Diophantine Approximation on Abelian varieties in characteristic p José Felipe Voloch

1. Introduction

Let A be an abelian variety over a function field K in one variable over a finite field k. Let v be a place of K. In this paper we will study the topology induced on A(K) by the v-adic topology on A(K_{v}). In many cases this will lead to bounds for the v-adic distance between points in A(K) in terms of their height and to abelian analogues of Leopoldt’s conjecture. This paper also studies the question of integral points on affine open subsets of Abelian varieties in positive characteristics. In the classical case of number fields, Lang has conjectured and Faltings [F] proved that for A be an abelian variety over the number field K, if V is an affine open subset of A and S is a finite set of places of K, then the set of S-integral points of V is finite. Faltings has also a non-effective bound. Buium and the author [BV] obtained a similar result for function fields of characteristic zero, but in positive characteristic the problem has not been studied, except in dimension 1, i.e., for elliptic curves. In this case we have obtained, in [V], results on this problem. In this paper we will obtain a result under restrictive, but quite general, hypotheses for abelian varieties of arbitrary dimension and deduce the finiteness of integral points on affine subsets under these hypotheses. Our strategy will be similar to [BV]. The question of integral points is related to estimates for the v-adic distance from rational points to a divisor and this in turn will be estimated by the distance to a subvariety of smaller dimension. This inductive procedure, in the present situation, is provided by a general result of Hrushovski ([H], see theorem 1 below) and reduces the problem to our basic question of estimating v-adic distance between points.

We do not solve our problem in full generality. Our results are restricted to the cases when either A has ”sufficiently general moduli”(in a sense that will be precised below) or is an elliptic curve defined over a finite field. The intermediate cases are still open.

Acknowledgments

This paper was planned as a collaboration with D. Abramovich. We had jointly obtained results similar to Hrushovski’s theorem under the same restrictive hypotheses and using the same methods as in [AV]. We also had some ideas to deal with the open intermediate cases mentioned above, but there were some gaps in the argument that we hope to fix eventually. I thank him very much for the many conversations we had on this subject. I would like to thank also J.-P. Serre and J. Tate for pointers to the literature and the NSF (grant DMS-9301157) and the University of Texas’s URI for financial support.

2. Local results

Let A/K be an abelian variety of dimension n. For any closed subscheme X ⊂ A there is a function λ_{v}(X, .) : A(K_{v}) → [0, ∞] which satisfies the following property: for any affine open set U ⊂ A and any system of generators g_{1}, …, g_{m} ∈ O(U) of the ideal defining X ∩ U in U, we may write λ_{v}(X, P) = min{v(g_{1}(P)), …, v(g_{m}(P))} + b(P) with b bounded on any bounded subset of U(K_{v}). The function λ_{v}(X, .) is uniquely determined by the above property up to the addition of a bounded function and is called the local height function associated to X. This notion is developed in detail in [Si]. In the analogous situation for number fields, Lang has conjectured [L] that λ_{v}(X, P) ≪ log h(P) + O(1), for all P ∈ A(K) \ X, where h(P) is the logarithmic height, and Faltings [F] proved that λ_{v}(X, P) = o(h(P)). The following result is theorem 6.3 of [H].

Theorem 1 (Hrushovski). Let A and X be as above and assume that the K/k-trace of A is zero. Then there exists a subvariety Y of X, defined over K̄, which is a finite union of translates of abelian subvarieties of A, such that λ_{v}(X, P) ≪ λ_{v}(Y, P) + 1 for all P ∈ A(K).

Let A be an abelian variety over a local field of characteristic p with valuation v. Let A_{1} be the formal group of A, that is the kernel of reduction modulo v. Then A_{1} is naturally a Z_{p}-module, since p^{n}P → O, n → ∞ for P in A_{1}. We will be studying this Z_{p}-module structure and will begin by lemma 1 below. Let us remark that it is not hard to check that A_{1} has infinite Z_{p}-rank. We will not, strictly speaking, need this fact but it is helpful to put some of our results in perspective. Lemma 1. Let A be an ordinary Abelian variety over a local field of characteristic p with valuation v. There exists a neighbourhood U of O in the v-adic topology, such that λ_{v}(O, pP) = pλ_{v}(O, P) + O(1) for all P in U.

Proof: Let n = dim A and t_{1}, …, t_{n} be local parameters at O , so that λ_{v}(O, P) = min{v(t_{1}(P)), …, v(t_{n}(P))} and the parameters can be chosen so that t_{i}(pP) = a_{i}t_{i}(P)^{p}+ ⋯, i = 1, …, n, because A has an ordinary formal group, where the ⋯ represent a power series in t_{1}(P), …, t_{n}(P) of degree greater than p and the a_{i} are non-zero v-adic integers. The result now follows. Note that the implied constant depends on the valuation of the a_{i}’s, which in turn depend on the reduction type of A at v, in particular this constant is zero if A has good, ordinary reduction at v.

