Lang’s conjecture in characteristic p: an explicit bound

Alexandru Buium and José Felipe Voloch

Let K be a function field in one variable with constant field k and denote by K_{a}, K_{s} its algebraic and separable closures, respectively. Let X/K be an algebraic curve of genus at least two. The function field analogue of Mordell’s conjecture states that X(K) is finite unless X is K_{a}- isomorphic to a curve defined over k, in which case X is called isotrivial. This was first proved by Manin [Man] in characteristic zero and, shortly after, another proof was given by Grauert [Gra] and this proof was then adapted by Samuel [Sa] to positive characteristic. Since then several different proofs were given for Mordell’s conjecture over function fields. In particular Szpiro [Sz] was the first to prove an effective version of Mordel’s conjecture in characteristic p.

Mordell’s conjecture was generalized by Lang [L] (and proved through work of Raynaud [R] and Faltings [F]). An analogue of Lang’s conjecture over function fields of characteristic p was proved by the second author and Abramovich [V] [AV]. The aim of the present paper is to prove an effective version of Lang’s conjecture in characteristic p. Our approach consists of combining the approaches in [B1] and [V] [AV] which in their turn were motivated by Manin’s work [Man]. Here is our main result:

Theorem. Let K be a function field in one variable and characteristic p > 0. Let X be a smooth projective curve of genus g ≥ 2 over K embedded into its Jacobian J. Assume X has non-zero Kodaira-Spencer class (equivalently, X is not defined over K^{p}). If Γ is a subgroup of J(K_{s}) such that Γ/pΓ is finite, then:

♯(X ∩ Γ) ≤ ♯(Γ/pΓ) · p^{g} · 3^{g} · (8g − 2) · g!

We stress the fact that we do not assume Γ is finitely generated, which is the main feature in Lang’s conjecture that distinguishes it from the Mordell conjecture .

A similar result in characteristic zero was obtained by the first author [B4] but the bound there is huge as compared to the bound here. This is a reflection of the fact that the characteristic p case is in some sense “easier” than the characteristic zero case.

The question of the existence of this type of bounds for Lang’s conjecture was raised by Mazur in [Maz] and is quite different from what one understands by “effective Mordell”. In particular, even in the special case when Γ in our Theorem is finitely generated, our bound is not a consequence of Szpiro’s [Sz]. Indeed, assuming we are in the hypothesis of the Theorem above with Γ finitely generated, let K_{1} ⊂ K_{s} be the field generated over K by the coordinates of the points in Γ. What Szpiro’s “effective Mordell” yields is a bound for the height of the points in X(K_{1}) that depends on g =genus of X, p =characteristic of k, q_{1} =genus of K_{1} and s_{1} =number of points of bad reduction of a semistable model of X ⊗K_{1}/K_{1}. It follows that ♯(X ∩Γ) is bounded by a constant that depends on g, p, q_{1}, s_{1}. But of course q_{1}, s_{1} are not bounded by a constant that depends on ♯(Γ/pΓ) only; we may always keep ♯(Γ/pΓ) constant and vary Γ so that both q_{1} and s_{1} go to infinity.

In order to prove the Theorem let us start by recalling a construction from [B1]. Assume we have fixed a derivation δ = ∂/∂t of K where t ∈ K is a separable transcendence basis of K/k. Then for any K−scheme X one defines the “first jet scheme along δ” by the formula

X^{1} := Spec(S(Ω_{X/k})/I)

where I is the ideal generated by sections of the form df − δf (f ∈ O_{X}). This object was analysed in [B2], [B3] where the characteristic zero case only was considered. But many of the facts proved there extend, with identical proofs, in positive characteristic. In particular the following hold. Assume X above is a smooth variety over K. Then exactly as in [B1], p.1396, X^{1} identifies with the torsor for the tangent bundle T X := Spec(S(Ω_{X/K})) corresponding to the Kodaira Spencer class

ρ(δ) ∈ H^{1}(X, T_{X/K})

(where ρ : Der_{k}K → H^{1}(X, T_{X/K}) is the Kodaira Spencer map; this map played various roles in virtually all approaches to the Mordell and Lang conjectures over function fields.) So exactly as in [B2], section 1, we may write X^{1} as the complement of a divisor in a projective bundle:

X^{1} = P(E)\P(Ω_{X/K})

where E is the vector bundle defined by the extension

0 → O_{X} → E → Ω_{X/K} → 0

corresponding to ρ(δ) ∈ H^{1}(X, T_{X/K}) ≃ Ext^{1}(Ω_{X/K}, O_{X}).

