Weil I §6: A rationality theorem

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 06 July 2012


Let 0 be a projective space of dimension 1 on 𝔽q, X00 a nonsingular projective variety, A00 a linear subspace of codimension two, D00 the dual line, 𝔽q an algebraic closure of 𝔽q and ,X,A,D on 𝔽q obtained from 0,X0,A0,D0 by extension of scalars. The diagram [De; 5.1.1] of [De; 5.6] provides an analogue of the diagram over 𝔽q: X0 X0˜ D0 π0 f0 (W6 6.1.1)

Suppose X is connected of even dimension n+1=2m+2, and the pencil (Xt) tD of hyperplane sections of X defined by D is a Lefschetz pencil. The set S of tD such that Xt is singular is defined over 𝔽q, i.e. providing S0D0. We set U0=D0-S0 and U=D-S.

Let uU. The vanishing part of the cohomology EHn( Xu,l ) is stable under π1(U,u), thus defines over U a local system of Rnf*l. This last object is defined over 𝔽q: Rif*l is the reciprocal image of the l-sheaf Rif0*l on D0 and, on U,  is the reciprocal image of a local system 0 Rnf0*l.

The cup product is an alternating form ψ: Rnf0*lRnf0*l l(-n). Denoting the orthogonal complement to 0 relative to ψ by 0, in Rnf0*l |U0 we see that ψ induces a perfect duality ψ: 0 00 0 00 l(-n).

For every x|U0|, the polynomial det( 1- Fx*t, 0 00 ) has rational coefficients.

Let j0 be the inclusion of U0 in D0, and j that of U in D. The eigenvalues of F* acting on H1(D, j* ) are algebraic numbers such that all their complex conjugates satisfy q n+12 -12 |α| q n+12 +12 .

By [De; 5.10] and Theorem 1.1 [De; 6.2], the hypotheses of [De; 3.2] are in fact verified for ( U0, 0 00 , ψ ) for β=n, and we apply [De; 3.9].

Let 𝒢0 be a locally constant l-sheaf on U0 such that its reciprocal image 𝒢 on U is a constant sheaf. Then there exist units αi in l such that for every x|U0|, we have det( 1-Fx*t ,𝒢0 ) = i( 1-αi deg(x) t ).

This lemma expresses 𝒢0 as the reciprocal image of a sheaf on Spec(𝔽q) knowing its direct image on Spec(𝔽q). This last is identified with an l-adic representation G0 of Gal( 𝔽q /𝔽q ) and we take det(1-Ft, G0) = i (1-αit).

The Lemma 1.3 [De; 6.4] applies to Rif0*l (in), to Rnf0*l / 0 and to 00.

For x|U0|, the fibre Xx = f0-1 (x) is a variety over the finite field k(x). If x is a point of U over x, Xx is deduced from Xx by extension of scalars of k(x) to its algebraic closure k(x) = 𝔽q, and Hi( Xx, l ) is the fibre of Rif*l at x. The formula [De; 1.5.4] for the variety Xx over k(x) is thus written Z(Xx,t) = idet( 1-Fx*t , Rif0*l ) (-1) i+1 , and Z(Xx,t) is the product of Zf = det( 1-Fx*t , Rnf0*l/0 ) det( 1-Fx*t , 00 ) in det( 1-Fx*t , Rif0*l ) (-1)i+1 with Zm = det( 1-Fx*t , 0 00 ).

Putting 0 = 0 00 ,   = and applying Lemma 1.3 [De; 6.4] to the factors Zf and Zm we find that there exist l-adic units αi  (1iN) and βj  (1jM) in l such that for every x|U0| Z(Xx,t) = i( 1-αi degx t ) j( 1-βj degx t ) det( 1-Fx*t ,0 ) and, in particular, the second factor is in (t). If αi coincides with βj, it is reasonable to simultaneously remove this αi from the list of α and this βj from the list of the β. Thus we may assume, and we will assume, that αiβj for all i and all j.


It suffices to prove that the polynomials i (1-αit) and j (1-βjt) have rational coefficients, i.e. that the family of αi (resp. the family of βj) is defined over . We will deduce the following propositions.

Let (γi)  (1iP) and (δj)  (1jQ) be two families of l-adic units in l. We assume that γi δj. If K is a very large finite set of integers 1, and L is a very large subset of |U0| of density 0, then, if x|U0| satisfies kdeg(x) (for every kK) and xL, the denominator of det( 1-Fx*t , 0 ) i( 1-γi deg(x) t ) j( 1-δj deg(x) t ) , (W6 6.6.1) written in irreducible form, is j( 1-δj deg(x) t ).

