Weil Conjectures I (translation of La Conjecture de Weil I by Pierre Deligne)
Evgeny Goncharov


Julian Schwinger — Recollections from many decades
    Stanley Deser
    
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I attempted to write the full translation of this article to make the remarkable proof of Pierre Deligne available to a greater number of people. Overviews of the proofs can be found elsewhere. I especially recommend the notes of James Milne on Etale Cohomology that also contain a justification for the theory underlying this article and proofs of the results used by Deligne. The footnotes are mostly claims that some details appear in Milne, clarifications of some of the terminology or my personal struggles. I have also made a thorough overview of the proof together with more detailed explanations - <https://arxiv.org/abs/1807.10812>. Enjoy!


In this article I prove the Weil conjecture about the eigenvalues of Frobenius endomorphisms. The precise formulation is given in (1.6). I tried to make the demonstration as geometric and elementary as possible and included reminders: only the results of paragraphs 3, 6, 7 and 8 are original. 

In the article following this one I will give various refinements of the intermediate results and the applications, including the hard Lefschetz theorem (on the iterated cup products by the cohomology class of a hyperplane section).

The text faithfully follows from the six lectures given at Cambridge in July 1973. I thank N.Katz for allowing me to use his notes.






§ THE THEORY OF GROTHENDIECK: A COHOMOLOGICAL INTERPRETATION OF L-FUNCTIONS

(1.1) Let X be a scheme of finite type over ℤ, |X| be the set of closed points of X and for x ∈ |X| we denote by N(x) the number of elements in the residue field k(x) of X at x. The Hasse-Weil zeta function of X is 
    ζ_X(s)=∏_x ∈ |X|(1-N(x)^-s)^-1  (1.1.1)
 (this product converges absolutely for Re(s) large enough). For X=Spec(ℤ), ζ_X(s) is the Riemann zeta function. 

We will consider exclusively the case when X is a scheme over a finite field 𝔽_q. 

For x ∈ |X| we will write g_x instead of N(x). Denoting (x)=[k(x) : 𝔽_q] we have q_x=q^(x). It makes sense to introduce a new variable t=q^-s. Let 
    Z(X; t)=∏_x ∈ |X|(1-t^(x))^-1  (1.1.2);
 this product converges for |t| small enough and we have 
    ζ_X(s)=Z(X;q^-s)   (1.1.3)


(1.2) Dwork (On the rationality of the zeta function of an algebraic variety, Amer. J. Math., 82, 1960, p. 631-648) and Grothendieck ([1] and SGA5) have demonstrated that Z(X; t) is a rational function of t. 

For Grothendieck, this is a corollary of general results in l-adic cohomology (where l is a prime number not equal to the characteristic p of 𝔽_q). These provide a cohomological interpretation of the zeros and poles of Z(X; t), and a functional equation when X is proper and smooth. The methods of Dwork are p-adic. For X a non-singular hypersurface in a projective space they also provided him with a cohomological interpretation of zeros and poles, and the functional equation. They inspired the crystalline theory of Grothendieck and Berthelot, which for X proper and smooth provides a p-adic cohomological interpretation of zeros and poles, and the functional equation. Based on Washnitzer ideas, Lubkin created a variant of this theory, valid only for X proper, smooth and liftable to characteristic 0 (A p-adic proof of Weil's conjectures, Ann of Math, 87, 1968, pp. 125-255).

We will make essential use of Grothendieck's results and recall them below. 

(1.3) Let X be an algebraic variety over an algebraically closed field k of characteristic p, i.e. a separated scheme of finite type over k. We do not exclude the case p=0. For any prime number l ≠ p, Grothendieck defined l-adic cohomology groups H^i(X, ℚ_l). He also defined cohomology groups with compact support H_c^i(X, ℚ_l). For X proper the two coincide. H_c^i(X, ℚ_l) are vector spaces of finite dimension over ℚ_l, zero for i>2dim(X).

(1.4) Let X_0 be an algebraic variety over 𝔽_q, 𝔽̅_q the algebraic closure of 𝔽_q and X the algebraic variety over 𝔽̅_q obtained from X_0 by extension of scalars of 𝔽_q to 𝔽̅_q. In the language of Weil and Shimura we would express this situation by: "Let X be an algebraic variety defined over 𝔽_q". Let F: X → X be the Frobenius morphism; it sends a point with coordinates x to the point with coordinates x^q; in other words, for U_0 a Zariski open subset of X_0, defining an open subset U of X, we have F^-1(U)=U; for x ∈ H^0(U_0, 𝒪) we have F^*x=x^q. Let us identify the set |X| of closed points of X with X_0(𝔽̅_q) (all the points Hom_𝔽_q(Spec(𝔽̅_q), X_0) of X_0 with coefficients in 𝔽̅_q) and let φ∈ Gal(𝔽̅_q / 𝔽_q) be the substitution of Frobenius: φ(x)=x^q. The action of F on |X| identifies with the action of φ on X_0(𝔽̅_q). Then:

a) The set X^F of closed points of X fixed under F is identified with the set X_0(𝔽_q) ⊂ X_0(𝔽̅_q) of points of X defined over 𝔽_q. This simply expresses the fact that for x ∈𝔽̅_q we have x ∈𝔽_q ⇔ x^q=x.

b) Similarly, the set X^F^n of closed points of X fixed under the n-th iteration of F is identified with X_0(𝔽_q^n).

c) The set |X| of closed points of X is identified with the set |X|_F of orbits of F (or φ) on |X|. The degree (x) of x ∈ |X_0| is the number of elements in the corresponding orbit.

d) From b) and c) we see that 
    #X^F^n=#X_0(𝔽_q^n)=∑_(x)|n(x)   (1.4.1)
 (for x ∈ |X_0| and (x)|n, x defines (x) points with coordinates in 𝔽_q^n all conjugate over 𝔽_q).

(1.5) The morphism F is finite, in particular, proper. Therefore, it induces morphisms 
    F^*: H_c^i(X, ℚ_l) → H_c^i(X, ℚ_l).
 Grothendieck proved the formula of Lefschetz 
    #X^F=∑_i (-1)^iTr(F^*, H_c^i(X, ℚ_l));
 the right side, that is a priori an l-adic number is an integer and is equal to the left side. We should note that such a formulation is only reasonable because dF=0, even at infinity (X is not assumed to be proper); the relation dF=0 implies that fixed points of F have multiplicity one.

We have a similar formula for the iterations of F: 
    #X^F^n=X_0(𝔽_q^n)=∑_i (-1)^i Tr(F^*n, H_c^i(X, ℚ_l))   (1.5.1)


We take the logarithmic derivative of (1.1.2): 
    t d/dtlog Z(X_0, t)=td/dtZ(X_0, t)/Z(X_0, t)=∑_x ∈ |X_0| --(x)t^(x)/1-t^(x)=
 
    =∑_x ∈ |X_0|∑_n >0(x)t^n(x)(1.4.1) = ∑_n X_0(𝔽_q^n)t^n   (1.5.2)


For F an endomorphism of a vector space V we have a formal series identity 
    t d/dtlog((1-Ft, V)^-1)=∑_n > 0 Tr(F^n, V)t^n   (1.5.3)
 (check for V=1 and observe that both sides are additive in V when we take short exact sequences). By substituting (1.5.1) into (1.5.2) and applying (1.5.3) one finds 
    t d/dtlog Z(X_0, t)=∑_i (-1)^i t d/dtlog (1-F^*t, H_c^i(X, ℚ_l))^-1,
 or 
    Z(X, t)=∏_i  (1-F^*t, H_c^i(X, ℚ_l))^(-1)^i+1  (1.5.4)
 The right side is in ℚ_l(t). The formula implies that its Taylor expansion at t=0, a priori a formal series in ℚ_l [[t]] with constant coefficient one, is in ℤ[[t]] and is equal to the left side, also considered as a formal series in t. This formula is the Grothendiek's cohomological interpretation of the Z-function.

Our main result is the following:

Let X_0 be a projective nonsingular (= smooth) variety over 𝔽_q. For each i, the characteristic polynomial (1-F^*t, H^i(X, ℚ_l)) has integer coefficients independent of l (l ≠ p). The complex roots α of this polynomial (complex conjugates of the eigenvalues of F^*) are of absolute value |α|=q^i/2. 


We show that (1.6) is a consequence of the following apparently weaker statement:


For each i and each l ≠ p the eigenvalues of the Frobenius endomorphism F^* on H^i(X, ℚ_l) are algebraic numbers of absolute value |α|=q^i/2.


Proof of (1.7) ⇒(1.6): Let's look at Z(X_0, t) as a formal series with constant term 1, an element of ℤ[[t]]: Z(X_0, t)=∑_n a_n t^n. From (1.5.3), the image of Z(X_0, t) in ℚ_l[[t]] is a Taylor expansion of a rational function. This means that for N and M large enough (≥ the degrees of numerator and denominator) the Hankel determinants 
    H_k=((a_i+j+k)_0 ≤ i, j ≤ M)  (k > N)
 are zero. They vanish in ℚ_l if and only if they vanish in ℚ; Z(X_0, t) is a Taylor expansion of an element in ℚ(t). In other words, 
    Z(X_0, t) ∈ℤ[[t]] ∩ℚ_l(t) ⊂ℚ(t).


Let Z(X_0, t)=P/Q, with P, Q ∈ℤ[t] coprime and with positive constant terms. According to a lemma of Fatou, since Z(X_0, t) lies in ℤ[[t]] and has constant term 1, the constant terms of P and Q are 1[See the proof in James Milne's lectures on Etale Cohomology (Milne). Here and below - footnotes of the translator.]. Let 
    P_i (t)= (1-F^*t, H^i (X, ℚ_l)).
 (1.7) implies that P_i are coprime. The right hand side of (1.5.4) is therefore an irreducible fraction and 
    P(t)=∏_i  odd P_i(t)
 
    Q(t)=∏_i  even P_i(t).
 Let K be the subfield of the algebraic closure ℚ̅_l of ℚ_l generated over ℚ by the roots of R(t)=P(t)Q(t). The roots of P_i(t) are the roots of R(t) such that all their complex conjugates have absolute value q^-i/2. This set is stable under Gal(K / ℚ). Therefore, P_i(t) has rational coefficients. According to a lemma of Gauss (or because roots of P_i, being roots of R, are inverses of algebraic integers), it even has integer coefficients. The above description of the roots of P_i(t) is independent of l, therefore, the polynomial P_i(t) is also independent of l. 

The rest of the article is dedicated to the demonstration of (1.7). 

(1.8) The theory of Grothendieck provides cohomological interpretation not only of zeta functions but also of L-functions. The results are as follows.

