NÉRON-TATE PROJECTION OF ALGEBRAIC POINTS

BJORN POONEN

Abstract. Let X be a geometrically irreducible closed subvariety of an abelian variety A over a number field k such that X generates A. Let V be a finite-dimensional subspace of A(\overline{k}) ⊗ R, and let π : A(\overline{k}) → V be the orthogonal projection relative to a Néron-Tate pairing ⟨ , ⟩ : A(\overline{k}) × A(\overline{k}) → R. For V = A(k) ⊗ R, we prove that π(X(\overline{k})) = A(k) ⊗ Q, and moreover, there exist c, c′ > 0 such that for any a ∈ A(k) ⊗ Q, \{ x ∈ X(\overline{k}) : π(x) = a and h(x) < ch(a) + c′ } is Zariski dense in X.

1. Introduction

Let k be a number field, and let \overline{k} be its algebraic closure. Let A be an abelian variety over k, and let X be a geometrically irreducible closed subvariety of A. Several results describe the location of the rational or algebraic points of X within A. For example, the “Mordell-Lang conjecture” states that if Γ is a finite rank subgroup of A(\overline{k}) and if X is not a translate of an abelian subvariety, then X(\overline{k}) ∩ Γ is not Zariski dense in X. This version of the statement was proved by Hindry [Hi], after earlier work of Faltings, Raynaud, Vojta and others. A generalization to semiabelian varieties was proved by McQuillan [McQ].

If one defines the Néron-Tate canonical height h : A(\overline{k}) → R_{≥0} associated to a symmetric ample line sheaf on A, one can also state the “generalized Bogomolov conjecture:” If X is not a translate of an abelian subvariety by a torsion point, there exists ϵ > 0 such that \{ x ∈ X(\overline{k}) : h(x) < ϵ \} is not Zariski dense in X. The conjecture was proved by Zhang [Zh1], using ideas from an important special case (the original Bogomolov conjecture) proved by Ullmo [Ul] using an equidistribution theorem of Szpiro, Ullmo, and Zhang [SUZ]. There is also the combined “Mordell-Lang plus Bogomolov” result of [Po] and the further distribution result of [Zh2]. Moriwaki [Mo],[Mo2] has proved generalizations of most of these statements with k replaced by a finitely generated field extension of Q.

Define a Néron-Tate pairing for A to be a bilinear form ⟨ , ⟩ : A(\overline{k}) × A(\overline{k}) → R such that ⟨x, x⟩ = h(x) for a height function h as above. We may consider ⟨ , ⟩ also as an inner product on the vector space A(\overline{k})_{R} := A(\overline{k}) ⊗ R, which is infinite dimensional if dim A > 0. For any field extension L of k, define also A(L)_{Q} := A(L) ⊗ Q and A(L)_{R} := A(L) ⊗ R.

In this article, we study the image of X(\overline{k}) under the orthogonal projection π : A(\overline{k}) → V where V is a finite-dimensional subspace of A(\overline{k})_{R}. After possibly enlarging k, we have V ⊆ A(k)_{R}, and for our purposes, we lose no information in enlarging V to A(k)_{R}. Also, by translating X to assume 0 ∈ X (enlarging k if necessary), and then replacing A by the image of the Albanese homomorphism Alb X → A, we may reduce to the case in which X generates A, i.e., in which Alb X → A is surjective, or equivalently, the differences P − Q with P, Q ∈ X(\overline{k}) generate the group A(\overline{k}). We make these hypotheses, to simplify the statement of our result. Theorem 1. Let X be a geometrically irreducible closed subvariety of an abelian variety A over a number field k. Assume that X generates A. Let π : A(\overline{k}) → A(k)_{R} denote the orthogonal projection relative to a Néron-Tate pairing for A. Then

Date: February 21, 2000.

This research was supported by NSF grant DMS-9801104, a Sloan Fellowship, and a Packard Fellowship.

This paper is to appear in International Mathematics Research Notices.

(a) π(X(\overline{k})) = A(k)_{Q}.

(b) There exist c, c′ > 0 such that for any a ∈ A(k)_{Q},

\{ x ∈ X(\overline{k}) : π(x) = a and h(x) < ch(a) + c′ \}

is Zariski dense in X. Almost in contradiction with Theorem 1 we have the following, which is a formal consequence of the Mordell-Lang conjecture. Theorem 2. Let A be an abelian variety over \overline{Q} with dim A > 0. Then there exists a nonzero linear functional π : A(\overline{Q})_{Q} → Q such that for every geometrically irreducible closed subvariety X ⊆ A not containing a translate of a positive-dimensional abelian subvariety of A,

(a) π(X(\overline{Q})) is a discrete subset of Q in the archimedean topology.