If X/K is a smooth group scheme then so is X^{1}/K.

Also, since δ lifts to a derivation of K_{s}, there is an obvious “lifting map”

∇ : X(K_{s}) → X^{1}(K_{s})

which in case X/K is a group is a group homomorphism.

The following is the characteristic p analogue of a fact from [B3], (2.2):

Lemma. If X/K is a smooth projective curve of genus ≥ 2 with non zero Kodaira Spencer class then X^{1} is an affine surface.

Proof. By the discussion preceding the Lemma it is enough to check that the divisor P(Ω_{X/K}) is ample in P(E), equivalently that E is ample, which is the same as E_{a} being ample (where E_{a} is the pull back of E on X_{a} := X ⊗_{K} K_{a}). Let F : X_{a} → X_{a} be the absolute Frobenius (viewed as a scheme morphism over the integers). Assume E_{a} is not ample and seek a contradiction. By the characteristic p analogue of “Gieseker’s Theorem”

[Gie] due to Martin-Deschamps [MD] it follows that there exists a power F^{m} : X_{a} → X_{a} of F such that the pull back of the sequence

(∗) 0 → O_{X_{a}} → E → Ω_{X_{a}/K_{a}} → 0

via F^{m} splits. Now, since Ω_{X_{a}/K_{a}} has degree 2g − 2 > (2g − 2)/p, a result of Tango [T] Theorem 15 p. 73 implies that the sequence (*) itself must be split, which contradicts the fact that the Kodaira-Spencer class of X/K is non zero. This completes the proof of the Lemma.

Proof of the Theorem. The closed immersion X ⊂ J induces a closed immersion X^{1} ⊂ J^{1}. For any point P ∈ X(K_{s}) ∩ pJ(K_{s}) we have ∇(P) ∈ X^{1}(K_{s}) ∩ pJ^{1}(K_{s}). Since J^{1} is an extension of J by a vector group (same argument as in [B2] (2.2)) the algebraic group B = pJ^{1} coincides with the maximum abelian subvariety of J^{1} and the projection B → J is an isogeny. Moreover by [Ro], p.704, Lemma 2, the natural isogeny (the Verschiebung) J^{(p)} → J factors through B → J. Since Verschiebung is of degree p^{g}, B → J has degree at most p^{g}.

In order to prove the Theorem it is obviously enough to prove that, over K_{a}, X^{1} ∩ B is finite, of cardinality at most p^{g} · 3^{g} · (8g − 2) · g!. Finiteness follows trivially from our Lemma above: X^{1} is affine and B is complete and both are closed in J^{1} so their intersection is closed in both X^{1} and B, so X^{1} ∩ B is both affine and complete, hence it is finite over K. To estimate its cardinality we use Bézout’s theorem in Fulton’s form, along the lines of [B4] (except that here we do not need any “iteration” and we do not have to take multiplicities into account !).

Recall that X^{1} and J^{1} are Zariski locally trivial principally homogeneous spaces for the tangent bundles of X and J respectively. Let η_{X} ∈ H^{1}(X, T_{X/K}) and η_{G} ∈ H^{1}(J, T_{J/K}) be the corresponding cohomology classes defining these homogeneous spaces and let