(1.9) Let X be an algebraic variety over a field k. For the definition of a constructible ℚ_l-sheaf on X consult SGA 5 VI[Or Milne.]. It suffices to say that: 

a) If ℱ is a constructible ℚ_l-sheaf, there exists a finite partition of X into locally closed parts such that ℱ|X_i is locally constant.

b) Assume that X is connected and let x̅ be a geometric point of X. For ℱ locally constant, π_1(X, x̅) acts on the stalks ℱ_x̅; the map ℱ→ℱ_x̅ defines an equivalence of categories (locally constant ℚ_l-sheaves on X) → (continuous representations of π_1(X, x̅) on ℚ_l vector spaces of finite dimension). Such a representation in general does not factor through a finite quotient.

c) If k=ℂ, the constructible ℚ_l-sheaves over X are identified with the sheaves of ℚ_l vector spaces ℱ on X^an and their exists a finite partition of X into Zariski locally closed parts and for each i a local system[We will also call it a locally constant sheaf (french. constant tordu). This is an abuse of terminology. A sheaf ℳ=(ℳ_n) of ℤ_l-modules is called locally constant if each ℳ_n is locally constant. It is not, in general, locally constant in the classical sense. Similar remarks apply elsewhere to ℤ_l and ℚ_l-sheaves.] of free of finite type ℤ_l-modules ℱ_i on X_i such that 
    ℱ|X_i=ℱ_i ⊗_ℤ_lℚ_l.


We will only consider constructible ℚ_l-sheaves and call them just ℚ_l-sheaves.

(1.10) Assume that k is algebraically closed and let ℱ be a ℚ_l-sheaf on X. Grothendieck defined the l-adic cohomology groups H^i(X, ℱ) and H_c^i(X, ℱ). H_c^i(X, ℱ) are vector spaces of finite dimension over ℚ_l, zero for i > 2 (X). For k=ℂ,   H^i(X, ℱ) and H_c^i(X, ℱ) are the usual cohomology groups (resp. groups with compact support) of X^an with coefficients in ℱ. 

(1.11) Let X_0 be an algebraic variety over 𝔽_q, X the corresponding variety over 𝔽̅_q and ℱ_0 a sheaf of sets on X_0 (for the etale topology). We denote by ℱ its inverse image on X. In addition to the Frobenius isomorphism F: X → X, we have a canonical isomorphism F^*: F^* ℱ∼→ℱ. Here is a description. We regard ℱ_0 as an etale space over X_0, i.e. we identify ℱ_0 with an algebraic space [ℱ_0], equipped with an etale morphism f: [ℱ_0] → X_0 such that ℱ_0 is the sheaf of local sections of [ℱ_0]. The similar etale space [ℱ] over X is obtained from [ℱ_0] by extension of scalars. So we have a commutative diagram 

    [ℱ]rF[swap]df   [ℱ]df
    
    X rF    X

and a morphism [ℱ] → X ×_(F, X, f) [ℱ]=[F^* ℱ], that is an isomorphism because f is etale. The inverse of this isomorphism defines the isomorphism F^* ℱ∼→ℱ that we seek. 

This construction is generalized to ℚ_l-sheaves.

(1.12) Let X_0 be an algebraic variety over 𝔽_q, ℱ_0 a ℚ_l-sheaf on X_0, (X, ℱ) is obtained by extension of scalars of 𝔽_q to 𝔽̅_q, F: X → X and F^*: F^*ℱ→ℱ. Finite morphisms F and F^* define an endomorphism 
    F^*: H_c^i(X, ℱ) → H_c^i(X, F^*ℱ) → H_c^i(X, ℱ).
 For x ∈ |X|, F^* defines a morphism F_x^*:ℱ_F(x)→ℱ_x. For x ∈ X^F it is an endomorphism of ℱ_x. Grothendieck proved the formula of Lefschetz 
    ∑_x ∈ X^F Tr(F_x^*, ℱ_x)=∑_i (-1)^i Tr(F^*, H_c^i(X, ℱ)).
 A similar formula holds for the iterations of F: n-th iteration of F^* defines morphisms 
 F_x^*n: ℱ_F^n(x)→ℱ_x; for x fixed under F^n, F_x^*n is an endomorphism and 
    ∑_x ∈ X^F^n Tr(F_x^*n, ℱ_x)=∑_i (-1)^i Tr(F^*n, H_c^i(X, ℱ))   (1.12.1)


(1.13) Let x_0 ∈ |X|, Z be the orbit corresponding to F in |X| and x ∈ Z. The orbit Z has (x_0) elements (1.4). We denote by F_x_0^* the endomorphism F_x^*(x_0) of ℱ_x and let 
    (1-F_x_0^*t, ℱ_0)=(1-F_x_0^*t, ℱ_x).
 Because of the local isomorphism[The Frobenius F is a local isomorphism and hence an isomorphism on stalks.], (ℱ_x, F_x_0^*) does not depend on the choice of X. This justifies omitting x in the notation. We will use a similar notation for other functions of (ℱ_x, F_x_0^*).

(1.14) Define Z(X_0, ℱ_0, t) ∈ℚ_l[[t]] by the product 
    Z(X_0, ℱ_0, t)=∏_x ∈ |X_0|(1-F_x^*t^(x), ℱ_0)^-1  (1.14.1)
 For the constant sheaf ℚ_l we recover (1.1.2). According to (1.5.3), the logarithmic derivative of Z is 
    t d/dtlog Z(X_0, ℱ_0, t) def = t d/dtZ(X_0, ℱ_0, t)/Z(X_0, ℱ_0, t) = ∑_n ∑_x ∈ X^F^n=X_0(𝔽_q^n) Tr(F_x^*n, ℱ_0)t^n   (1.14.2)
 Substituting (1.12.1) in (1.14.2) we find by a calculation similar to (1.5) the generalization of (1.5.4) 
    Z(X_0, ℱ_0, t)=∏_i (1-F^*t, H_c^i(X, ℱ))^(-1)^i+1  (1.14.3)
 This formula is an identity in ℚ_l[[t]].

(1.15) It is sometimes convenient to use Galois language instead of the geometric one. Here is the dictionary. 

If 𝔽̅_q^1 and  𝔽̅_q^2 are two algebraic closures of 𝔽_q, (X_0, ℱ_0) over 𝔽_q defines by extension of scalars (X_1, ℱ_1) over 𝔽̅_q^1 and (X_2, ℱ_2) over 𝔽̅_q^2. All 𝔽_q-isomorphisms σ: 𝔽̅_q^1 ∼→𝔽̅_q^2 define isomorphisms 
    H_c^*(X_1, ℱ_1) ∼→ H_c^*(X_2, ℱ_2).
 In particular, for 𝔽̅_q^1 =𝔽̅_q^2 (denote by 𝔽̅_q), we find that Gal(𝔽̅_q / 𝔽_q) acts on H_c^*(X, ℱ) (action by transport of structure [This phrase is commonly used to state the principle that any isomorphism Y_1 → Y_2 extends canonically to an isomorphism of objects constructed from Y_1 and Y_2 (cohomology groups, sheaves, etc). When Y_1=Y_2, automorphisms also extend.]). Let φ∈ Gal(𝔽̅_q / 𝔽_q) be the substitution of Frobenius. We verify that 
    F^*=φ^-1  (in  End(H_c^*(X, ℱ))).
 This leads to the definition of the geometric Frobenius F ∈ Gal(𝔽̅_q / 𝔽_q) as φ^-1. We have 
    F^*=F   (1.15.1)
 Let x be a geometric point of X_0, localized to x_0 ∈ |X_0|. By transport of structure, the group Gal(k(x)/k(x_0)) acts on the stalk (ℱ_0)_x of ℱ_0 at x; in particular, we have a geometric Frobenius relative to k(x_0): F_x_0∈ Gal(k(x)/k(x_0)). For x defined by a closed point, still denoted by x, in X we have ℱ_x=(ℱ_0)_x and 
    F_x_0^* def = F_x^*(x_0)=F_x_0   (in  End(ℱ_x))   (1.15.2)
 In the Galois notation, (1.14.3) looks like 
    ∏_x ∈ |X_0|(1-F_xt^(x), ℱ_0)^-1=∏_i (1-Ft, H_c^i(X, ℱ))^(-1)^i+1.





§ THE THEORY OF GROTHENDIECK: POINCARE DUALITY

(2.1) To explain the relationship between the roots of unity and orientations I will first repeat the two classical cases in a wacky language.

a) Differentiable manifolds. - Let X be a differentiable manifold purely of dimension n. The orientation sheaf ℤ' on X is the sheaf locally isomorphic to the constant sheaf ℤ, whose invertible sections on an open U in X correspond to the orientations of U. An orientation of X is an isomorphism of ℤ' with the constant sheaf ℤ. The fundamental class of X is a morphism Tr: H_c^n(X, ℤ') →ℤ; if X is orientable, it is identified with a morphism Tr: H_c^n(X, ℤ) →ℤ. The Poincare duality is expressed using the fundamental class. 

b) Complex varieties. - Let ℂ be the closure of ℝ. A smooth complex algebraic variety or rather the underlying differentiable variety is always orientable. To justify this it suffices to orient ℂ itself. This amounts to a choice: 

a) choosing one of the two roots of the equation X^2=-1; we call it +i;

b) choosing an isomorphism from ℝ/ℤ to U^1={z ∈ℂ|  |z|=1}; +i is the image of 1/4;

c) choosing one of the two isomorphisms x →exp(± 2π ix) from ℚ/ℤ to the group of the roots of unity of ℂ, which extends continuously to an isomorphism from ℝ/ℤ to U^1.

We denote by ℤ(1) a free ℤ-module of rank one whose set of generators has two elements canonically corresponding to one of the two-element sets a), b), c). The simplest is to take ℤ(1)=Ker(exp: ℂ→ℂ^*). The generator y=± 2 π i corresponds to the isomorphism c): x →exp(xy). Let ℤ(r) be the r-th tensor power of ℤ(1). If X is a smooth complex algebraic variety purely of complex dimension r, the orientation sheaf on X is the constant sheaf of value ℤ(r). 

(2.2) To "orient" an algebraic variety over an algebraically closed k of characteristic zero, we must choose an isomorphism from ℚ/ℤ to the group of the roots of unity of k. The set of such isomorphisms is the principal homogeneous space for Ẑ^* (no longer for ℤ^*). When one is only interested in the l-adic cohomology, it suffices to consider the roots of unity of order a power of l, and to assume that the characteristic p of k differs from l. We denote by ℤ/l^n(1) the group of the roots of unity of k of order dividing l^n. For various n,  ℤ/l^n(1) form a projective system with transition maps 
    σ_m, n: ℤ/l^m(1) →ℤ/l^n(1): x → x^l^m-n.
 We let ℤ_l=lim proj ℤ/l^n(1) and ℚ_l(1)=ℤ_l(1) ⊗_ℤ_lℚ_l. Denote by ℚ_l(r) the r-th tensor power of ℚ_l(1); for r ∈ℤ negative we put ℚ_l(r)=ℚ_l(-r  ). 

As a vector space over ℚ_l, ℚ_l(1) is isomorphic to ℚ_l. However, the automorphism group of k acts non-trivially on ℚ_l(1): it acts via the character with values in ℤ_l^*, which gives its action on the roots of unity. In particular, for k=𝔽̅_q, the substitution of Frobenius φ: x → x^q acts by multiplication by q. 

Let X be an algebraic variety purely of dimension n over k. The orientation sheaf of X for the l-adic cohomology is the constant ℚ_l-sheaf ℚ_l(n). The fundamental class is a morphism 
    Tr: H_c^2n(X, ℚ_l(n)) →ℚ_l,
 or rather even 
    Tr: H_c^2n(X, ℚ_l) →ℚ_l(-n).



For X proper and smooth, purely of dimension n, the bilinear form 
    Tr(x ∪ y): H^i(X, ℚ_l) ⊗ H^2n-i(X, ℚ_l) →ℚ_l(-n)
 is a perfect paring (it identifies H^i(X, ℚ_l) with the dual of H^2n-i(X, ℚ_l(n))).


