⋃_{n≥1} nX(\overline{𝔽}_{q}) = SL_{2}(\overline{𝔽}_{q}), even though one can easily construct curves that generate SL_{2} as a group.

1.4. Projections of subvarieties. If ℓ is a prime and A is a torsion abelian group, let A{ℓ} = ⋃_{e≥1} A[ℓ^{e}] be the ℓ-primary part of A. If S is a set of primes, let A{S} = ⨁_{ℓ∈S} A{ℓ} ⊆ A. By the density of S, we mean natural density: see Section 6 for the definition.

The paper [BT04] proves, under the assumptions of Theorem 1.4 (for X a curve), that there exists a set S of positive density such that the restriction of the canonical projection G(\overline{𝔽}_{q}) ↠ G(\overline{𝔽}_{q})\{S\} to X(\overline{𝔽}_{q}) is surjective. In fact they can find S containing any given finite set of primes. Their work extends earlier results for a curve in its Jacobian in [AI85] (where it is assumed that #S = 1) and [PS03, §2] (where it is observed that the method of [AI85] works also for finite S). The paper [PS03] also considers (for finite S) the case of an arbitrary positive-dimensional subvariety X in a simple abelian variety A, by reducing to the case of curves: because their A is simple, they can sidestep the issue we must face, of whether a curve in A generates A.

Building on the ideas in [BT04], but with more work, we strengthen the result by constructing an S of density 1 for which the conclusion still holds:

Theorem 1.8. Let G be a semiabelian variety over \overline{𝔽}_{q}. Let X ⊆ G be an irreducible subvariety that generates G. Let S_{0} be any finite set of primes. Then there exists a set of primes S ⊇ S_{0} of density 1 such that the composition X(\overline{𝔽}_{q}) → G(\overline{𝔽}_{q}) ↠ G(\overline{𝔽}_{q})\{S\} is surjective.

This will be proved in Section 6.

2. Images of a curve under endomorphisms of a generalized Jacobian

The following statement generalizes Theorem 1.1 but we call it a lemma, because it will be used to prove the more general Theorem 1.2.

Lemma 2.1. Let X′ be a smooth projective integral curve over \overline{𝔽}_{q}. Let 𝔪 be a modulus, and let J be the generalized Jacobian of X′ with respect to 𝔪, as defined in [Ser88, Chapter I]. Let X be the image of the map ι: X′ − 𝔪 → J sending x to the class of x − D, where D is a fixed divisor of degree 1 supported on X′ − 𝔪. Then ⋃_{ϕ∈End J} ϕ(X(\overline{𝔽}_{q})) = J(\overline{𝔽}_{q}).

Proof of Theorem 1.1. We may assume dim J > 0; then ι is an embedding. Suppose a ∈ J(\overline{𝔽}_{q}); we must find ϕ ∈ End J and x ∈ X(\overline{𝔽}_{q}) such that ϕ(x) = a. Without loss of generality, X, J, ι, and a are defined over 𝔽_{q}. Let F be the q-power Frobenius endomorphism of J. We will seek x ∈ X(𝔽_{q^{n}}) for some large n; in this case f_{n} := F^{n} − 1 would kill x, while F − 1 kills a, so we try ϕ := 1 + F + ⋯ + F^{n−1}, which is an isogeny since its derivative is nonzero. In particular, we can choose b ∈ ϕ^{−1}(a). Then (F^{n} − 1)b = (F − 1)a = 0, so b ∈ J(𝔽_{q^{n}}). We hope X(𝔽_{q^{n}}) meets ϕ^{−1}(a) = b + ker ϕ = b + (F − 1)J(𝔽_{q^{n}}). Equivalently, if we define g: J → J (over 𝔽_{q^{n}}) by g(z) = b + (F − 1)z, we hope that Y_{n} := g^{−1}(X) has an 𝔽_{q^{n}}-point.

Over \overline{𝔽}_{q}, the curve Y_{n} is isomorphic to the inverse image of X under the separable isogeny F −1: J → J, so Y_{n} is geometrically irreducible by Proposition 10 in [Ser88, VI.§2.11]. Since its \overline{𝔽}_{q}-isomorphism type is independent of n and b, the Weil conjectures prove Y_{n}(𝔽_{q^{n}}) ≠ ∅ if n is large enough. □

3. The subgroup generated by a subvariety

Lemma 3.1. Let G be a nontrivial algebraic group over an algebraically closed field k. Then G(k) is not cyclic.