3. Global results Lemma 2. Let A be an abelian variety over a function field K over a finite field of characteristic p > 0 and v a place of K. Assume that A[p^{∞}] ∩ A(K_{s}) = Γ is finite, where K_{s} is the separable closure of K. Then there is a neighbourhood U of O in the v-adic topology, such that, if P ∈ A(K) ∩ U then P ∈ pA(K).

Proof: Assume first that Γ = {O}. Let ker[p] be the group subscheme of A which is the kernel of multiplication by p. For any field extension L/K, there is a coboundary map in flat cohomology of group-schemes, δ_{L} : A(L)/pA(L) → H^{1}(L, ker[p]) which is injective. The image of δ_{K} is finite, by the Mordell-Weil theorem. Also δ_{K_{v}} is continuous in the v-adic topology. ([M2], Lemma III.6.5). It follows that there is a v-adic neighbourhood U of O such that if P ∈ A(K) ∩ U then δ_{K_{v}}(P) = 0. We will show that, in fact P ∈ pA(K), which will prove the lemma. From the above there exists Q ∈ A(K_{v}), such that pQ = P. Let L = K(Q), then L/K is a finite extension and L ⊂ K_{v} so, in particular, v is unramified in L and, a fortiori, L/K is separable. Let M/K be the Galois closure of L/K. Then P ∈ ker δ_{M}. Hence δ_{K}(P) is in the kernel of the restriction map H^{1}(K, ker[p]) → H^{1}(M, ker[p]). But, by the Hochschild-Serre spectral sequence ([M1], Remark II.2.21 (a)), the kernel of restriction is the Galois cohomology group H^{1}(Gal(M/K), ker[p](M)). However, ker[p](M) = O by hypothesis, since M/K is separable. Hence δ_{K}(P) = 0, which shows that Q ∈ A(K). Note that since Γ = {O}, Q is uniquely determined by P. We claim that, as P → O v-adically, we have Q → O also. Indeed, if this were not so, there would be a sequence P_{n} → O with P_{n} = pQ_{n} and Q_{n} bounded away from O. Since A(K_{v}) is compact, after passing to a subsequence, we can assume that Q_{n} → Q ≠ O and pQ = O, but Q is separable over K, since is defined over K_{v} and this contradicts Γ = {O}.

In general, let ϕ : A → A/Γ = B and ψ : B → A, ϕ ∘ ψ = [p^{m}]. If P ∈ A(K) is sufficiently close to O v-adically, then so is ϕ(P) and therefore, by the above there exists Q ∈ B(K), ϕ(P) = p^{m+1}Q = ϕ(ψ(pQ)). Therefore P − pψ(Q) ∈ Γ, but if we choose P sufficiently close to O then Q will be close to O also and therefore, P − pψ(Q) = O since Γ is finite. This completes the proof.

Theorem 2. Let A be an ordinary abelian variety over a function field K of characteristic p > 0 and v a place of K and assume that the K/k-trace of A is zero and that A[p^{∞}]∩A(K_{s}) is finite. Let X be a subvariety of A. Then λ_{v}(X, P) ≪ h(P)^{1/2} for all P ∈ A(K), P ∉ X.

Corollary 1. Hypotheses as in Theorem 2. Assume further that X is an ample divisor. Then for any finite set S of places of K, the set of S-integral points of A\X is finite.

Proof of corollary 1: In this case, h(P), for an S-integral point of A\X, is the sum of λ_{v}(X, P) over the elements of S, and it follows that the height is bounded, which proves the corollary.

Proof of theorem 2: By theorem 1, we can replace X by Y which is a finite union of translates of abelian subvarieties of A and by taking a component of Y, we can assume that Y is an abelian subvariety of A. Finally passing to A/Y we are reduced to showing that λ_{v}(O, P) ≪ h(P)^{1/2} for all P ∈ A(K).