(2.4) Let X_0 be a proper and smooth algebraic variety over 𝔽_q, purely of dimension n and we obtain X over 𝔽̅_q from X_0 by extension of scalars. The morphism (2.3) is compatible with the action of Gal(𝔽̅_q/ 𝔽_q). If (α_j) are the eigenvalues of the geometric Frobenius acting on H^i(X, ℚ_l), the eigenvalues of F acting on H^2n-i(X, ℚ) are (q^n α_j^-1).

(2.5) Assume for simplicity that X is connected. The proof of (2.4) goes as follows, once we transpose to the geometric language instead of the Galois one (see (1.15)).

a) The cup-product puts H^i(X, ℚ_l) and H^2n-i(X, ℚ_l) into perfect duality with values in H^2n(X, ℚ_l) that has dimension one.

b) The cup product commutes with the inverse image of F^* by the Frobenius morphism 
 F: X → X.  

c) The morphism F is finite of degree q^n: on H^2n(X, ℚ_l) F^* is multiplication by q^n. 

d) Therefore, the eigenvalues of F^* satisfy the property (2.4).

(2.6) We let χ(X)=∑_i (-1)^i  H^i(X, ℚ_l). For n odd, the form Tr(x ∪ y) on H^n(X, ℚ_l) is skew-symmetric; the integer n χ(X) is always even. It is easy to deduce from (1.5.4) and (2.3), (2.4) that 
    Z(X_0, t)=ε q^-n χ(X)/2t^-χ(X)Z(X_0, q^-nt^-n)
 for ε=± 1. If n is even, let N denote the multiplicity of the eigenvalue q^n/2 of F^* acting on H^n(X, ℚ_l) (i.e the dimension of the corresponding invariant subspace). We have 
    ε= 1,     n
     (-1)^N,     n
 This is the Grothendieck's formulation of the functional equation for Z-functions. 

(2.7) We will need other forms of the duality theorem. The case of curves will be enough for our purposes. If ℱ is a ℚ_l-sheaf on an algebraic variety X over an algebraically closed k, we denote by ℱ(r) the sheaf ℱ⊗ℚ_l(r). This sheaf is (not canonically) isomorphic to ℱ.


Let X be smooth purely of dimension n and ℱ be a locally constant sheaf. We denote by ℱ the dual of ℱ. The bilinear form 
    Tr(x ∪ y): H^i(X, ℱ) ⊗ H_c^2n-i(X,  ℱ(n)) → H_c^2n(X, ℱ⊗ ℱ(n)) → H_c^2n(X, ℚ_l(n)) →ℚ_l
 is a perfect pairing. 


(2.9) Assume that X is connected and that x is a closed point of X. The functor ℱ→ℱ_x is an equivalence of the category of locally constant ℚ_l-sheaves with that of l-adic representations of π_1(X, x). Via this equivalence, H^0(X, ℱ) is identified with the invariants of π_1(X, x) acting on ℱ_x: 
    H^0(X, ℱ) ∼→ℱ_x^π_1(X, x).
 According to (2.8), for X smooth and connected of dimension n we have 
    H_c^2n(X, ℱ)=H^0(X,  ℱ(n)  )=(( ℱ_x(n))^π_1(X, x) ).
 The duality exchanges invariants (the largest invariant subspaces) with coinvariants (the largest invariant quotients)[I haven't seen the terms "invariant" and "coinvariant" used in this setting. For a group G acting on M the "invariants" (resp. "coinvariants") are denoted M^G (resp M_G).]. The formula takes form 
    H_c^2n(X, ℱ)=(ℱ_x)_π_1(X, x)(-n).
 We will only use it for n=1. 


Let X be a connected smooth curve over an algebraically closed field k, x a closed point of X and ℱ a locally constant ℚ_l-sheaf. We have 

(i)H_c^0(X, ℱ)=0 if X is affine.

(ii)H_c^2(X, ℱ)=(ℱ_x)_π_1(X, x)(-1).
 Assertion (i) simply states that ℱ does not have sections with finite support.

(2.11) Let X be a connected smooth projective curve over an algebraically closed k, U an open set in X, the complement of the finite set S of closed points of X, j the inclusion U X and ℱ a locally constant ℚ_l-sheaf on U. Let j_*ℱ be the constructible ℚ_l-sheaf - the direct image of ℱ. Its stalk at x ∈ S has rank less or equal to the rank of the stalk at a general point; it is the space of invariants of the local monodromy group. 


The bilinear form 
    Tr(x ∪ y): H^i(X, j_*ℱ) ⊗ H^2-i(X, j_* ℱ(1)) → H^2(X, j_*ℱ⊗ j_* ℱ(1)) →
 
    → H^2(X, j_*(ℱ⊗ ℱ)(1)) → H_c^2(X, ℚ_l(1)) →ℚ_l
 is a perfect pairing.


(2.13) It will be convenient to have ℚ_l-sheaves ℚ_l(r) on any scheme X where l is invertible. The point is to define ℤ/l^n(1). By definition, ℤ/l^n(1) is the etale sheaf of the l^n-th roots of unity. 



(2.14) Bibliographical notes on paragraphs 1 and 2. 

A) All the important results in etale cohomology are first proved for torsion sheaves. The extension to ℚ_l-sheaves is done by passing to formal limits. In what follows, for each theorem mentioned I will not refer to the reference where it is proved, but to the reference where a similar statement for a torsion sheaf is proved. 

B) With the exception of the Lefschetz formula and (2.12), results in etale cohomology used in this article are all proved in SGA 4. For those already stated, the references are: definition of H^i: VII; definition of H_c^I: XVII 5.1; finiteness theorem: XIV 1, completed in XVII 5.3; cohomological dimension: X; Poincare duality: XVIII.

C) The relation between the various Frobeniuses ((1.4), (1.11), (1.15)) is explained in detail in SGA 5, XV, par. 1, 2.

D) The cohomological interpretation of the Z-functions is clearly exposed in [1]; however, Lefschetz formula (1.12) for X a smooth projective curve is used, but not proved. For the proof, one has to consult SGA 5. 

E) The form (2.12) of the Poincare duality theorem follows from the general result SGA 4, XVIII (3.2.5) (for S=Spec(k),  X=X,  K=j_*ℱ,  L=ℚ_l) by a local calculation that is not difficult. The statement will be explicitly included in the final version of SGA 5. For the case interesting to us (tame ramification of ℱ), we could obtain it transcendentially[Transcendential algebraic geometry deals with varieties defined over ℂ and concentrates on their structure of holomorphic manifolds, that allows one to use powerful techniques of topology, analysis, differential equations, etc.] by lifting X and ℱ to characteristic 0. 



§ THE MAIN LEMMA (LA MAJORATION FONDAMENTALE)

The result of this paragraph was catalyzed by reading the lecture of Rankin [3][For a slightly different exposition of the main lemma using the equivalence of (1.9) b) consult Milne.]. 

(3.1) Let U_0 be a curve on 𝔽_q, complement in ℙ^1 to a finite set of closed points, U be the curve over 𝔽̅_q corresponding to it, u a closed point of U, ℱ_0 a locally constant sheaf on U_0 and ℱ its inverse image on U. 

Let β∈ℚ. We say that ℱ_0 is of weight β if for all x ∈ |U_0|, the eigenvalues of F_x acting on ℱ_0 (1.13) are algebraic numbers all of which complex conjugates are of absolute value q_x^β/2. For example, ℚ_l(r) is of weight -2r.


Let's make the following hypotheses:

(i) ℱ_0 is equipped with a bilinear skew-symmetric nondegenerate form 
    ψ: ℱ_0 ⊗ℱ_0 →ℚ_l(-β)   (β∈ℤ).


(ii) The image of π_1(U, u) in GL(ℱ_u) is an open subgroup of the symplectic group Sp(ℱ_u, ψ_u).

(iii) For all x ∈ |U_0|, the polynomial (1-F_x t, ℱ_0) has rational coefficients. 

Then ℱ is of weight β.


We may and do assume that U is affine and that ℱ≠ 0. 


Let 2k be an even integer and denote by 2k⊗ℱ_0 the 2k-th tensor power of ℱ_0. For x ∈ |U_0| the logarithmic derivative 
    t d/dtlog((1-F_x t^(x), 2k⊗ℱ_0)^-1)
 is a formal series with positive rational coefficients.  


Hypothesis (iii) ensures that for all n  Tr(F_x^n, ℱ_0) ∈ℚ. The number 
    Tr(F_x^n, 2k⊗ℱ_0)=Tr(F_x^n, ℱ_0)^2k
 is a positive rational and we apply (1.5.3). 


The local factors (1-F_x t^(x), 2k⊗ℱ_0)^-1 are formal series with positive rational coefficients.


The formal series log((1-F_x t^(x), 2k⊗ℱ_0)^-1) has constant term zero, from (3.3) all the coefficients are ≥ 0; the coefficients of the exponentiation are therefore also positive.


Let f_i=∑_n a_i, nt^n be a sequence of formal series with positive real coefficients. We assume that the order of f_i-1 tends to infinity with i; and we denote f=∏_i f_i. Then the radius of convergence of f_i is greater or equal to that of f. 


If f=∑_n a_n t^n, we have a_i, n≤ a_n.


Under the assumptions of (3.5), if f and the f_i are Taylor expansions of meromorphic functions, then 
    inf { |z|  | f(z)=∞}≤ inf { |z|  | f_i(z)=∞}



Indeed, those numbers are the radii of convergence. 

(3.7) For each partition P of [1, 2k] [of {1, ⋯, 2k}] into the two element sets { i_α, j_α} (i_α≤ j_α), we define 
    ψ_P: 2k⊗ℱ_0 →ℚ_l(-kβ): x_1 ⊗⋯⊗ x_2k→∏_αψ(x_i_α, x_j_α).
 Let x be a closed point of X [Let u be a closed point of U.]. Hypothesis (ii) ensures that the coivariants of π_1(U, u) on 2k⊗ℱ_u are the coinvariants on 2k⊗ℱ_u of the entire symplectic group (π_1 is Zariski-dense in Sp). Let 𝒫 be the set of partitions P. From H.Weil (The classical groups, Princeton University Press, chap. VI, par. 1), for an appropriate 𝒫' ⊂𝒫, depending on (ℱ_u), ψ_P (for P ∈𝒫') defines an isomorphism 
    (2k⊗ℱ_u)_π_1=(2k⊗ℱ_u)_Sp∼→ℚ_l(-kβ)^𝒫'.
 Let N be the number of elements in 𝒫'. According to (2.10) the formula above gives 
    H_c^2(U, 2k⊗ℱ) ≃ℚ_l(-kβ-1)^N.
 Since H_c^0(U, 2k⊗ℱ)=0 [Again, see (2.10).], the formula (1.14.3) reduces to 
    Z(U_0, 2k⊗ℱ_0, t)=(1-F^*t, H^1(U, 2k⊗ℱ))/(1-q^kβ+1t)^N.
 This Z-function is therefore the Taylor series expansion of a rational function having only one pole at t=1/q^kβ+1. We will only use the fact that the poles are of modulus t=1/q^kβ+1 in ℂ. This could be concluded from the general arguments about reductive groups. If α is an eigenvalue of F_x on ℱ_0, then α^2k is an eigenvalue of F_x on 2k⊗ℱ_0. We now let α be any complex conjugate of [the original] α. The inverse power 1/α^2k/(x) is a pole of (1-F_x t^(x), 2k⊗ℱ)^-1. According to (3.4) and (3.6) we have 
    |1/q^kβ+1| ≤ |1/α^2k/(x)|,
 or 
    |α| ≤ q_x^β/2+1/2k.
 Letting k go to infinity we find that 
    |α| ≤ q_x^β/2.
 On the other hand, the existence of ψ ensures that q_x^βα^-1 is an eigenvalue, so we have the inequality 
    |q_x^βα^-1| ≤ q_x^β/2,
 or 
    q_x^β/2≤ |α|.
 This completes the proof.