Proof. If G(k) is cyclic, then G is commutative. By the structure theory of algebraic groups, G contains an algebraic subgroup H isomorphic to 𝔾_{a}, 𝔾_{m}, or a nontrivial abelian variety. In each case, H(k) has infinitely many elements of finite order, so H(k) is not cyclic. Thus G(k) is not cyclic. □

Lemma 3.2. Let G be a commutative algebraic group over an algebraically closed field k. Let X be an irreducible subvariety of G, not necessarily closed. Let X_{n} be the image of X^{n} under the addition morphism G^{n} → G. Then there exists an algebraic subgroup H ⊆ G and a point g ∈ G(k) such that X_{n} ⊆ ng + H for all n ≥ 1 with equality for n ≫ 1. Moreover, H is uniquely determined by X ⊆ G.

Proof. Translate X to reduce to the case 0 ∈ X. Then X_{n} ⊆ X_{n+1} for all n. Let H_{n} be the Zariski closure of X_{n} in G. Then H_{n} is the closure of the image of the irreducible variety X^{n}, so H_{n} is irreducible. Since dim G < ∞, the sequence H_{1} ⊆ H_{2} ⊆ ⋯ is eventually constant; say H_{n} = H_{n+1} = · · ·. Define H = H_{n}. Then H ⊆ H + H = H_{n} + H_{n} ⊆ H_{2n} = H, so H + H = H. If h ∈ H(k), then h + H ⊆ H, but h + H and H are closed subvarieties of G of the same dimension, so h + H = H. In particular, 0 ∈ h + H, so −h ∈ H. Thus H is an algebraic subgroup of G.

Let m ≥ 2n. Then given h ∈ H(k), the subvarieties X_{m−n} and h − X_{n} are dense in H = h − H so they intersect. Hence h ∈ X_{m−n} + X_{n} = X_{m}. Thus X_{m} = H for any m ≥ 2n. So the first conclusion holds, with g = 0.

Any H satisfying the first conclusion equals the image of X_{n} × X_{n} under the subtraction map G × G → G for n ≫ 1, so it is determined. □

Corollary 3.3. Let notation be as in Lemma 3.2. Then the following are equivalent: (i) The subvariety X is not contained in any translate of an algebraic subgroup H ⊊ G. (ii) There exists n ≥ 1 such that the addition map X^{n} → G is surjective. (iii) The set X(k) generates G(k) as an abstract group.

Proof.

(i) ⟹ (ii): Assuming (i), the H of Lemma 3.2 must be G.

(ii) ⟹ (iii): Trivial.

(iii) ⟹ (i): Suppose X(k) generates G(k). If X were contained in the translate of H ⊊ G by g, then (G/H)(k) would be generated by the image of g, contradicting Lemma 3.1. □

Definition 3.4. If the equivalent conditions of Corollary 3.3 hold, we say that X generates G. If X and G are defined over a field that is not necessarily algebraically closed, we say that X generates G if it is so after extending the base field to an algebraic closure.

4. Generating curves inside generating subvarieties

This section contains the work needed to reduce the case of X of arbitrary dimension to the case of curves.

Lemma 4.1. Let X be a geometrically irreducible quasi-projective variety over a finite field 𝔽_{q}. Assume dim X ≥ 1. There exists a finite extension k ⊇ 𝔽_{q} such that for every finite subset S ⊆ X(\overline{k}) there exists a geometrically irreducible curve Z ⊆ X_{k} containing S.

Proof. Without loss of generality, replace X by a projective closure. Replacing 𝔽_{q} by a finite extension and X by its associated reduced subscheme, we may assume X is geometrically integral. By [dJ96, Theorem 4.1], after passing to a finite extension k, there exists a smooth projective geometrically integral X′ ⊆ ℙ^{n} and a generically finite proper morphism π: X′ → X. Lift each s ∈ S to get a finite subset S′ ∈ X′(\overline{k}). By [Poo04, Corollary 3.4], there exists a geometrically irreducible curve Z′ ⊆ X′ passing through all points of S′. Let Z = π(Z′). □

Remark 4.2. Lemma 4.1 holds also if 𝔽_{q} is replaced by an arbitrary field, because the analogue of [Poo04, Corollary 3.4] is true (and easier to prove) over infinite fields.