Now, using theorem 1 and lemmas 1 and 2 we may assume that P ∈ U, where U is a neighbourhood of O in the v-adic topology such that for all P ∈ A(K) ∩ U we have P ∈ pA(K) and λ_{v}(O, pP) ≤ pλ_{v}(O, P) + O(1). Moreover, we can assume that U does not contain any non-zero torsion point. Let r be maximal with P = p^{r}Q, Q ∈ A(K) ∩ U, which exists since P is not torsion. Then λ_{v}(O, P) ≤ p^{r}(λ_{v}(O, Q) + O(1)). Also λ_{v}(O, Q) is bounded, since if Q were sufficiently close to O then Lemma 2 would imply that r is not maximal. It follows that there exists c > 0 such that λ_{v}(O, P) ≤ cp^{r}. On the other hand, since Q is not a torsion point, h(P) = p^{2r}h(Q) ≥ dp^{2r}, where d is the minimum height of a non-torsion point. Putting these inequalities together, we get λ_{v}(O, P) ≤ cp^{r} ≤ (c/d^{1/2})h(P)^{1/2}, as desired.

4. Moduli and p-torsion points

In this section we show that the condition that A[p^{∞}] ∩ A(K_{s}) is finite holds if A has sufficiently general moduli.

Suppose T is a scheme, S is a scheme over T and A is an abelian scheme over S. Let ω ω ω_{A}^{1} denote the sheaf on S of invariant relative one-forms on A/S and ^{t}A the dual of A. Then one has the Kodaira-Spencer pairing κ: ω ω ω_{A}^{1} ⊗ ω ω ω_{^{t}A}^{1} → Ω_{S/T}^{1} . Let S = Spec(R), where R is a complete local ring of characteristic p with residue field k̄ and T = Spec(k̄). Let f denote the map ^{t}A → S and Ω_{R/k̄}^{1} = Ω_{S/T}^{1} . Then the sequence of sheaves on ^{t}A

0 → f^{∗}Ω_{R/k̄}^{1} → Ω_{^{t}A/k̄}^{1} → Ω_{^{t}A/R}^{1} → 0

is exact. Let Kod denote the composition

ω ω ω_{^{t}A}^{1} ≅ f_{∗}Ω_{^{t}A/R}^{1} → R^{1}f_{∗}f^{∗}Ω_{R/k̄}^{1} ≅ R^{1}f_{∗}O_{^{t}A} ⊗ Ω_{R/k̄}^{1},

where the second map is the boundary map. Then, κ(ω,^{t}ω) = ω.Kod(^{t}ω).

For an object X over R, we let X̄ denote its special fibre. Suppose the residue field k̄ of R is algebraically closed. If m is the maximal ideal of R, a construction of Serre and Tate gives a pairing q: T_{p}Ā(k̄) × T_{p}^{t}Ā(k̄) → 1 + m for an ordinary abelian variety A, where T_{p}Ā(k̄) is the ”physical” Tate module of Ā, which in turn gives local parameters on the local moduli space of ordinary abelian varieties over an Artin local ring of residue characteristic p (see [K]). In [K], Katz gives formulas for the Serre-Tate parameters in terms of the Kodaira-Spencer pairing. We will need the following theorem which is a corollary of Katz’s results.

Suppose A and ^{t}A are dual ordinary abelian varieties over R. If α ∈ T_{p}A, we can view it as a homomorphism from the p-divisible group of ^{t}A to G_{m}, the formal multiplicative group. We define then, ω_{α} = α^{∗}(dt/t) ∈ ω ω ω_{^{t}A}^{1}, where dt/t is the canonical invariant form on G_{m}. For a ∈ R^{∗}, let d log(a) = da/a ∈ Ω_{R/k̄}^{1}.

Theorem 3 (Katz). Suppose R is as above, k̄ is algebraically closed and A is an ordinary abelian variety over R. If α ∈ T_{p}Ā(k̄) and ^{t}α ∈ T_{p}^{t}Ā(k̄), we have:

d log q(α,^{t}α) = κ(ω_{^{t}α}, ω_{α}) .

To justify the assertion, made in the introduction, that abelian varieties with sufficiently general moduli satisfy the hypotheses of Theorem 2, we will prove the following result.

Proposition. Let A be an ordinary abelian variety over a function field K of characteristic p > 0 such that the Kodaira-Spencer map has maximal rank, then A[p] ∩ A(K_{s}) = O.

Proof: Let us briefly recall the definition of the Serre-Tate pairing (see [K]). Given an element of T_{p}Ā(k̄) × T_{p}^{t}Ā(k̄) one constructs canonically a subgroup of the p-divisible group of A isomorphic to an extension of Z_{p} by T_{p}μ, where the latter is the p-divisible group of p-power roots of unity. Then the value of the Serre-Tate pairing is the class of this extension in Ext^{1}(Z_{p}, T_{p}μ). If we are interested (as is the case) only in the Serre-Tate pairing modulo p-th powers, we can restrict our attention to extensions of Z/pZ by μ_{p} contained in the group-scheme ker[p]. If A[p] ∩ A(K_{s}) contains a non-zero element, it gives rise to a trivial such extension for each subgroup of ker[p] isomorphic to μ_{p}. It follows that the Serre-Tate pairing modulo p-th powers is not of maximal rank and hence, from Katz’s theorem, that the Kodaira-Spencer map is also not of maximal rank, proving the proposition.