Let α be an eigenvalue of F^* acting on H_c^1(U, ℱ). Then α is an algebraic number all of which complex conjugates satisfy 
    |α| ≤ q^β+1/2+1/2.




The formula (1.14.3) for ℱ_0 reduces to 
    Z(U_0, ℱ, t)=(1-F^*t, H_c^1(U, ℱ)).
 The left hand side is a formal series with rational coefficients, given its representation as a product and hypothesis (iii). The right hand side is therefore a polynomial with rational coefficients, 1/α is its root. This already implies that α is algebraic. To complete the proof, it suffices to show that the infinite product that defines Z(U_0, ℱ_0, t) converges absolutely (thus it is nonzero) for |t| <  q^-β/2-1. 

Let N be the rank of ℱ and let 
    (1-F_xt, ℱ)=∏_i=1^N (1-α_i, xt).
 According to (3.2), |α_i, x|=q_x^β/2. The convergence of the infinite product for Z follows from that of the series 
    ∑_i, x |α_i, x t^(x)|.
 For |t|=q^-β/2-1-ε (ε >0) we have 
    ∑_i, x |α_i, x t^(x)|=N∑_x q_x^-1-ε.
 On the affine line there are q^n points with coordinate in 𝔽_q^n, so there are at most q^n closed points of degree n. So we have 
    ∑_x q_x^-1-ε≤∑_n q^nq^n(-1-ε)=∑_n q^-nε < ∞,
 which completes the proof.


Let j_0 be the inclusion of U_0 in ℙ_𝔽_q^1, j that of U into ℙ^1 and α an eigenvalue of F^* acting on H^1(𝒫^1, j_*ℱ). Then α is an algebraic number all of which complex conjugates satisfy 
    q^β+1/2-1/2≤ |α| ≤ q^β+1/2+1/2.



The segment of the long exact sequence in cohomology defined by the short exact sequence  
    0 → j_! ℱ→ j_*ℱ→ j_*ℱ/j_! ℱ→ 0
 (j_! is the extension by 0) is 
    H_c^1(U, ℱ) → H^1( ℙ, j_* ℱ) → 0.
 So the eigenvalue α already appears in H_c^1(U, ℱ) [We will repeatedly use the following facts: 

A) If H^i(X') → H^i(X) is a surjection, then (1.7) for H^i(X') ⇒ (1.7) for H^i(X).  

B) If H^i(X)  H^i(X”) is an embedding, then (1.7) for H^i(X”) ⇒ (1.7) for H^i(X).] and so by (3.8) we have: 
    |α| ≤ q^β+1/2+1/2.
 The Poincare duality (2.12) implies that q^β+1α^-1 is an eigenvalue, so we have the inequality 
    |q^β+1α^-1| ≤ q^β+1/2+1/2
 and the corollary is proved.



§ LEFSCHETZ THEORY: LOCAL THEORY


(4.1) On ℂ Lefshietz local results are as follows. Let D={z|  |z| < 1} be the unit disk, D^*=D-{0} and f:X → D be a morphism of analytic spaces. We assume that

a) X is nonsingular and purely of dimension n+1;

b) f is proper;

c) f is smooth outside the point x of the special fiber X_0=f^-1(0);

d) At x f has a nondegenerate double point.

Let t ≠ 0 in D and X_t=f^-1(t) "the" general fiber. To the previous data we associate:

α) Specialization morphisms sp: H^i(X_0, ℤ) → H^i(X_t, ℤ): X_0 is a deformation retract of X and sp is the composition arrow 
    H^i(X_0, ℤ) ∼← H^i(X, ℤ)→ H^i(X_t, ℤ)


β) The monodromy transformations T: H^i(X_t, ℤ) → H^i(X_t, ℤ), which describe the effect on the singular cycles of X_t of "rotating t around 0". This is even an action on H^i(X_t, ℤ), the stalk at t of the local system R^i f_* ℤ|D^*, of the positive generator of π_1(D^*, t). 

Lefschetz theory describes α) and β) in terms of the vanishing cycle[We can define δ to be the unique (up to sign) generator of H^n(X_0, ℤ)^⊥⊂ H^n (X_t, ℤ) (under the pairing induced by Poincare duality).] δ∈ H^n (X_t, ℤ). This cycle is well-defined up to sign. For i ≠ n,  n+1 we have 
    H^i(X_0, ℤ) ∼→ H^i(X_t, ℤ)  (i ≠ n,  n+1).
  For i=n,  n+1 we have an exact sequence 
    0 → H^n(X_0, ℤ) → H^n(X_t, ℤ) x → (x, δ)→ℤ→ H^n+1(X_0, ℤ) → H^n+1(X_t, ℤ) → 0.
 For i ≠ n, the monodromy T is the identity. For i=n we have 
    Tx=x ± (x, δ)δ.
 
The values of ±, Tδ and (δ, δ) are as follows: 



 n  4     0     1     2     3 

 Tx=x±(x, δ)δ     -     -     +     + 
  
(δ, δ)     2     0     -2     0 
 Tδ     -δ     δ     -δ     δ    



The monodromy transformation preserves the intersection form Tr(x ∪ y) on H^n(X_t, ℤ). For n odd, it is the symplectic transvection. For n even, it is the symmetric orthogonal. 

(4.2) There is an analog of (4.1) in abstract algebraic geometry. The disk D is replaced by the spectrum of a henselian discrete valuation ring A with an algebraically closed residue field. Let S be the spectrum, η its generic point (spectrum of the field of fractions of A), s the closed point (spectrum of the residue field). The role of t is played by the geometric generic point η̅ (spectrum of the closure of the field of fractions of A). 

Let f: X → S be a proper morphism, with X regular purely of dimension n+1. We assume that f is smooth except for an ordinary double point x of the special fiber X_s. Let l be a prime number different from the residual characteristic[Characteristic of the residue field of A.]  p of S. Denoting by X_η̅ the generic geometric fiber, we have the specialization morphism 
    sp: H^i(X_s, ℚ_l) ∼← H^i(X, ℚ_l) → H^i(X_η̅, ℚ_l)   (4.2.1)
 The role of T is played by the action of the inertia group I=Gal(η̅/ η) on H^i(X_η̅, ℚ_l) by transport of structure (see (1.15)): 
    I=Gal(η̅/η) → GL(H^i(X_η̅, ℚ_l))   (4.2.2)
 The data (4.2.1), (4.2.2) fully determines the sheaf R^i f_*ℚ_l on S. 

(4.3) Let n=2m for n even and n=2m+1 for n odd. (4.2.1) and (4.2.2) can still be described in terms of the vanishing cycle 
    δ∈ H^n(X_η̅, ℚ_l)(m)   (4.3.1)
 This cycle is well-defined up to sign. 

For i ≠ n,  n+1 we have 
    H^i(X_s, ℚ_l) ∼→ H^i(X_η̅, ℚ_l)  (i ≠ n,  n+1)   (4.3.2)
  For i=n,  n+1 we have an exact sequence 
    0 → H^n(X_s, ℚ_l) → H^n(X_η̅, ℚ_l) x → Tr(x ∪δ)→ℚ_l(m-n) → H^n+1(X_s, ℚ_l) → H^n+1(X_η̅, ℚ_l) → 0   (4.3.3)


The action (4.2.2) (local monodromy) is trivial for i ≠ n. For i=n, it is described as follows.

A) n odd. - We have a canonical homomorphism 
    t_l: I →ℤ_l(1),
 and the action of σ∈ I is 
    x → x± t_l(σ)(x, δ)δ.


B) n even. - We will not need this case. Let's just say that if p ≠ 2, there exists a unique character of order two 
    ε: I →{±1},
 and we have 



 σ x=x     if     ε(σ)=1 
 σ x=x±(x, δ)δ     if     ε(σ)=-1    



The signs ± in A) and B) are the same as in (4.1).

(4.4) These results imply the following information[Derivation of some of the results can be found in Milne.] about R^if_*ℚ_l.

a) If δ≠ 0:

    1) For i ≠ n the sheaf R^if_*ℚ_l is constant.

    2) Let j be the inclusion of η in S. We have 
    R^if_*ℚ_l=j_* j^* R^if_*ℚ_l.


b) If δ=0: (This is an exceptional case. Since (δ, δ)=± 2 for n even, it can only happen for n odd.)

    1) For i ≠ n+1 the sheaf R^if_*ℚ_l is constant.

    2) Let ℚ_l(m-n)_s be the sheaf ℚ_l(m-n) on {s}, extended by zero on S. Then we have an exact sequence 
    0 →ℚ_l(m-n)_s → R^n+1f_* ℚ_l → j_* j^*R^n+1f_* ℚ_l → 0,
 where j_* j^*R^n+1f_* ℚ_l is a constant sheaf. 



§ LEFSCHETZ THEORY: GLOBAL THEORY

(5.1) On ℂ the results of Lefshietz are as follows. Let ℙ be a projective space of dimension ≥ 1 and ℙ the dual projective space; its points parameterize the hyperplanes of ℙ and we denote by H_t the hyperplane defined by t ∈ ℙ. If A is  a linear subspace of codimension 2 in ℙ, the hyperplanes containing A are parameterized by points of the line D ⊂ ℙ, the dual of A. These hyperplanes (H_t)_t ∈ D form the pencil with axis A. 

Let X ⊂ℙ be a connected nonsingular projective variety of dimension n+1. Let X̃⊂ X × D be the set of pairs (x, t) such that x ∈ H_t. Projections to the first and second coordinates form a diagram[In the original paper the letter f is absent.] 
    X [r, leftarrow, "π"]
       X̃[d, "f"]
        D
      (5.1.1)
 The fiber of f at t ∈ D is the hyperplane section X_t=X ∩ H_t of X.

Fix X and take a general enough A. Then:

A) A is transverse to X and X̃ is the blowing up of X along A ∩ X. In particular, X̃ is nonsingular.

B) There is a finite subset S of D and for each s ∈ S a point x_s ∈ X_s such that f is smooth outside x_s.

C) The x_s are critical nondegenerate points of f. 

Therefore, for each s ∈ S local Lefschetz theory (4.1) applies to a small disk D_s around s and f^-1(D_s). 

(5.2) Let U=D-S. Let u ∈ U and choose disjoint loops (γ_s)_s ∈ S starting from u, with γ_s turning once around s: 


These loops generate the fundamental group π_1(U, u). This group acts on H^i(X_u, ℤ), the stalk at u of the local system R^if_*ℤ| U. According to the local theory (4.1), to each s ∈ S corresponds a vanishing cycle δ_s ∈ H^n(X_u, ℤ); these cycles depend on the choice of γ_s. For i ≠ n, the action of π_1(U, u) on H^i(X_u, ℤ) is trivial. For i=n we have 
    γ_s x=x ± (x, δ_s)δ_s   (5.2.1)


Let E be the subspace of H^n(X_u, ℚ) generated by the δ_s (vanishing part of the cohomology). 