Lemma 4.3. Let G be an d-dimensional commutative algebraic group over 𝔽_{q}. Then

(\sqrt{q} − 1)^{2d} ≤ #G(𝔽_{q}) ≤ (\sqrt{q} + 1)^{2d}.

Proof. Using exact sequences and Lang’s theorem on the vanishing of H^{1}(𝔽_{q}, A) for any algebraic group A, we reduce to the cases where G is an abelian variety, a torus, or 𝔾_{a}. If G is an abelian variety, the Weil conjectures give the bounds. If G is a torus, #G(𝔽_{q}) = det (q − F |Ĝ) where Ĝ is the character group of G, and F is the q-power Frobenius acting on it; since some power of F acts trivially on Ĝ, the eigenvalues of F are roots of unity, and the bound follows. If G = 𝔾_{a}, #G(𝔽_{q}) = q. □

Lemma 4.4. Let G be a commutative algebraic group over \overline{𝔽}_{q}. Let X ⊆ G be an irreducible subvariety. Suppose that dim X ≥ 1, and X generates G. Then there is an irreducible curve Z ⊆ X that generates G.

Proof. Without loss of generality, X and G are over 𝔽_{q}. Replace 𝔽_{q} by the k of Lemma 4.1 for X. Enlarge 𝔽_{q} again if necessary so that (\sqrt{q} − 1)^{2d} > (\sqrt{q} + 1)^{2d−2}, where d = dim G. For some n ≥ 1, we may express each point in G(𝔽_{q}) as x_{1} + ⋯ + x_{n} with x_{i} ∈ X(\overline{𝔽}_{q}). Lemma 4.1 gives an irreducible curve Z ⊆ X passing through the points of X occurring in all these expressions. Let H be as in Lemma 3.2 for Z ⊆ G. By choice of Z, H(𝔽_{q}) = G(𝔽_{q}). If H ≠ G, Lemma 4.3 applied to G and H gives

(\sqrt{q} − 1)^{2d} ≤ #G(𝔽_{q}) = #H(𝔽_{q}) ≤ (\sqrt{q} + 1)^{2d−2},

contradicting our earlier inequality. Thus H = G, so Z generates G. □

5. Multiples of subvarieties

If X is any integral curve, let \overline{X} be the smooth projective curve birational to X, and let g_{X} be the genus of \overline{X}.

Proof of Theorem 1.2. If dim X = 0, then dim G = 0 and we are done. Otherwise Lemma 4.4 lets us reduce to the case where X is an irreducible curve. Let X̃ ⊆ \overline{X} be the normalization of X. By Theorems 1 and 2 of [Ser88], the composition X̃ → X → G factors through a generalized Jacobian; i.e., there is a commutative square

\xymatrix{X̃ \ar@{.>}[r]^{ι} \ar@{->>}[d] & J_{𝔪} \ar@{->>}[d]^{π} \\ X \ar[r] & G}

where J_{𝔪} is the generalized Jacobian of \overline{X} with respect to some modulus 𝔪 supported on \overline{X} − X̃, the morphism ι is the usual one associated to some divisor of degree 1 supported on X̃, and π is a homomorphism. Since X generates G, the homomorphism π must be surjective. The result for X ↪ G now follows from Lemma 2.1 for X̃ → J_{𝔪}. □

The following lemma is more general than what we need to prove Theorem 1.4; we write it this way so that it can be used also in the proof of Theorem 1.8.

Lemma 5.1. Let X → J be the usual map from a curve over 𝔽_{q} to its generalized Jacobian with respect to some modulus 𝔪. Assume that it is an embedding. Let H be a subgroup of J(𝔽_{q}), and let a ∈ J(𝔽_{q}). If q is sufficiently large relative to (J(𝔽_{q}) : H), g_{X}, and deg 𝔪, then X(𝔽_{q}) meets H + a.