0 → O_{X} → E_{X} → Ω_{X/K} → 0
0 → O_{J} → E_{J} → Ω_{J/K} → 0

be the extension corresponding to η_{X}, η_{J} respectively. Consider the divisors D_{X} = P(Ω_{X/K}) ⊂ P(E_{X}) and D_{J} = P(Ω_{J/K}) ⊂ P(E_{J}). Since Ω_{J/K} ≃ O_{J}^{g} we have D_{J} ≃ J × P^{g−1}. Recall also that these divisors belong to the linear systems associated to O_{P(E_{X})}(1) and O_{P(E_{J})}(1) respectively and that we have identifications X^{1} ≃ P(E_{X})\D_{X} and J^{1} ≃ P(E_{J})\D_{J}. Let α : X → J be the inclusion. We claim there is a natural restriction homomorphism α^{∗}E_{J} → E_{X} prolonging the natural homomorphism α^{∗}Ω_{J/K} → Ω_{X/K}. Indeed E_{X}, E_{J} are subsheaves of the direct image sheaves π_{X∗}O_{X^{1}} and π_{J∗}O_{J^{1}} where π_{X} : X^{1} → X and π_{J} : J^{1} → J are the natural projections. These direct image sheaves have natural filtrations induced by the G_{m}−action on the tangent bundles, and E_{X}, E_{J} identify with the first piece of this filtration. Now we have a natural map α^{∗}π_{J∗}O_{J^{1}} → π_{X∗}O_{X^{1}}. This map is compatible with the G_{m}−actions in an obvious way so it preserves filtrations; in particular it sends α^{∗}E_{X} into E_{J}. (Cf. [B3], section 1, for details in an analogous situation.) The homomorphism α^{∗}E_{J} → E_{X} is clearly surjective so it induces a closed embedding P(E_{X}) ⊂ P(E_{J}) prolonging the embedding X^{1} ⊂ J^{1}. By abuse we shall still denote by π_{X}, π_{J} the projections P(E_{X}) → X, P(E_{J}) → J.

Claim. The line bundle H := π_{J}^{∗}O_{J}(3Θ) ⊗ O_{P(E_{J})}(1) is very ample on P(E_{J}). (Here Θ is the theta divisor on J.)

To check the Claim, note first that the trace of the linear system |H| on D_{J} is very ample. Indeed

H ⊗ O_{D_{J}} = H ⊗ O_{P(Ω_{J/K})} = p_{1}^{∗}O_{J}(3Θ) ⊗ p_{2}^{∗}O_{P^{g−1}}(1)

where p_{1}, p_{2} are the two projections of D_{J} = J × P^{g−1} onto the factors. So H ⊗ O_{D_{J}} is very ample on D_{J}, cf. [Mum] p. 163. Furthermore we have an exact sequence

H^{0}(P(E_{J}), H) → H^{0}(D_{J}, H ⊗ O_{D_{J}}) → H^{1}(P(E_{J}), π_{J}^{∗}O_{J}(3Θ))

But the H^{1} above is zero (use the Leray spectral sequence and the vanishing theorem in [Mum] p.150) so the trace of |H| on D_{J} is a complete linear system and hence is very ample. In particular |H| separates points of D_{J} and “vectors tangent to D_{J}”. Since |H| has no base points outside D_{J} either, it follows that |H| is base point free on P(E_{J}). Hence |H| restricted to the fibres of π_{J} is base point free. Since any base point free linear subsystem of |O_{P^{g}}(1)| equals actually the whole of |O_{P^{g}}(1)| it follows that |H| separates points in each fibre of π_{J} and separates “vectors tangent to each fibre”. All these imply that |H| separate points and tangent vectors on the whole of P(E_{J}) and our Claim is proved.

Our last step is to compute the degrees deg_{H}P(E_{X}) and deg_{H}B of P(E_{X}) and B respectively, as subvarieties of P(E_{J}) with respect to the embedding defined by H. Note that

H ⊗ O_{P(E_{X})} = π_{X}^{∗}O_{X}(3Θ) ⊗ O_{P(E_{X})}(1)

We may compute the selfintersection

(O_{P(E_{X})}(1) · O_{P(E_{X})}(1))_{P(E_{X})} = deg Ω_{X/K} = 2g − 2

and since (Θ · X)_{J} = g we get

deg_{H}P(E_{X}) = 2g − 2 + 6g = 8g − 2

On the other hand we have

H ⊗ O_{B} ≃ π^{∗}O_{J}(3Θ)

where π : B → J is the projection which we already know has degree at most p^{g}. So we get, using (Θ^{g})_{J} = g!, that

deg_{H}B = p^{g} · 3^{g} · g!

Now Bezout’s theorem in Fulton’s form [Fu] p.148, says that the number of irreducible components in the intersection of two projective varieties of degrees d_{1}, d_{2} cannot exceed d_{1}d_{2}. In particular

♯(X^{1} ∩ B) ≤ deg_{H}P(E_{X}) · deg_{H}B ≤ (8g − 2) · p^{g} · 3^{g} · g!

and our Theorem is proved.

Acknowledgements The authors would like to thank the Institute for Advanced Study for the hospitality during the preparation of this paper and, in particular, Prof. E. Bombieri. The authors would also like to thank the NSF for financial support through grants DMS 9304580 (A. Buium) and DMS-9301157 (J. F. Voloch).

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Institute for Advanced Study, Princeton, NJ 08540

Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA e-mail: voloch@math.utexas.edu