E is stable under the action of the monodromy group π_1(U, u). The orthogonal E^⊥ of E (for the intersection form Tr(x ∪ y)) is the space of the invariants of the monodromy in H^n(X_u, ℚ). 

The γ_s generate the monodromy group, so this is clear from (5.2.1).


The vanishing cycles ±δ_s are conjugate (up to sign)[That is, given s,  s' ∈ S, there exists a σ∈π_1 such that σδ_s=±δ_s.] under the action of π_1(U, u). 


Let X ⊂ ℙ be the dual variety of X; it is the set of t ∈ ℙ such that H_t is tangent to X, i.e. such that X_t is singular or X ⊂ H_t. The variety X is irreducible. Let Y ⊂ X × ℙ be the space of pairs (x, t) such that x ∈ H_t. We have a diagram 
    X [r, leftarrow]
        Y [d, "g"]
        ℙ
 The fiber of g at t ∈ ℙ is the hyperplane section X_t= X ∩ H_t of X and g is smooth on the complement of the inverse image of X. 

We retrieve the situation of (5.1) by replacing ℙ by the line D ⊂ ℙ and Y by g^-1(D). We have S=D ∩ X. According to a theorem of Lefschetz, for D general enough, the map 
    π_1(D-S, u) →π_1( ℙ- X, u)
 is surjective. It suffices to show that ±δ_s are conjugate under π_1( ℙ-  X). 

For x in the smooth locus of codimension 1 of X, let ch be the path from t to x in ℙ-  X and γ_x the loop that follows ch until the neighborhood of X, turns once around X and then returns to t by ch. The loops γ_x (for various ch) are mutually conjugate. Since X is irreducible, two points in the smooth locus of X can always be joined, in X, by a path that does not leave the smooth locus. It follows that the conjugation class of γ_x does not depend on x. In particular, γ_s are mutually conjugate. We see from (5.2.1) that this implies the conjugacy of ±δ_s. 


The action of π_1(U, u) on E/(E ∩ E^⊥) is absolutely irreducible[The action on the k-vector space V is called absolutely irreducible if the corresponding action on V ⊗_k k̅ is irreducible.].


Let F ⊂ E ⊗ℂ be the subspace stable under the monodromy. If F ⊄(E ∩ E^⊥) ⊗ℂ, there exists an x ∈ F and s ∈ S such that (x, δ_s) ≠ 0. We then have 
    γ_s x-x=± (x, δ_s)δ_s ∈ F
 and δ_s ∈ F. According to (5.4), all the δ_s are then in F and F=E. This proves (5.5). 

(5.6) These results transpose as follows into abstract algebraic geometry. 
Let ℙ be a projective space of dimension >1 over an algebraically closed field k of characteristic p and X ⊂ℙ a connected projective nonsingular variety of dimension n+1. For A a linear subspace of codimension 2 we define D, the pencil (H_t)_t ∈ D, X̃ and the diagram (5.1.1) as in (5.1). We say that (H_t)_t ∈ D form a Lefschetz pencil of hyperplane sections if the following conditions are satisfied:

A) The axis A is transverse to X. X̃ is obtained by blowing up X along A ∩ X and is smooth. 

B) There is a finite subset S of D and for each s ∈ S a point x_s ∈ X_s such that f is smooth outside x_s.

C) The x_s are ordinary double singular points of f. 

For each s ∈ S the local Lefschetz theory of par. 4 applies to the spectrum D_s of the henselization of the local ring of D at s and to X̃_D_s=X̃×_D D_s. 

(5.7) Let N be the dimension of ℙ, r an integer ≥ 1 and ι_(r) the embedding of ℙ into the projective space of dimension N+r  N -1, the homogeneous coordinates of which are monomials of degree r in the homogeneous coordinates of ℙ [Deligne describes the Veronese mapping.]. The hyperplane sections of ι_(r)(ℙ) are the hypersurfaces of degree r of ℙ.

For p ≠ 0 it might happen that there is no such pencil of hyperplane sections of X that is Lefschetz. However, if r ≥ 2 and we replace the projective embedding ι_1: X ℙ by ι_r=ι_(r)∘ι_1, then, in this new embedding, any general enough pencil of hypersurface sections of degree r on X is still Lefschetz.

(5.8) For the rest of this discussion, we are studying the Lefschetz pencil of hyperplane sections of X, excluding the case p=2, n even. The case of n odd will suffice for our purposes. We put U=D-S. Take u ∈ U and l a prime number ≠ p. The local results of par. 4 show that R^n f_* ℚ_l is tamely ramified at each s ∈ S. The tame fundamental group of U is a quotient of the profinite completion of the corresponding transcendential[That is, taken over ℂ, see footnote 7.] fundamental groups (lifting to characteristic 0 of the tame coverings and the Riemann existence theorem). The algebraic situation is therefore similar to the transcendential situation and the transfer of Lefschetz's results is done by standard arguments. In the proof of (5.4) the theorem of Lefschetz for π_1 is replaced by a theorem of Bertiny and we have to invoke Abhyankar's lemma to control the ramification of R^*g_*ℚ_l along the smooth locus of codimension one in X [See the sketch of this proof in Milne.]. 

The results are as follows[Again, proofs of some of the results appear in Milne.]:

a) If the vanishing cycles are nonzero: 

    1) For i ≠ n the sheaf R^if_*ℚ_l is constant.

    2) Let j be the inclusion of U in D. We have 
    R^nf_*ℚ_l=j_* j^* R^nf_*ℚ_l.


    3) Let E ⊂ H^n(X_u, ℚ_l) be the subspace of the cohomology generated by the vanishing cycles. This subspace is stable under π_1(U, u) and 
    E^⊥=H^n(X_u, ℚ_l)^π(U, u).
 The representation of π(U, u) on E/(E ∩ E^⊥) is absolutely irreducible and the image of π_1 in GL(E/(E ∩ E^⊥)) is generated (topologically[They generate a dense subgroup.]) by the x → x±(x, δ_s)δ_s  (s ∈ S) (the ± sign is determined as in (4.1)). 

b) If the vanishing cycles are zero: (This is an exceptional case. Since (δ, δ)=± 2 for n even, it can only happen for n odd: n=2m+1. Note that if one vanishing cycle is zero, they all are because of conjugacy.)

    1) For i ≠ n+1 the sheaf R^if_*ℚ_l is constant.

    2) We have an exact sequence 
    0 →s ∈ S⊕ℚ_l(m-n)_s → R^n+1f_*ℚ_l →ℱ→ 0
 with ℱ constant. 

    3) E=0.

(5.9) The subspace E ∩ E^⊥ of E is the kernel of the restriction to E of the intersection form Tr(x ∪ y). Therefore, this form induces a bilinear nondegenerate form 
    ψ: E/(E ∩ E^⊥) ⊗ E/ (E ∩ E^⊥) →ℚ_l(-n),
 skew-symmetric for n odd and symmetric for n even. This form is preserved by the monodromy; for n odd, therefore, the monodromy representation induces 
    ρ: π_1(U, u) → Sp(E/(E ∩ E^⊥), ψ).



[See a somewhat more detailed exposition of this part in Milne.] The image of ρ is open. 


The image of ρ  is a compact, therefore, analytic l-adic[A Lie group over ℚ_l.] subgroup of Sp(E/(E ∩ E^⊥), ψ). It suffices to show that its Lie algebra 𝔏 equals 𝔰𝔭(E/(E ∩ E^⊥), ψ). The transcendential analog of this Lie algebra is the Lie algebra of the Zariski closure of the monodromy group. 

We deduce from (5.8) that 𝔏 is generated by transformations with zero square 
    N_s: x → (x, δ_s)δ_s   (s ∈ S)
 and that E/(E ∩ E^⊥) is an absolutely irreducible representation of 𝔏. The theorem comes from the following lemma. 


Let V be a finite dimensional vector space over the field k of characteristic 0, ψ a nondegenerate skew-symmetric form on a Lie subalgebra 𝔏 of the Lie algebra 𝔰𝔭(V, η). We assume that: 

    (i) V is a simple representation of 𝔏. 

    (ii) 𝔏 is generated by the family of endomorphisms of V of the form x →ψ(x, δ)δ. 

Then 𝔏=𝔰𝔭(V, ψ).

We may and do assume that V, and thus 𝔏 are nonzero. Let W ⊂ V be the set of δ∈ V such that N(δ): x →ψ(x, δ)δ is in 𝔏. 

a) W is stable under homotheties (since 𝔏 is a vector subspace of 𝔤𝔩(V)). 

b) If δ∈ W, exp(λ N(δ)) is an automorphism of (V, ψ, 𝔏), therefore, transforms W to itself. If δ',  δ”∈ W, we have exp(λ N(δ'))δ”=δ”+λψ(δ”, δ')δ' ∈ W [We use the nilpotence of N(δ) first to define exp(λ N(δ)) and then to obtain the formula.]; if ψ(δ', δ”) ≠ 0, then the vector subspace spanned by δ' and δ” lies in W. 

c) It follows that W is the union of its maximal linear subspaces W_α and that those are pairwise orthogonal.  Each W_α is therefore stable under the N(δ)   (δ∈ W), so it is stable under 𝔏. By hypothesis (i), W_α=V and 𝔏 contains all N(δ) for δ∈ V.  We conclude by noting that Lie algebra 𝔰𝔭(V, ψ) is generated by the N(δ)  (δ∈ V). 

Remark (5.12) (not necessary for the exposition). - It is now easy to prove (1.6) for a hypersurface of odd dimension n in ℙ_𝔽_q^n+1. 

Let X_0 be such a hypersurface and X̅_0 the hypersurface over 𝔽̅_q, which is obtained by extension of scalars. We have 
    H^i(X̅_0, ℚ_l)=ℚ_l(-i)   (0 ≤ i ≤ n);
 H^i(X̅_0, ℚ_l(i)) is generated by the i-th cup power of η, the cohomology class c_1(𝒪(1)) of a hyperplane section. Therefore, we have 
    Z(X_0, t)=(1-F^*t, H^n(X̅_0, ℚ_l)/∏_i=0^n(1-q^it)
 and (1-F^*t, H^n(X̅_0, ℚ_l) is a polynomial with integer coefficients independent of l. 

Let's vary X_0 within the Lefschetz pencil of hypersurfaces that is defined over 𝔽_q (see (5.7) for X=ℙ^n+1; the existence of such a pencil is not clear; if we wanted to complete the argument sketched here, we would have to use the arguments that will be given in (7.1). One verifies that E coincides here with the whole H^n and (3.2) provides the Weil conjecture for all the hypersurfaces of the pencil, in particular for X_0.

(5.13) Bibliographical notes on paragraphs 3 and 4.

A) The results of Lefschetz (4.1) and (5.1) to (5.5) are contained in this book [2]. For the local theory (4.1), it may be more handy to consult SGA 7, XIV (3.2).

B) The results of paragraph 4 are proved in parts XIII, XIV and XV of SGA 7.

C) (5.7) is proved in SGA 7, XVII[Also see the sketch of the proof presented in Milne.].