The proof will be given in [De; 6.10-13]: By Lemma 2.2 [De; 6.7] below, Proposition 2.1 [De; 6.6] furnishes an intrinsic description of the family of δj in terms of the rational fractions (W6 6.6.1) for x|U0|.

Let K be a finite set of integers 1 and (δj)  (1jQ) and (εj)  (1jQ) two families of elements of a field. If, for n very large, not divisible by any of the kK the family of δjn coincides with that of the εjn (up to rearrangement), then the family of δj coincides with the family of εj (up to rearrangement).

We proceed by induction on Q. The set of integers n such that δQn = εjn is an ideal (nj). We prove that there exists j0 such that δQ = εj0. If not the nj will be different from 1, and there exist arbitrarily large integers n, not divisible by any of the nj, nor by any of the kK. We will have δQn εjn, and this will contradict the hypothesis. Thus there exists j0 such that δQ = εj0. We conclude by applying the induction hypothesis to the families (δj)  (jQ) and (εj)  (jj0).

Let (γi)  (1iP) and (δj)  (1jQ) be two families of l-adic units in l, R(t) = i (1-γit) and S(t) = j (1-δjt). We assume that for every x|U0| i ( 1-δideg(x)t ) divides i ( 1-γideg(x)t ) det(1-Fx*t,0). Then S(t) divides R(t).

Disregarding the pairs of common elements in the families (γi) and (δj) verify the hypothesis of Proposition 2.1 [De; 6.6]. Apply Proposition 2.1 [De; 6.6]. By hypothesis, the rational fractions (W6 6.6.1) are polynomials. Thus no δ remains, which means that S(t) divides R(t).

This proposition furnishes an intrinsic characterisation of R(t) in terms of the family of polynomials i ( 1-γideg(x)t ) det(1-Fx*t,0). It is the lcm of the polynomials S(t) = j (1-δjt) which satisfies the hypothesis of Proposition 2.3 [De; 6.8].


We prove [De; 6.5] and thus Theorem 1.1 [De; 6.2] (modulo Proposition 2.1 [De; 6.6]). Take (γi) = (αi) and (δj) = (βj) in Proposition 2.1 [De; 6.6]. We find an intrinsic characterization of the family of βj in terms of the family of rational fractions Z(Xx,t)  ( x|U0| ). Since these are in (t), the family of βj is defined over .

The polynomials i ( 1-αideg(x)t ) det(1-Fx*t,0) are thus in [t]. Proposition 2.3 [De; 6.8] provides an intrinsic description of the family of αi in terms of this family of polynomials. The family of αi is thus defined over .


Preliminaries: Let uU and u be the fibre of at u. The arithmetic fundamental group π1 (U0,u), extension of ^ = Gal( 𝔽q / 𝔽q ) (generator: φ) by the geometric fundamental group π1 (U,u), acts on u by symplectic similitudes: ρ: π1(U0,u) CSp(u,ψ). We denote by μ(g) the multiplier of a symplectic similitude g. Let H ^×CSp(u,ψ) be the subgroup defined by the equation q-n = μ(g) (q being an l-adic unit, qn l* is defined for all n^). The fact that ψ has values in l(-n) can be expressed by saying that the map from π1 into ^×CSp, with coordinates the canonical projection on ^ and ρ, factors ρ1: π1(U0,u) H.

The image H1 of ρ1 is open in H.

In fact, π1(U0,u) projects onto ^, and the image of π1(U,u) = Ker( π1(U0,u)^ ) in Sp(u,ψ) = Ker(H^) is open [De; 5.10].

For δl an l-adic unit, the set Z of (n,g) H1 such that δn is an eigenvalue of g is closed of measure 0.

It is clear that Z is closed. For each n^, let CSpn be the set of g CSp(u,ψ) such that μ(g) = q-n, and let Zn be the set of gCSpn such that δn is an eigenvalue of g. Then CSpn is a homogeneous space for Sp, and we check that Zn is a proper algebraic subspace, thus of measure 0. By Lemma 4.1 [De; 6.11], H1 ( {n}×Zn ) is thus of measure 0 in the inverse image of n in H1, and we apply Fubini to the projection H1^.


We prove Proposition 2.1 [De; 6.6]. For each i and j, the set of integers n such that γin=δjn is the set of multiples of a fixed integer nij (we don't exclude nij=0). By hypothesis, nij1.

By Lemma 4.2 [De; 6.12] and the density theorem of Čebotarev, the set of x|U0| such that a δjdeg(x) is an eigenvalue of Fx* acting on 0 is of density 0. We take for K the set of nij and for L the set of x as above.


[De] P. Deligne, La conjecture de Weil: I, Publications mathématiques de l'I.H.É.S., tome 43, (1974), p. 273-307.

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