(b) \{ x ∈ X(\overline{Q}) : π(x) = a \} is finite for every a ∈ Q. There is no contradiction, however, since ker π in Theorem 2 need not be the orthogonal complement of a finite-dimensional subspace of A(\overline{Q})_{Q}. Remarks.

(1) Borrowing terminology from the field of medical imaging, Theorem 1 implies that X cannot be recovered from its Néron-Tate CAT scan!

(2) Analogues of Theorems 1 and 2 where the number fields are replaced by any field finitely generated over Q can be formulated using the height functions defined by Moriwaki [Mo],[Mo2], and their proofs are the same as in the number field case.

2. Proofs

Lemma 3. Let X be a geometrically irreducible projective variety over an infinite field k, with dim X ≥ 1. Then there exists a geometrically irreducible closed curve Y ⊆ X such that the induced morphism Alb Y → Alb X is surjective. Moreover, the union of such Y is Zariski dense in X.

Proof. Let A = Alb X. Choose a prime ℓ not equal to the characteristic of k. Let A[ℓ] denote the kernel of multiplication by ℓ on A. For each P ∈ A[ℓ](\overline{k}), choose a zero-cycle of degree zero on X\overline{k} representing P. Let S′ ⊆ X(\overline{k}) be the set of points appearing in these zero-cycles together with one extra point Q ∈ X(\overline{k}). Let S be the image of S′ in X.

The blow-up α : X′ → X at S is projective; embed X′ in some P^{N}. Bertini’s theorem [Da, p. 249] gives a dense open subset U of the Grassmannian of linear subspaces L ⊆ P^{N} of codimension dim X′ − 1 such that any point of U(k) corresponds to L ⊆ P^{N} for which X′ ∩ L is geometrically irreducible. Choose such an L, and let Y′ = X′ ∩ L. For dimension reasons, Y′ meets every exceptional fiber of α. Let Y = α(Y′). Then Y is a geometrically irreducible curve passing through the points of S′, so the image of Alb Y → A contains A[ℓ]. The only abelian subvariety of A containing all ℓ^{2 dim A} points of order dividing ℓ is A itself, so Alb Y → A = Alb X is surjective. (This last trick is due to O. Gabber [Ka].) The final statement follows, since Y also passes through Q ∈ X(\overline{k}), which was arbitrary. □

Remark. It follows from [Ka] or alternatively [Po2] that the conclusion of Lemma 3 holds even if k is a finite field. Lemma 4. Let A, k, ⟨ , ⟩, and π be as in Theorem 1. Then π(A(\overline{k})) ⊆ A(k)_{Q}.

Proof. Given P ∈ A(\overline{k}), let L be a Galois extension of k such that P ∈ A(L). Any σ ∈ Gal(L/k) acts as an isometry of A(L)_{R} with ⟨ , ⟩) and preserves A(k)_{R}, so π(^{σ}P) = π(P). Thus

π(P) = \frac{1}{[L : k]} π ( ∑_{σ∈Gal(L/k)} ^{σ}P ) ∈ A(k)_{Q}.

□

If X is a curve over a perfect field k and n ≥ 1, denote by X^{(n)} the quotient of X^{n} by the action of the symmetric group S_{n} permuting the coordinates. Points in X^{(n)}(k) will be identified with G_{k}-stable unordered n-tuples of points in X(\overline{k}), where G_{k} := Gal(\overline{k}/k). Lemma 5. Let X be a smooth projective geometrically integral curve of genus g ≥ 1 over F_{q}. Let U be a dense open subset of X^{(g)}. Then there exist infinitely many u ∈ U(\overline{F}_{q}) such that Gal(\overline{F}_{q}/F_{q}(u)) acts transitively on the g-tuple corresponding to u.