D) (5.8) is proved in SGA 7, XVIII. The irreducibility theorem is proved there for E but only under the hypothesis that E ∩ E^⊥={0}. Proof of the general case (for E/(E ∩ E^⊥)) is similar. 



§ THE RATIONALITY THEOREM

(6.1) Let ℙ_0 be a projective space of dimension ≥ 1 over 𝔽_q, X_0 ⊂ℙ_0 a projective nonsingular variety, A_0 ⊂ℙ_0 a linear subspace of codimension two, D_0 ⊂ ℙ_0 the dual line, 𝔽̅_q the algebraic closure of 𝔽_q and ℙ, X, A, D over 𝔽̅_q obtained from ℙ_0, X_0, A_0, D_0 by extension of scalars. The diagram (5.1.1) from (5.6) comes from a similar diagram over 𝔽_q:

    X_0 [r, leftarrow, "π_0"]
       X̃_0 [d, "f_0"]
        D_0
      (6.1.1)


We assume that X is connected of even dimension n+1=2m+2 and that the pencil of hyperplane sections of X defined by D is a Lefschetz pencil. The set S of t ∈ D such that X_t is singular and defined over 𝔽_q comes from S_0 ⊂ D_0. We denote U_0=D_0-S_0 and U=D-S.

Let u ∈ U. The vanishing part of the cohomology E ⊂ H^n(X_u, ℚ_l) is stable under π_1(U, u), so it is defined over U by a local subsystem ℰ of R^nf_*ℚ_l. The latter is defined over 𝔽_q: R^if_*ℚ_l is the inverse image of the ℚ_l sheaf R^if_0*ℚ_l on D_0 and, on U, ℰ is the inverse image of a local subsystem 
    ℰ_0 ⊂ R^nf_0*ℚ_l.
 The cup product is a skew-symmetric form 
    ψ: R^nf_0*ℚ_l ⊗ R^nf_0*ℚ_l →ℚ_l(-n).
 Denoting by ℰ_0^⊥ the orthogonal of ℰ_0 relative to ψ, on R^nf_0*ℚ_l | U_0 we see that ψ induces a a perfect pairing 
    ψ: ℰ_0/(ℰ_0 ∩ℰ_0^⊥) ⊗ℰ_0/(ℰ_0 ∩ℰ_0^⊥) →ℚ_l(-n).
 


For all x ∈ |U_0| the polynomial (1-F_x^*t, ℰ_0/(ℰ_0 ∩ℰ_0^⊥)) has rational coefficients. 



Let j_0 be the inclusion of U_0 in D_0 and j that of U in D. The eigenvalues of F^* acting on H^1(D, j_*ℰ_0/(ℰ_0 ∩ℰ_0^⊥)) are algebraic numbers all of which complex conjugates α satisfy 
    q^n+1/2-1/2≤ |α| ≤ q^n+1/2+1/2.



According to (5.10) and (6.2), the hypotheses of (3.2) are in fact verified for (U_0, ℰ_0/(ℰ_0 ∩ℰ_0^⊥), ψ) for β=n and we apply (3.9). 


Let 𝒢_0 be a locally constant ℚ_l-sheaf on U_0 such that its inverse image 𝒢 on U is a constant sheaf. Then there exist units α_i in ℚ̅_l such that for each x ∈ |U_0| we have 
    (1-F_x^*t, 𝒢_0)=∏_i (1-α_i^(x) t).



The lemma expresses the fact that 𝒢_0 is the inverse image of a sheaf on Spec(𝔽_q), namely, its direct image on Spec(𝔽_q) [Indeed, if ε: U_o → Spec 𝔽_q is the canonical morphism, the assumption implies that ε^*ε_* 𝒢_0 →𝒢_0 becomes an isomorphism once we base change to 𝔽̅_q.]. The latter identifies with an l-adic representation G_0 of Gal(𝔽̅_q/𝔽_q) and we have[To establish the lemma, we also use the relationship between the stalks of the sheaf and those of its inverse/direct image (see Milne).] 
    (1-Ft, G_0)=∏_i (1-α_i t).
 

Lemma (6.4) applies to R^if_0*ℚ_l  (i ≠ n), to R^n f_0*ℚ_l/ℰ_0 and to ℰ_0 ∩ℰ_0^⊥[To see that the inverse images of the latter two are constant, use the equivalence of categories of locally constant sheaves and continuous representations of π_1 and the fact that the monodromy action acts by deforming the cohomology by the vanishing cycles.] . 

For x ∈ |U_0| the fiber X_x=f_0^-1(x) is a variety over the finite field k(x). If x̅ is a point of U above x, X_x̅ is obtained from X_x by extension of scalars of k(x) to the algebraic closure k(x̅)=𝔽̅_q and H^i(X_x̅, ℚ_l) is the stalk of R^if_*ℚ_l at x̅. The formula (1.5.4) for the variety X_x over k(x) gives 
    Z(X_x, t)=∏_i (1-F_x^* t, R^i f_0*ℚ_l)^(-1)^i+1
 and Z(X_x, t) is a product of 
    Z^f=(1-F_x^* t, R^n f_0*ℚ_l/ℰ_0)(1-F_x^* t, ℰ_0 ∩ℰ_0^⊥) ∏_i ≠ n(1-F_x^* t, R^i f_0*ℚ_l)^(-1)^i+1
 and 
    Z^m=(1-F_x^* t, ℰ_0/(ℰ_0 ∩ℰ_0^⊥).
 Put ℱ_0=ℰ_0/(ℰ_0 ∩ℰ_0^⊥), ℱ=ℰ/(ℰ∩ℰ^⊥) and apply (6.4) to the factors of Z^f. We find that there exist l-adic units α_i  (1 ≤ i 
≤ N) and β_j  (1 ≤ j ≤ M) in ℚ̅_l such that for all x ∈ |U_0| 
    Z(X_x, t)=i∏ (1-α_i^(x)t)/j∏ (1-β_j^(x)t)(1-F_x^*t, ℱ_0)
 and in particular the right side is in ℚ(t). If some α_i coincides with a β_j, we can simultaneously delete α_i from the family of α and β_j from the family of β. Therefore, we may and do assume that α_i ≠β_j for all i and all j. 

(6.5) It suffices to prove that polynomials i∏ (1-α_i t) and j∏ (1-β_j t) have rational coefficients, i.e. the family of α_i (resp. the family of β_j) is defined over ℚ. We will deduce that from the following propositions. 


Let (γ_i)  (1 ≤ i ≤ P) and (δ_j)  (1 ≤ j ≤ Q) be two families of l-adic units in ℚ̅_l. Assume that γ_i ≠δ_j. If K is a large enough set of integers ≠ 1, and L is a large enough nowhere dense subset of |U_0|, then, if x ∈ |U_0| satisfies k ∤(x) (for all k ∈ K) and x ∉ L, the denominator of 
    (1-F_x^*t, ℱ_0) i∏ (1-γ_i^(x)t)/ j∏ (1-δ_j^(x)t)   (6.6.1)
 written in irreducible form, is j∏ (1-δ_j^(x)t). 


The proof will be given in (6.10-13). According to (6.7) below, (6.6) provides an intrinsic description of the family of δ_j in terms of the family of rational fractions (6.6.1) for x ∈ |U_0|. 


Let K be a finite set of integers ≠ 1 and (δ_j)  (1 ≤ j ≤ Q) and (ε_j)  (1 ≤ j ≤ Q) be two families of elements of a field. If, for all n large enough, not divisible by any of the k ∈ K, the family of δ_j^n coincides with that of ε_j^n (up to order), then the family of δ_j coincides with that of ε_j (up to order). 


We proceed by induction on Q. The set of integers n such that δ_Q^n=ε_j^n is an ideal (n_j). Let's prove that there exists a j_0 such that δ_Q=ε_j_0. Otherwise the n_j would be distinct from 1 and there would be arbitrarily large integers n, not divisible by any of the n_j nor by any of the k ∈ K. We would have δ_Q^n ≠ε_j^n and this contradicts the hypothesis. So there exists a j_0 such that δ_Q=ε_j_0.We conclude by applying the induction hypothesis to the families (δ_j)  (j ≠ Q) and (ε_j)  (j ≠ j_0).


Let (γ_i)  (1 ≤ i ≤ P) and (δ_j)  (1 ≤ j ≤ Q) be two families of p-adic units in ℚ̅_l, R(t)=i∏ (1-γ_i t) and S(t)=j∏ (1-δ_j t). Assume that for all x ∈ |U_0|    j∏ (1-δ_j^(x) t)  [In the original paper i appears instead of j.] divides 
    i∏ (1-γ_i^(x)t) (1-F_x^*t, ℱ_0).
 Then S(t) divides R(t). 

Remove from the families (γ_i) and (δ_j) pairs of common elements until they verify the hypothesis of (6.6). Apply (6.6). By hypothesis, the rational fractions (6.6.1) are polynomials. Therefore, no δ survives, which means that[By the remarks after Proposition (6.6) and Lemma (6.7).] S(t) divides R(t). 

(6.9) We prove (6.5) and (6.2) (modulo (6.6)). Let's put (γ_i)=(α_i) and (δ_i)=(β_i) in (6.6). We get an intrinsic characterization of the family of β_j in terms of the family of rational functions Z(X_x, t)  (x ∈ |U_0|). These being in ℚ(t), the family of β_j is defined over ℚ. 

Polynomials[for various x] i∏ (1-α_j^(x)t)  (1-F_x^* t, ℱ_0) are therefore in ℚ[t]. Proposition (6.8) provides an intrinsic description of the family of α_i in terms of this family of polynomials[Since a polynomial is uniquely characterized by its quotients.]. The family of α_i is thus defined over ℚ. 

(6.10)
Let u ∈ U and ℱ_u the stalk of ℱ at u. The arithmetic fundamental group π_1(U_0, u), the extension of Ẑ=Gal( 𝔽̅/𝔽) (generator: φ) by the geometric fundamental group[Both of the aforementioned groups are also called etale fundamental groups (of U_0 and U respectively).] π_1(U, u),  acts on ℱ_u by symplectic similitudes[i.e. a group A(V) of linear transformations g of a vector space V equipped with a nondegenerate bilinear form <, >: V × V → k such that <gv, gw>=μ(g)<v, w> for a multiplicative character μ:A(V) → k^× called the similitude multiplier] 
    ρ: π_1(U_0, u) → CSp(ℱ_u, ψ).
 We denote by μ(g) the multiplier of the symplectic similitude g. Let 
    H ⊂Ẑ× CSp(ℱ_u, ψ)
 be the subgroup defined by the equation 
    q^-n=μ(g)
 (q being an l-adic unit, q^n ∈ℚ_l^* is defined for all n ∈Ẑ). The fact that ψ has values in ℚ_l(-n) implies that the map from π_1 to Ẑ× CSp, the coordinates of the canonical projection to Ẑ and ρ factor through 
    ρ_1: π_1(U_0, u) → H.
  

The image H_1 of ρ_1 is open in H. 

Indeed, π_1(U_0, u) projects onto Ẑ and the image of π_1(U, u)=Ker(π_1(U_0, u) →Ẑ) in 
 Sp(ℱ_u, ψ)=Ker(H →Ẑ) is open (5.10).   


