RATIONAL POINTS ON FANO VARIETIES

HÉLÈNE ESNAULT

En souvenir d’un séminaire de Yuri I. Manin à Moscou, où il était question de points rationnels.

Let X ⊂ ℙ^{n} be a smooth projective variety defined over a field k of characteristic 0, with complement U := ℙ^{n} \ X. Let κ ∈ ℕ. Then the condition

(0.1) Hodge Type H_{c}^{m}(U) ≥ κ, for all m ≥ 0

on the Hodge type of relative de Rham cohomology, that is the largest integer a for which the Hodge filtration fulfills

F^{a}H_{c}^{m}(U) = H_{c}^{m}(U), (0.2)

is implied by the condition that H_{c}^{m}(U) is supported in codimension κ. The converse is one example of the Hodge conjecture: if (0.2) is fulfilled, then there should be a codimension ≥ κ subvariety Z ⊂ ℙ^{n} with codim_{X}X ∩ Z ≥ κ fulfilling

(0.3) H_{c}^{m}(U) = H^{m}(ℙ^{n}, X) ⟶^{restriction=0} H^{m}(ℙ^{n} \ Z, X \ Z ∩ X).

Using a decomposition of the diagonal as initiated by S. Bloch in [1], (0.3) is implied by the condition

CH_{0}(X ×_{k} K) = … = CH_{κ−1}(X ×_{k} K) = ℚ (0.4)

on the Chow groups for any algebraically closed field extension K ⊃ k. Thus

(0.4) ⇒ (0.3) ⇒ (0.1). (0.5)

The converse

(0.1) ⇒ (0.4) (0.6)

is one example of the Bloch-Beilinson conjectures asserting that Hodge theory controls the Chow groups even after base extension. (In particular, it implies the Hodge conjecture in this specific instance).

Date: June 17, 2003.

Let now X ⊂ ℙ^{n} be a projective variety over the finite field 𝔽_{q}, with complement U := ℙ^{n} \ X. Let κ ∈ ℕ. Then the congruence condition

|X(𝔽_{q})| ≡ |ℙ^{n}(𝔽_{q})| mod q^{κ} (0.7)

on the number of rational points is equivalent to the condition that the reciprocal roots and poles of the zeta function ζ(U, t) defined by

\frac{ζ′(U, t)}{ζ(U, t)} = ∑_{s≥1} |U(𝔽_{q^{s}})|t^{s−1} (0.8)

be divisible by q^{κ} as algebraic integers. If X is smooth, then the Weil conjectures proven by P. Deligne together with Grothendieck-Lefschetz trace formula imply that this condition is equivalent to the eigenvalues of the geometric Frobenius acting on ℓ-adic cohomology H_{c}^{m}(\overline{U}, ℚ_{ℓ}) being divisible by q^{κ} as algebraic integers for all m ≥ 0

(0.9) Eigenvalues Frob. on H_{c}^{m}(\overline{U}, ℚ_{ℓ}) ≅ q^{κ} · alg. int.

This is implied by the condition (0.3) in étale cohomology

H_{c}^{m}(\overline{U}, ℚ_{ℓ}) = H^{m}(\overline{ℙ^{n}}, \overline{X}, ℚ_{ℓ}) ⟶^{restriction=0} H^{m}(\overline{ℙ^{n} \ Z}, \overline{X \ Z ∩ X}, ℚ_{ℓ}). (0.10)

The converse is one example of the Tate conjectures, which asserts then that this should be equivalent to the condition that H_{c}^{m}(\overline{U}, ℚ_{ℓ}) be supported in codimension κ. Thus

(0.10) ⇒ (0.9) ⇒ (0.7). (0.11)

Again, using (a generalization of) S. Bloch’s decomposition of the diagonal, one has

(0.10) ⇐ (0.4). (0.12)

The converse

(0.7) ⇒ (0.4) (0.13)

would then be an extension of the Bloch-Beilinson conjectures saying that over finite fields, the zeta function controls the Chow groups even after base extension.

One applies this hierarchy of ideas to obtain the following theorem:

Theorem 0.1 ([4]). Let X be a smooth projective variety defined over the finite field 𝔽_{q}. Let us assume that for all algebraically closed fields K ⊃ 𝔽_{q} one has CH_{0}(X ×_{k} K) = ℚ. Then the number of 𝔽_{q}-rational points of X is equal to 1 modulo q.

One applies this to smooth Fano varieties, that is smooth geometrically irreducible varieties X over a field with ω_{X}^{−1} ample. According to Campana and Kollár-Miyaoka-Mori, they are rationally connected over any such K, thus a fortiori CH_{0}(X ×_{k} K) = ℚ.

Corollary 0.2 ([4]). If X is a smooth Fano variety over the finite field 𝔽_{q}, then the number of its 𝔽_{q}-rational points is equal to 1 modulo q.

That a smooth Fano variety over a finite field has a rational point is a positive answer to a conjecture by S. Lang and Yu. Manin. The original proof in [4] was done using rigid cohomology instead of étale cohomology, and is nicely reported on in [3], together with other applications of slope arguments. However, our proof has two weaknesses: it does not address higher κ’s nor does it address singular varieties. For higher κ’s, given (0.11) and (0.13), the problem is that one is not able to compute Chow groups. More precisely, Campana/Kollár-Miyaoka-Mori’s theorem linking negativity of ω_{X} and triviality of CH_{0} does not have a version for higher CH_{m}. For singular varieties, one no longer has the Weil conjectures at one’s disposal, thus (0.9) implies (0.7) but in general the converse is false. The theorem of Ax-Katz provides a large class of singular varieties for which (0.7) is fulfilled: let X ⊂ ℙ^{n} be defined by r equations of degrees d_{1} ≥ … ≥ d_{r} and let κ be the integral part of the rational number \frac{n−d_{1}−...−d_{r}}{d_{1}} . Then (0.7) holds true. A first step is

Theorem 0.3 ([5]). (0.9) holds true as well.

Yet the proof uses both ℓ-adic and p-adic methods, which is not too encouraging. Except again for the q congruence, that is for κ = 1, where the correct motivic cohomology is defined in [2] and a motivic statement implying (0.9) is proven.

References

[1] Bloch, S. : Lectures on algebraic cycles, Duke University Mathematics Series, IV, Durham, N. C. (1980).

[2] Bloch, S.; Esnault, H.; Levine, M.: Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces, preprint 2003, 16 pages.

[3] Chambert-Loir, A.: Points rationnels et groupes fondamentaux: applications de la cohomologie p-adique [ d’après P. Berthelot, T. Ekedahl, H. Esnault..], Séminaire Bourbaki Mars 2003.

[4] Esnault, H.: Varieties over finite fields with trivial Chow group of 0-cycles have a rational point, Invent. math. 151 (2003), 187-191.

[5] Esnault, H.; Katz, N.: Cohomological divisibility and point count divisibility, preprint 2003, 10 pages.

4 HÉLÈNE ESNAULT

Universität Essen, FB6, Mathematik, 45117 Essen, Germany

E-mail address: esnault@uni-essen.de