Proof. For F_{r} ⊇ F_{q}, let σ : X → X denote the r-th power Frobenius morphism. The set S_{r} := X(F_{r^{g}}) − ⋃_{d|g,d<g} X(F_{r^{d}}) has size r^{g} + o(r^{g}) as r → ∞ through powers of q, by the Weil bounds. The map S_{r} → X^{(g)}(F_{r}) sending x to the g-tuple {x, x^{σ}, …, x^{σ^{g−1}}} (on which Gal(\overline{F}_{q}/F_{r}) acts transitively) is a g-to-one map, so the image has r^{g}/g + o(r^{g}) points as r → ∞. By the Weil bounds again, at most O(r^{g−1}) of these lie outside U. Hence there remain r^{g}/g + o(r^{g}) points in U(F_{r}) corresponding to desired g-tuples. Finally, r^{g}/g + o(r^{g}) is unbounded as r → ∞. □

Lemma 6. Let f : X → X′ be a finite morphism between quasiprojective varieties over a number field k, and let h and h′ denote height functions on X(\overline{k}) and X′(\overline{k}), respectively, defined (up to O(1)) using embeddings of X and X′ in projective spaces. Then there exist constants c_{1}, c_{2} > 0 such that h(x) ≤ c_{1}h′(f(x)) + c_{2} for all x ∈ X(\overline{k}).

Proof. If we change the embedding of X, then h and the new height h̃ are bounded by linear polynomials in each other, since the isomorphisms between the two copies of X are given locally by rational functions. Hence the question is independent of embeddings. In particular, we may reduce to the case where X = Spec B is embedded in A^{m} and X′ = Spec A is embedded in A^{n} for some m, n ≥ 0. By finiteness, each of the m coordinate functions t on X satisfies a monic polynomial

(1)  t^{r} + a_{1}t^{r−1} + ⋯ + a_{r} = 0

with a_{i} ∈ A. For x ∈ X(\overline{k}), (1) shows that the height of t(x) is bounded by a linear polynomial in the heights of the a_{i}(f(x)), which in turn are bounded by a linear polynomial in h′(f(x)). □

Proof of Theorem 1. By Lemma 4, π(X(\overline{k})) ⊆ A(k)_{Q}, so it remains to prove that for any a ∈ A(k)_{Q}, \{ x ∈ X(\overline{k}) : π(x) = a \} is Zariski dense in X.

Lemma 3 lets us reduce to the case where X is a geometrically integral curve. (I learned this method for reducing to curves from Shou-Wu Zhang.) We may enlarge k in order to assume that X(k) contains a smooth point P_{0} of X. Translating X and a by −P_{0}, we may assume that P_{0} = 0 in A. Let J be the Albanese (Jacobian) variety of the normalization X̃ of X. Then we have a commutative diagram

\begin{matrix}X̃ & ⟶^{j} & J \\ ^{α}↓ & & ↓^{ϕ} \\ X & ⟶ & A \end{matrix}

where ϕ is a surjection and j is the Abel map sending the point P̃_{0} ∈ X̃(k) above P_{0} to 0 ∈ J(k). Choose a quotient abelian variety B of J such that the induced homomorphism J → A × B is an isogeny. Define ⟨ , ⟩_{J} and π_{J} for J by tensoring the pullbacks of symmetric ample line sheaves on A and B. Then we have isomorphisms

J(\overline{k})_{Q} ≅ A(\overline{k})_{Q} ⊕ B(\overline{k})_{Q},
J(\overline{k})_{R} ≅ A(\overline{k})_{R} ⊕ B(\overline{k})_{R},

respecting the pairings. If we find a Zariski dense set of points S in X̃(\overline{k}) with π_{J}(S) = \{(a, 0)\} under the isomorphism above, then α(S) is a Zariski dense set of points in X(\overline{k}) with π(α(S)) = {a}, and the heights of the latter points are bounded by a linear polynomial in the heights of the former points, as is true for images under any morphism. Hence from now on, we may assume that X is a geometrically integral smooth projective curve of genus g ≥ 1 embedded in its Jacobian A by the Albanese map determined by P_{0} ∈ X(k).

Since A(\overline{k}) is divisible, a ∈ A(k)_{Q} is represented by a point in A(\overline{k}), which we again call a. Enlarge k to assume that a ∈ A(k). (This changes π as well, but it only makes the problem harder.) Choose a prime 𝔭 of good reduction for A, and let F_{q} denote the residue field. Extend 𝔭 to a place of \overline{k}. Let X̄, ā ∈ Ā(F_{q}), etc. denote the mod 𝔭 reductions of X, a ∈ A(k), etc. Let ϕ denote the birational morphism X̄^{(g)} → Ā sending {x_{1}, …, x_{g}} to x_{1} + ⋯ + x_{g}, using the embedding X̄ ↪ Ā. Let U and V denote dense open subsets of X̄^{(g)} and Ā, respectively, such that ϕ induces an isomorphism U → V. Let ū ∈ U(\overline{F}_{q}) be one of the infinitely many points given by Lemma 5, let b̄ = ϕ(ū) − gā ∈ A(\overline{F}_{q}), and lift b̄ to a torsion point b ∈ A(\overline{k}).