For δ∈ℚ̅_l an l-adic unit, the set Z of (n, g) ∈ H_1 such that δ^n is an eigenvalue of g is closed of measure 0  [With respect to the Haar measure on Ẑ× CSp.]. 

It is clear that Z is closed. For each n ∈Ẑ let CSp_n be the set of g ∈ CSp(ℱ_u, ψ) such that μ(g)=q^-n and let Z_n be the set of g ∈ CSp_n such that δ^n is an eigenvalue of g. Then CSp_n is a homogeneous space for Sp and we check that Z_n is a proper algebraic subspace, thus, of measure 0. According to (6.11), H^1 ∩ ({n }× Z_n) is of measure 0 in the inverse image in H_1 of n and we apply Fubini to the projection H_1 →Ẑ. 

(6.13) Let us prove (6.6). For each i and j, the set of integers n such that γ_i^n=δ_j^n is the set of multiples of a fixed integer n_ij (we do not exclude n_ij=0). By hypothesis, n_ij≠ 1. 

According to (6.12) and the Chebotarev's density theorem, the set of x ∈ |U_0| such that β_j^(x) is an eigenvalue of F_x^* acting on ℱ_0 is nowhere dense. We take for K the set of n_ij and for L the set of x as above. 


§ COMPLETION OF THE PROOF OF (1.7)



Let X_0 be a nonsingular absolutely irreducible[or geometrically irreducible] projective variety of even dimension over 𝔽_q. Let X over 𝔽̅_q be obtained from X_0 by extension of scalars and α an eigenvalue of F^* acting on H^d(X, ℚ_l). Then α is an algebraic number all of which complex conjugates, still denoted α, satisfy 
    q^d/2-1/2≤ |α| ≤ q^d/2+1/2  (7.1.1)
 

We proceed by induction on d (always assumed even). The case d=0 is trivial even without assuming that X_0 is absolutely irreducible; we assume from now on d ≥ 2. We put d=n+1=2m+2. 

If 𝔽_q^r is an extension of degree r of 𝔽_q and X_0'/𝔽_q^r is obtained from X_0/𝔽_q by extension of scalars, the statement (7.1) for X_0/𝔽_q is equivalent to (7.1) for X_0'/𝔽_q^r; in the same way as q is replaced by q^r, the eigenvalues of F^* are replaced by their r-th powers. 

According to (5.7), in a suitable projective embedding i: X →ℙ, X admits a Lefschetz pencil of hyperplane sections. The previous remark allows us to assume that the pencil is defined over 𝔽_q (once we replace 𝔽_q by a finite extension). 

Therefore, assume that there exists a projective embedding X_0 →ℙ_0 and a subspace A_0 ⊂ℙ_0 of codimension two that defines the Lefschetz pencil. We recall the notations of (6.1) and (6.3). A new extension of scalars allows us to assume that: 

a) The points of S are defined over 𝔽_q. 

b) The vanishing cycles for x_s  (s ∈ S) are defined over 𝔽_q (since only ±δ is intrinsic, they can only be defined over quadratic extensions).

c) There exists a rational point u_0 ∈ U_0. We take the corresponding point u of U as the base point. 

d) X_u_0=f_0^-1(u_0) admits a smooth hyperplane section Y_0 defined over 𝔽_q. We let Y=Y_0 ⊗_𝔽_q𝔽̅_q.

Since X̃ is obtained from X by blowing up along a smooth subvariety A ∩ X of dimension two, we have 
    H^i(X, ℚ_l)  H^i(X̃, ℚ_l)
 (in fact, H^i(X̃, ℚ_l)= H^i(X, ℚ_l) ⊕ H^i-2 (A ∩ X, ℚ_l)(-1))[This is by the Thom isomorphism theorem. See a (rather technical) proof in Milne.]. It suffices to prove (7.1.1) for the eigenvalues α of F^* acting on H^d(X̃, ℚ_l). 

The Leray spectral sequence for f is 
    E_2^pq=H^p(D, R^q f_*ℚ_l) ⇒ H^p+q(X̃, ℚ_l).
 It suffices to prove (7.1.1) for the eigenvalues of F^* acting on E_2^pq for p+q=d=n+1. Those are[I highly recommend consulting Milne's book for the explanations of the steps in A), B), C).]:

A) E_2^2, n-1. According to (5.8), R^n-1f_*ℚ_l is constant. From (2.10) we have 
    E_2^2, n-1=H^n-1(X_u, ℚ_l)(-1).
 Applying the weak Lefschetz theorem (corollary of SGA 4, XIV (3.2)) and the Poincare duality (SGA 4, XVIII), we have 
    H^n-1(X_u, ℚ_l)(-1)  H^n-1(Y, ℚ_l)(-1)
 and we apply the induction hypothesis to Y_0. 

B) E_2^0, n+1. If the vanishing cycles are nonzero, R^n+1f_* ℚ_l is constant and 
    E_2^0, n+1=H^n+1(X_u, ℚ_l).
 The Gysin map 
    H^n-1(Y, ℚ_l)(-1) → H^n+1(X_u, ℚ_l)
 is surjective (by an argument dual to that of A)) and we apply the induction hypothesis to Y_0. 

If the vanishing cycles are zero, the exact sequence of (5.8) b) gives the following exact sequence 
    s ∈ S⊕ℚ_l (m-n) → E_2^0, n+1→ H^n+1(X_u, ℚ_l).
 The eigenvalues of F acting on ℚ_l (m-n) are q^d/2 and for H^n+1 everything is as above. 

C) E_2^1, n. If we had the hard Lefschietz theorem, we would know that ℰ∩ℰ^⊥ is zero and that R^n f_* ℚ_l is the direct sum of j_*ℰ and a constant sheaf. The H^1 of a constant sheaf on ℙ^1 is zero and it would suffice to apply (6.3).

Since we have not proved the hard Lefshetz theorem yet, we will have to figure a way out. If the vanishing cycles are zero, R^nf_* ℚ_l is constant ((5.8) b)) and E_2^1, n=0. Therefore we may and do assume that the vanishing cycles are nonzero. Filter R^nf_* ℚ_l=j_*j^* R^n f_* ℚ_l (5.8) by the subsheafs j_*ℰ and j_*(ℰ∩ℰ^⊥). If the vanishing cycles δ are not in ℰ∩ℰ^⊥  [More precisely, not in E ∩ E^⊥.] we have exact sequences[The constant sheaf in (7.1.2) is j_*(R^n f_* ℚ_l/ ℰ). To see that this (and the next few) sheaves are constant, reason as in footnote 31 and use that the inverse/direct images of constant sheaves via j are constant (the second claim is false in general).]: 
    0 → j_*ℰ→ R^n f_* ℚ_l → constant  sheaf → 0   (7.1.2)
 
    0 → constant  sheaf  j_*(ℰ∩ℰ^⊥)) → j_*ℰ→ j_*(ℰ/(ℰ∩ℰ^⊥)) → 0   (7.1.3)

If, God forbid, the δ are in ℰ∩ℰ^⊥, we have ℰ⊂ℰ^⊥ and exact sequences[For ℱ=R^nf_*ℚ_l/j_* ℰ^⊥.]: 
    0 → the  constant  sheaf  j_*ℰ^⊥→ R^n f_* ℚ_l → a  sheaf  ℱ→ 0   (7.1.4)
 
    0 →ℱ→ the  constant  sheaf  j_*j^* ℱ→s ∈ S⊕ℚ_l(n-m)_s → 0   (7.1.5)
 

In the first case the long exact sequences in cohomology give 
    H^1(D, j_* ℰ) → H^1(D, R^n f_* ℚ_l) → 0   (7.1.2')
 
    0 → H^1(D, j_*ℰ) → H^1(D, j_*(ℰ/(ℰ∩ℰ^⊥)))   (7.1.3')
 and we apply (6.3).

In the second case, they give 
    0 → H^1(D, R^n f_*ℚ_l) → H^1(D, ℱ)   (7.1.4')
 
    s ∈ S⊕ℚ_l(n-m) → H^1(D, ℱ) → 0   (7.1.5')
 and we remark that F acts on ℚ_l(n-m) by multiplication by q^d/2.   


Let X_0 be a nonsingular projective absolutely irreducible variety of dimension d over 𝔽_q. Let X over 𝔽̅_q be obtained from X_0 by extension of scalars and α be an eigenvalue of F^* acting on H^d(X, ℚ_l). Then α is an algebraic number all of which complex conjugates, still denoted α,  satisfy 
    |α|=q^d/2.



We first prove that (7.2) ⇒ (1.7). For X_0 projective nonsingular over 𝔽̅_q we have to prove the following statements: 

W(X_0, i). Let X be obtained from X_0 by extension of scalars of 𝔽_q to 𝔽̅_q. If α is an eigenvalue of F^* acting on H^i(X, ℚ_l), then α is an algebraic number all of which complex conjugates, still denoted α, satisfy |α|=q^i/2.

a) If 𝔽_q^n is an an extension of degree n of 𝔽_q and X_0'/𝔽_q^n is obtained from X_0/𝔽_q by extension of scalars, then W(X_0, i) is equivalent to W(X_0', i): the extension of scalars replaces α by α^n and q by q^n.

b) If X_0 is purely of dimension n, W(X_0, i) is equivalent to W(X_0, 2n-i); this follows from Poincare duality[If α is an eigenvalue of F^* acting on H^i(X, ℚ_l), then q^n/α is an eigenvalue of F^* acting on H^2n-i(X, ℚ_l). See Milne.].

c) If X_0 is a an union of irreducible X_0^α, W(X_0, i) is equivalent to the collection of W(X_0^α, i).

d) If X_0 is purely of dimension n, Y_0 is a smooth hyperplane section of X_0 and i < n, then W(Y_0, i) ⇒ W(X_0, i): this follows from the weak Lefschetz theorem[Depending on what one takes for the weak Lefschetz theorem, this follows either directly from it and the properties of Frobenius (if we assume that the weak Lefschetz theorem provides us with surjectivity of the Gysin map in cohomology) or one has to also invoke the Gysin sequence to establish surjectivity (if we only assume H^i(X, ℚ_l)=0 for i>d and X affine). One can also avoid using the weak Lefschetz theorem and apply a Kunneth formula argument instead (due to A. Mellit). See Milne for clarifications.]. 

To prove the statements W(X_0, i) we move in succession:

-by c) we assume that X_0 is purely of dimension n;

-by b) we also assume that 0 ≤ i ≤ n;

-by a) and d) we also assume i=n;

-by a) and c) we also assume that X_0 is absolutely irreducible. 

Now the case satisfies the conditions of (7.2). 

(7.3) We prove (7.2). For every integer k, α^k is an eigenvalue of F^* acting on H^kd(X^k, ℚ_l) (Kunneth's formula). For k even, X^k satisfies the conditions of (7.1), so we have 
    q^kd/2-1/2≤ |α^k| ≤ q^kd/2+1/2
 and 
    q^d/2-1/2k≤ |α| ≤ q^d/2+1/2k.
 Letting k go to infinity, we establish (7.2). 


§ FIRST APPLICATIONS

Let X_0 ⊂ℙ_0^n+r be a nonsingular complete intersection over 𝔽_q of dimension n and of multidegree (d_1, ⋯ d_r). Let b' be the n-th Betti number of the complex nonsingular complete intersection with the same dimension and multidegree. Put b=b' for n odd and b=b'-1 for n even. Then 
    |# X_0(𝔽_q)-#ℙ^n(𝔽_q)| ≤ b q^n/2.