Proof. Let A = J/H. As usual, F will denote a q-power Frobenius endomorphism on A or J. Since H is contained in the kernel of F − 1 acting on J, we get the diagonal map ϕ in the following commutative diagram of separable isogenies:

\xymatrix{J \ar[r]^{F −1} \ar@{->>}[d] & J \ar@{->>}[d] \\ A \ar@{.>}[ur]^{ϕ} \ar[r]_{F −1} & A.}

The kernel A(𝔽_{q}) of the bottom F −1 map is mapped by ϕ to the kernel H of the right vertical map. If we define ψ: A → J by ψ(z) = ϕ(z) + a, then ψ(A(𝔽_{q})) = H + a. Let Y = ψ^{−1}(X). By Proposition 10 in [Ser88, I.§2.11], Y is geometrically irreducible. Then g_{Y} and #(\overline{Y} − Y) do not depend on a, and are bounded in terms of g, deg 𝔪, and deg ψ = deg ϕ = (J(𝔽_{q}) : H). Since q is sufficiently large relative to these quantities, the Weil conjectures give #Y (𝔽_{q}) > 0. Choose y ∈ Y (𝔽_{q}), and let x = ψ(y). Then x ∈ X(𝔽_{q}) and x ∈ ψ(A(𝔽_{q})) = H + a. □

If G is a commutative algebraic group over 𝔽_{q} and m ≥ 1, let G[m] be the kernel of G(\overline{𝔽}_{q}) →^{m} G(\overline{𝔽}_{q}). If S ⊆ G(\overline{𝔽}_{q}), let 𝔽_{q}(S) be the extension of 𝔽_{q} generated by the coordinates of all the points in S. If P ∈ G(\overline{𝔽}_{q}), let 𝔽_{q}(P) = 𝔽_{q}(\{P \}).

Proof of Theorem 1.4. We get a square as in the proof of Theorem 1.2. Because G is semiabelian, Hom(𝔾_{a}, G) = 0, so we can replace 𝔪 by its associated reduced modulus, and hence J_{𝔪} by its quotient by its unipotent radical, in order to assume that J_{𝔪} is semiabelian. Thus we reduce to the case where X ↪ G is the usual inclusion ι: \overline{X} − 𝔪 ↪ J_{𝔪} of a curve with respect to a reduced modulus into its generalized Jacobian.

Let a ∈ G(\overline{𝔽}_{q}). We want to find n ≥ 1 and x ∈ X(\overline{𝔽}_{q}) such that nx = a. Without loss of generality, enlarge q so that X, G, ι, a are all defined over 𝔽_{q}. Choose m ≥ 1 so that G(𝔽_{q}) ⊆ G[m]. Since G is semiabelian, G[m^{2}] is finite. Let d = [𝔽_{q}(G[m^{2}]) : 𝔽_{q}]. Choose ℓ ≫ 1 with gcd(ℓ, d) = 1.

If P ∈ G(𝔽_{q^{ℓ}}) and mP ∈ G(𝔽_{q}), then m^{2}P = 0, so P ∈ G(𝔽_{q^{d}}), and hence P ∈ G(𝔽_{q^{ℓ}}) ∩ G(𝔽_{q^{d}}) = G(𝔽_{q}). In other words, G(𝔽_{q^{ℓ}})/G(𝔽_{q}) has order prime to m and hence also to #G(𝔽_{q}). Thus G(𝔽_{q^{ℓ}}) ≃ G(𝔽_{q}) × H.

If ℓ ≫ 1, Lemma 5.1 applied to 𝔽_{q^{ℓ}} gives x ∈ X(𝔽_{q^{ℓ}}) and h ∈ H such that x = h+a. Choose n ∈ ℤ_{≥1} with n ≡ 1 (mod m) and n ≡ 0 (mod #H). Then nx = nh + na = 0 + a = a. □

Remark 5.2. In the corresponding result of Bogomolov and Tschinkel for curves (Theorem 1 of [BT04]), it is shown that the result remains true if ⋃_{n≥1} nX(\overline{𝔽}_{q}) is replaced by ⋃_{\begin{matrix}n≥1 \\ n≡1 (mod d) \end{matrix}} nX(\overline{𝔽}_{q}) for any d ≥ 1. Our proof gives this as well: when m is chosen in the proof

of Theorem 1.4, choose it to be a multiple of d.

This can be generalized to an arbitrary arithmetic progressions:

Corollary 5.3. Let notation be as in Theorem 1.4, and let b, d ∈ ℤ_{≥1}. Then

⋃_{\begin{matrix}n≥1 \\ n≡b (mod d) \end{matrix}} nX(\overline{𝔽}_{q}) = G(\overline{𝔽}_{q}).