By choice of U, we can write ga + b = x_{1} + ⋯ + x_{g} for x_{i} ∈ X(\overline{k}), which are uniquely determined up to permutation. Moreover, Gal(\overline{k}/k(b)) acts transitively on the x_{i}, since the choice of ū guarantees Galois-transitivity on the reductions. Hence π(x_{i}) = π(x_{1}) for all i, and

π(x_{1}) = \frac{1}{g} ∑_{i=1}^{g} π(x_{i}) = \frac{1}{g} π(ga + b) = a,

since π(a) = a and π(b) = 0. There were infinitely many choices for ū, hence infinitely many distinct possibilities for b̄, for b, and for x_{1}. In particular, the x_{1} with π(x_{1}) = a are Zariski dense in X. Finally, let E denote the largest open subset of A above which the summing morphism s : X^{g} → A is finite, i.e., above which X^{(g)} → A is an isomorphism.

Lemma 6 applied to s^{−1}(E) → E shows that h(x_{1}) is bounded by a linear polynomial in h(ga + b) = g^{2}h(a). □

Proof of Theorem 2. Let X_{1}, X_{2}, … be a complete list of the countably many possibilities for X. Choose a flag of subspaces

0 = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ …

of A(\overline{Q})_{Q} such that dim V_{n} = n and ⋃ V_{n} = A(\overline{Q})_{Q}. Let S_{n}(X_{j}) denote the set of x ∈ X_{j}(\overline{Q}) whose image in A(\overline{Q})_{Q} lies in V_{n}. The Mordell-Lang conjecture guarantees that S_{n}(X_{j}) is finite for each n ≥ 0 and j ≥ 1. Starting with the zero map π_{0} : V_{0} → Q, by induction on n ≥ 1, we can define Q-linear maps π_{n} : V_{n} → Q such that π_{n}|_{V_{n−1}} = π_{n−1} and |π_{n}(x)| ≥ n for any x in the finite set ⋃_{j≤n} (S_{n}(X_{j}) − S_{n−1}(X_{j})).

The π_{n} glue to give π : A(\overline{Q})_{Q} → Q. For each j, n ≥ 1, \{x ∈ X_{j}(\overline{Q}) : π(x) ∈ (−n, n)\} is contained in S_{n−1}(X_{j}), so it is finite. This implies, for each j ≥ 1, that π(X_{j}(\overline{Q})) is discrete and that \{x ∈ X_{j}(\overline{Q}) : π(x) = a\} is finite for each a ∈ Q. □

Acknowledgements

I thank Thomas Scanlon for a brief conversation which led to an improvement of my original Theorem 2, and especially Shou-Wu Zhang for several discussions on related questions in Néron-Tate geometry.

References

[Da] Danilov, V. I., Algebraic varieties and schemes, in: Algebraic geometry I, edited by I. R. Shafarevich, Encyclopaedia of Mathematical Sciences 23, Springer-Verlag, Berlin, 1994.

[Hi] Hindry, M., Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575–603.

[Ka] Katz, N., Space filling curves over finite fields, Math. Res. Lett. 6 (1999), no. 5-6, 613–624.

[McQ] McQuillan, M., Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1, 143–159.

[Mo] Moriwaki, A., Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), 101–142.

[Mo2] Moriwaki, A., A generalization of conjectures of Bogomolov and Lang over finitely generated fields, to appear in Duke Math. J., preprint available at http://arXiv.org/abs/math.NT/9908092

[Po] Poonen, B., Mordell-Lang plus Bogomolov, Invent. Math. 137 (1999), no. 2, 413–425.

[Po2] Poonen, B., Bertini theorems over finite fields, in preparation.

[SUZ] Szpiro, L., Ullmo, E., and Zhang, S., Équirépartition des petits points, Invent. Math. 127 (1997), no. 2, 337–347.

[Ul] Ullmo, E., Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167–179.

[Zh1] Zhang, S., Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159–165.

[Zh2] Zhang, S., Distribution of almost division points, Duke Math. J. 103 (2000), no. 1, 39–46.

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

  E-mail address: poonen@math.berkeley.edu