5. Abelian analogues of Leopoldt’s conjecture in characteristic p

Leopoldt’s conjecture asserts that the Z_{p}-rank of the p-adic closure of the group of units of a number field is the same as the Z-rank of the group of units. This is known in a few cases but is open in general. An analogue of it for function fields of characteristic p has been shown by Kisilevsky [Ki]. These questions have obvious abelian analogues replacing the multiplicative group by an abelian variety and we will study this question now. Recall that A_{1}, the kernel of reduction in A(K_{v}), is naturally a Z_{p}-module. Also, since we assumed that K is a function field in one variable over a finite field, A_{1} ∩ A(K) is of finite index in A(K).

Theorem 4. Let A be an ordinary abelian variety over a function field K of characteristic p > 0 and v a place of K and assume that A[p^{∞}] ∩ A(K_{s}) = is finite. Then the Z_{p}-rank of the closure of A_{1} ∩ A(K) in A(K_{v}) in the v-adic topology is equal to the Z-rank of A(K).

Proof: Let P_{1}, …, P_{n} ∈ A(K) be Z-linearly independent generating A_{1} ∩ A(K) over Z and assume that they become Z_{p} dependent, say ∑ α_{i}P_{i} = 0. We may assume that not all the α_{i} are divisible by p. Let a_{i} be integers p-adically close to α_{i}. Then a_{i} − α_{i} are divisible by a high power of p and so ∑(a_{i} − α_{i})P_{i} is v-adically close to zero, hence ∑ a_{i}P_{i} is close to zero which implies, by lemma 2, that the a_{i} are divisible by p, but then so are the α_{i}, giving a contradiction that proves the theorem.

Remark: Theorem 4 is valid, with the same proof, for semi-abelian varieties satisfying the same hypotheses. In particular it is valid for the multiplicative group, giving a new proof of the characteristic p analogue of Leopoldt’s conjecture already proved by Kisilevsky [Ki].

The same proof as in Lemma 2 yields the following result:

Lemma 3. Let A be an abelian variety defined over a finite field k and F be the corresponding Frobenius automorphism of A. Let K/k be a function field of characteristic p > 0 and v a place of K. Then there is a neighbourhood U of O in the v-adic topology, such that, if P ∈ A(K) ∩ U then P ∈ F (A(K)).

It is also clear that points v-adically close to O have images under F even closer to O. It follows that if R is the completion of the ring Z[F] with respect to the maximal ideal generated by F then the R-rank of the v-adic closure of A(K) in A(K_{v}) is equal to the Z[F]-rank of A(K). If R happens to be contained in Z_{p} then the Z_{p}-rank of the v-adic closure of A(K) in A(K_{v}) will be smaller than the Z-rank of A(K). This happens for example if A is an ordinary elliptic curve defined over the finite field k. Indeed, let x^{2} −ax+q be the characteristic polynomial of Frobenius (hence #A(k) = q +1−a). Since A is ordinary, a is prime to q = #k, so the characteristic polynomial has a p-adic root ϕ ≡ 0 mod p and Z[F] = Z[ϕ] ⊂ Z_{p}. Hence the analogue of Leopoldt’s conjecture fails in this case. This example also shows that Theorem 2 does not hold with the hypothesis on the p-power torsion removed.

Corollary 2. Let A be an ordinary abelian variety over a function field K of characteristic p > 0 and v a place of K and assume that A[p^{∞}] ∩ A(K_{s}) is finite. If P ∈ A(K) ∩ A_{1} and α ∈ Z_{p} is irrational then αP is transcendental over K.

Proof: Assume by contradiction that αP is algebraic and extend K, if necessary, so that it is rational. However, if α is irrational, P and αP are Z-independent, but Z_{p}- dependent, contradicting Theorem 4 and remark (b) above.

Remarks:(c) The analogue of this corollary for the multiplicative group has been proved by Mendès France and van der Poorten [MP].

(d) There is an analogue of this corollary for ordinary elliptic curves defined over a finite field. However one needs to assume that α ∉ Z[F]. (See lemma 3 and the discussion following it).

References.

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[V] J. F. Voloch, Explicit p-descent for elliptic curves in characteristic p, Compositio Math. 74 (1990) 247-258. Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA e-mail: voloch@math.utexas.edu