Proof. Given a ∈ G(\overline{𝔽}_{q}), choose a′ ∈ G(\overline{𝔽}_{q}) with ba′ = a. Remark 5.2 gives n′ ≥ 1 and x ∈ X(\overline{𝔽}_{q}) with n′ ≡ 1 (mod d) and n′x = a′. Multiply by b to get bn′ ≡ b (mod d) and (bn′)x = a. Let n = bn′. □

6. Projection

This section proves Theorem 1.8. Throughout this section G is a semiabelian variety over \overline{𝔽}_{q}, and X ⊆ G is an irreducible subvariety that generates G. As usual, we reduce to the case where X is a curve in a generalized Jacobian. Enlarge q to assume that G and X are defined over 𝔽_{q}. Let F be the q-power Frobenius endomorphism of G. Let d = dim G. Let P be the set of prime numbers.

Lemma 6.1. Let ℓ ∈ P. Let P ∈ G(\overline{𝔽}_{q})\{ℓ\}. Let deg P = [𝔽_{q}(P) : 𝔽_{q}]. If ℓ ∤ deg P, then deg P divides ∏_{i=1}^{2d}(ℓ^{i} − 1).

Proof. Let ℓ^{s} be the order of P. Then deg P equals the size of the F -orbit of P, which divides the order of the image of F in Aut G[ℓ^{s}], which divides # Aut G[ℓ^{s}], which divides # GL_{2d}(ℤ/ℓ^{s}ℤ) (since G[ℓ^{s}] is a free ℤ/ℓ^{s}ℤ-module of rank ≤ 2d), and # GL_{2d}(ℤ/ℓ^{s}ℤ) is a power of ℓ times ∏_{i=1}^{2d}(ℓ^{i} − 1). Finally, gcd(deg P, ℓ) = 1. □

If r ∈ P, let 𝔽_{q^{r^{∞}}} = ⋃_{i≥0} 𝔽_{q^{r^{i}}}.

Lemma 6.2. If S ⊆ P is finite and r ∈ P − S, then G(𝔽_{q^{r^{∞}}}){S} is finite.

Proof. It suffices to consider S = {ℓ} for a single prime ℓ ≠ r. By Lemma 6.1, the degrees of points in G(𝔽_{q^{r^{∞}}}){ℓ} are bounded, say by N. Then G(𝔽_{q^{r^{∞}}}){ℓ} ⊆ ⋃_{n=1}^{N} G(𝔽_{q^{n}}), which is finite. □

If T ⊆ P, define the upper density of T as

\overline{μ}(T) = lim sup_{x→∞} \frac{#\{ℓ ∈ T : ℓ ≤ x\}}{#\{ℓ ∈ P : ℓ ≤ x\}}.

Define \underline{μ}(T) similarly using lim inf instead of lim sup. If \overline{μ}(T) = \underline{μ}(T), let μ(T) be the common value, and call it the density of T. Also define

η(T) = sup_{x∈ℤ_{≥1}} \frac{#\{ℓ ∈ T : ℓ ≤ x\}}{#\{ℓ ∈ P : ℓ ≤ x\}}.

Thus \overline{μ}(T) ≤ η(T).

Lemma 6.3. Let r be prime. Then the set T := {ℓ ∈ P : (∃n ≥ 0) such that ℓ | #G(𝔽_{q^{r^{n}}})} has density 0.

Proof. For m ≥ 1, let T_{m} be the subset of T obtained by discarding r (if it is in T) and all primes dividing #G(𝔽_{q^{r^{m}}}). Since only finitely many primes were discarded, \overline{μ}(T) = \overline{μ}(T_{m}).

Suppose ℓ ∈ T_{m}. Then there is a point in G(𝔽_{q^{r^{∞}}}){ℓ} of degree r^{n} for some n > m, so Lemma 6.1 gives

r^{m} | r^{n} | ∏_{i=1}^{2d}(ℓ^{i} − 1).

Factor the right hand side as a polynomial in \overline{ℚ}[ℓ] into ∏_{j=1}^{s}(ℓ−ζ_{j}) where s = 1+2+⋯+2d and each ζ_{j} is in \overline{ℚ}. Extend the r-adic valuation on ℚ to v_{r}: \overline{ℚ}^{×} → ℚ with v(r) = 1. Then ℓ ∈ T_{m} implies v_{r}(ℓ − ζ_{j}) ≥ m/s for some j. The set of integers ℓ satisfying v_{r}(ℓ − ζ_{j}) ≥ m/s for a particular j is either empty or a residue class modulo r^{⌈m/s⌉}, so T_{m} is contained in a union of at most s residue classes modulo r^{⌈m/s⌉}. Dirichlet’s theorem on primes in arithmetic progressions says that the set of primes in each residue class has density 1/ (r^{⌈m/s⌉}(1 − r^{−1})). Thus

\overline{μ}(T) = \overline{μ}(T_{m}) ≤ \frac{s}{r^{⌈m/s⌉}(1 − r^{−1})} → 0

as m → ∞. □

Lemma 6.4. Given a ∈ G(\overline{𝔽}_{q}), ϵ > 0, and a finite set S_{0} ⊆ P, there exists a finite set U ⊆ P − S_{0} with η(U) < ϵ and a ∈ X(\overline{𝔽}_{q}) + G(\overline{𝔽}_{q})\{U\}.

Proof. Enlarge q to assume a ∈ A(𝔽_{q}). Pick r ∈ P − S_{0} with 1/π(r) < ϵ, where π(r) is the number of primes ≤ r. The set T defined in Lemma 6.3 satisfies \overline{μ}(T) = 0. Define subsets τ := {ℓ ∈ T : ℓ ≤ t} − {r} and T′ := T − τ depending on a parameter t > 0. As t → ∞ we have

η(T′) ≤ η(\{r\}) + η(T′ − \{r\}) ⟶ η(\{r\}) + \overline{μ}(T) = \frac{1}{π(r)} + 0 < ϵ

so we can choose t > max S_{0} such that η(T′) < ϵ. Let B = #G(𝔽_{q^{r^{∞}}}){τ}, which is finite by Lemma 6.2 since r ∉ τ. For all n, the subgroup H := G(𝔽_{q^{r^{n}}}){T′} of G(𝔽_{q^{r^{n}}}) has index #G(𝔽_{q^{r^{n}}}){τ}, which is bounded by B. By Lemma 5.1, if n ≫ 1, then X(𝔽_{q^{r^{n}}}) meets H + a. Then a = x − h for some x ∈ X(𝔽_{q^{r^{n}}}) and h ∈ H. Let U be the (finite) set of primes dividing the order of h. Then U ⊆ T′ ⊆ P − S_{0}, since t > max S_{0} and r ∉ S_{0}. Now a = x − h ∈ X(\overline{𝔽}_{q}) + G(\overline{𝔽}_{q})\{U\}, and η(U) ≤ η(T′) < ϵ. □

Lemma 6.5. For i ≥ 1, let U_{i} ⊆ P be finite. If ∑ η(U_{i}) converges, then μ (⋃ U_{i}) = 0.

Proof. The definitions imply

\overline{μ} (⋃ U_{i}) = \overline{μ} ( ⋃_{i≥N} U_{i}) ≤ η ( ⋃_{i≥N} U_{i}) ≤ ∑_{i≥N} η(U_{i}) → 0

as N → ∞. □

Lemma 6.6. Given a finite set S_{0} ⊆ P, there exists U ⊆ P − S_{0} of density 0 such that X(\overline{𝔽}_{q}) + G(\overline{𝔽}_{q})\{U\} = G(\overline{𝔽}_{q}).

Proof. Let a_{1}, a_{2}, … be an enumeration of A(\overline{𝔽}_{q}). Let U_{i} be as in Lemma 6.4 for a = a_{i} and ϵ = 2^{−i} and S_{0}. By Lemma 6.5, U := ⋃ U_{i} has density 0. Each U_{i} is disjoint from S_{0}, so U is disjoint from S_{0}. By construction,

a_{i} ∈ X(\overline{𝔽}_{q}) + G(\overline{𝔽}_{q})\{U_{i}\} ⊆ X(\overline{𝔽}_{q}) + G(\overline{𝔽}_{q})\{U\}

for all i. □

Proof of Theorem 1.8. Let S = P − U where U is as in Lemma 6.6. □

Acknowledgements

I thank Yuri Tschinkel for pointing out the reference [PS03] and for pointing out that my notation in an earlier draft was ambiguous at one point.

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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

  E-mail address: poonen@math.berkeley.edu
  URL: http://math.berkeley.edu/~poonen