Let X/𝔽̅_q be obtained from X_0 and ℚ_l η^i be the line in H^2n(X, ℚ_l) generated by the i-th cup power of the cohomology class of the hyperplane section. On this line F^* acts by multiplication by q^i. The cohomology of X is the direct sum of the ℚ_l η^i  (0 ≤ i ≤ n) and the primitive part of H^n(X, ℚ_l) of dimension b. According to (1.5), therefore, there exist b algebraic numbers α_j, the eigenvalues of F^* acting on this primitive cohomology, such that 
    # X_0(𝔽_q)=∑_i=0^n q^i+(-1)^n ∑_j α_j.
 According to (1.7), |α_j|=q^n/2 and 
    |# X_0(𝔽_q)-#ℙ^n(𝔽_q)|=|# X_0(𝔽_q)-∑_i=0^n q^i|= |∑_j α_j| ≤∑_j |α_j|=b q^n/2.



Let N be an integer ≥ 1, ε: (ℤ/N)^* →ℂ^* a character, k an integer ≥ 2 and f a holomorphic modular form on Γ_0(N) of weight k and with character ε : f is  a holomorphic function on the the Poincare half-plane X such that for ([ a b; c d ]) ∈ SL(2, ℤ), with c ≡ 0  (N) we have 
    f (az+b/cz+d)=ε(a)^-1(cz+d)^kf(z).
 We assume that f is cuspidal and primitive ("new" in the sence of Arkin-Lehner and Miyake), in particular an eigenvector of the Hecke operators T_p  (p ∤ N). Let f=∑_n=1^∞ a_nq^n with q=e^2π i z (and a_1=1). Then for p prime not dividing N 
    |a_p| ≤ 2p^k-1/2.
 In other words, the roots of the equation 
    T^2-a_p T+ε(p)p^k-1
 are of absolute value p^k-1/2. 


These roots are indeed the eigenvalues of the Frobenius acting on H^k-1 of a nonsingular projective variety of dimension k-1 defined over 𝔽_p.

Under restrictive assumptions, this fact is proved in my Bourbaki expose (Formes modulaires et representations l-adiques, expose 355, February 1969, in: Lecture Notes in Mathematics, 179). The general case is not much more difficult. 

Remark (8.3) J.P.Serre and myself have recently proved that (8.2) remains true for k=1. The proof is quite different. 

The following application was suggested to me by E.Bombiery.

Let Q be a polynomial in n variables and of degree d over 𝔽_q, Q_d a homogeneous part of degree d of Q and ψ: 𝔽_q →ℂ^* an additive nontrivial character on 𝔽_q. We assume that: 

(i) d is coprime to p [(the characteristic of 𝔽_q)]

(ii) The hypersurface H_0 in ℙ_𝔽_q^n-1 defined by Q_d is smooth. 

Then 
    |∑_x_1, ⋯, x_n ∈𝔽_qψ(Q(x_1, ⋯, x_n))| ≤ (d-1)^n q^n/2.



After replacing Q by a scalar multiple, we may (and do) assume that[For q=p^f we have Tr_𝔽_q/𝔽_p(x)=x+x^p+⋯+x^p^f-1.] 
    ψ(x)=exp(2 π i Tr_𝔽_q/𝔽_p(x)/p)   (8.4.1)
 Let X_0 be an etale covering of the affine space 𝔸_0 of dimension n over 𝔽_q with equation T^p-T=Q and let σ be the projection of X_0 to 𝔸_0: 
    σ: X_0 →𝔸_0
 
    X_0=Spec(𝔽_q [x_1, ⋯, x_n, T]/(T^p-T-Q)).
 The covering X_0 is Galois with Galois group ℤ/p; i ∈ℤ/p=𝔽_p acts by T → T+i. 

We let x ∈𝔸_0(𝔽_q) and compute the Frobenius endomorphism on the fiber of X_0/𝔸_0 at x. Let q=p^f and let 𝔽̅_q be the algebraic closure of 𝔽_q. For (x, T) ∈ X_0(𝔽̅) above x we have F((x, T))=(x, T^q) and 
    T^q=T+∑_i=1^f(T^p^i-T^p^i-1)=T+∑ Q(x)^p^i-1=T+Tr_𝔽_q/𝔽_p(Q(x)).
 This is the action of the element Tr_𝔽_q/𝔽_p(Q(x)) of the Galois group. 

Let E be the field of the p-th roots of unity and λ a finite place of E coprime to p. We will work in λ-adic cohomology. For j ∈ℤ/p, let ℱ_j, 0 be a E_λ local system of rank one on 𝔸_0 defined by X_0 and ψ(-jx): ℤ/p → E^* → E_λ^*: we have ι:X_0 →ℱ_j, 0 and ι(i ⋆ x)=ψ(-ij)ι(x). 
Denote without _0 objects obtained from 𝔸_0, X_0, ℱ_j, 0 by extension of scalars to 𝔽̅_q. The trace formula (1.12.1) for ℱ_j, 0 gives: 
    ∑_x_1, ⋯, x_n ∈𝔽_qψ(Q(x_1, ⋯, x_n))=∑_i Tr(F^*, H_c^i(𝔸, ℱ_1))   (8.4.2)
 

We have σ_* E_λ=j⊕ℱ_j and so 
    H_c^*(X, ℚ_l) ⊗_ℚ_l E_λ=j⊕ H_c^*(𝔸, ℱ_j)   (8.4.3)
 For j=0, ℱ_j is the constant sheaf E_λ; this factor corresponds to inclusion, by taking the inverse image, of the cohomology of 𝔸 in that of X.


(i) For j ≠ 0, H_c^i(𝔸, ℱ_j) is zero for i ≠ n, for i=n, the cohomology space has dimension (d-1)^n. 

(ii) For j ≠ 0, the cup product 
    H_c^n(𝔸, ℱ_j) ⊗ H_c^n(𝔸, ℱ_-j) → H^2n(𝔸, E_λ) Tr→ E_λ(-n)
 is a perfect pairing. 

(iii) X_0 is open in a nonsinqular projective variety Z_0. 


Let's deduce (8.4) from (8.5). Let j_0: X_0  Z_0 and j: X  Z be obtained by extension of scalars of 𝔽̅_q. According to (8.4.2), (i) and (1.7) for Z_0, it suffices to prove the injectivity of 
    H_c^n(𝔸, ℱ_1) σ^*→ H_c^n(X, ℱ_1)=H_c^n(X, E_λ) j_!→ H^n(Z, E_λ).
 We have Tr(a ∪ b)=1/pTr(j_! σ^* a ∩ j_! σ^* b), so injectivity follows from (ii)[By the same reasoning as in the proof of (7.1) A) (for which I referred to Milne).]. 

(8.6) We prove (8.5) (iii). Let ℙ_0 be the projective space over 𝔽_q, obtained from 𝔸_0 by adding a hyperplane at infinity ℙ_0^∞, H_0 ⊂ℙ_0^∞ with equation Q_d=0 and Y_0 the covering of ℙ_0 normalizing ℙ_0 along X_0. 

    X_0 [r, hook] [d, "σ"]
         Y_0 [d]
    𝔸_0 [r, hook]    ℙ_0 [r, hookleftarrow]    ℙ_0^∞[r, hookleftarrow]     H_0
      (8.7.1)

Let's study Y_0/ℙ_0 near the infinity, locally for the etale topology. 


Y_0 is smooth outside the inverse image of H_0.


The divisor of a rational function Q on ℙ_0 is the sum of the finite part div(Q)_f and (-d) times the hyperplane at infinity. We have: 
    div(Q)=div(Q)_f-dℙ_0^∞  (8.7.1)
 
    div(Q)_f ∩ℙ_0^∞=H_0
 At a finite distance, Y_0=X_0 is etale over 𝔸_0, so smooth. At the infinity but outside the inverse image of H_0 there exist local coordinates (z_1, ⋯, z_n) such that Q=z_1^-d (here we use (d, p)=1). In these coordinates, Y_0 appears as a product of a curve and a smooth space (corresponding to coordinates z_2, ⋯, z_n). By normality it is smooth.  


In the etale neighborhood of a point above H_0, Y_0 is smooth on a normal singular surface, always the same. 


This time we can find local coordinates such that Q=z_1^-dz_2. Indeed, since H_0 is smooth, div(Q)_f is smooth in the neighborhood of infinity and crosses ℙ_0^∞ transversely. This form is independent of the chosen point and uses only two coordinates, hence the assertion.

(8.9) The following method (due to Zariski) allows one to resolve singularities on surfaces: alternately, we normalize and we blow up the (reduced) singular locus. Operators in play commute with etale localization and taking products by a smooth space. The method of Zariski, therefore, allows one to resolve singularities on a space that (like Y_0) is, locally for the etale topology, smooth on a surface. The resolution obtained from Y_0 is the Z_0 we seek. 

If T is a curve on a surface S containing the singular locus and T' is the inverse image of T in the Zariski resolution S' of S, we know that if we repeatedly blow up the (reduced) singular locus of (T')_red in S', we obtain a surface S” such that the inverse reduced image (T”)_red of T in S” is a divisor with normal crossings. Again, operations in play commute with etale localization and taking products by a smooth space. Reasoning as above and observing that (Y_0, infinity) is locally smooth in (S, T), we can find Z_0 such that Z_0-X_0 is a divisor with normal crossings. 


(8.10) We prove (8.5) (i), (ii). These assertions are geometric; this allows us to work from now on in 𝔽̅_q. Let S' be the affine space over 𝔽̅_q that parametrizes polynomials in n variables of degree ≤ d and let S be an open in S' corresponding to the polynomials, which homogeneous part of degree d has nonzero discriminant. We denote by Q_S∈ H^0(S, 𝒪[x_1, ⋯, x_n]) the universal polynomial[∑_i_1+ ⋯+ i_n=da_i_1, ⋯, i_nx_1^i_1⋯ x_n^i_n] of S and by X_S the Galois etale covering of 𝔸_S=𝔸^n × S with equation T^p-T=Q_S and Galois group ℤ/p. Let ℙ_S=ℙ^n × S be the projective completion of 𝔸_S and Y_S the normalization of ℙ_S along X_s. We have, for S, a diagram similar to (8.6.1). 

Expressions of Q in local coordinates (8.7) and (8.8) remain valid in this situation with parameters such that locally for the etale topology on Y_S, Y_S/S is isomorphic to the product of S (that is smooth) with one fiber. The method of canonical resolution used in (8.9) gives us a relative compactification Z_S/S of X_S/S with Z_S-X_S a divisor with normal crossings relative to S 

    X_S [r, hook, "u"] [d, "σ"]
         Z_S [d, "f"]
    𝔸_S [r, hook, "a"]     S
 (f is proper and smooth, u is an open immersion, Z_s-X_s a divisor with relative normal crossings).

Let ℱ_j, S be a E_λ-sheaf on 𝔸_S obtained as in (8.4) from X_S/𝔸_S. We have σ^*E_λ=⊕ℱ_j, S, so 
    R^*(fu)_! (E_λ)=j⊕ R^* a_! ℱ_j, S.
 The properties of Z_S ensure that R^i(fu)_!E_λ=R^if_*(u_!E_λ) is a locally constant sheaf on S. 
Therefore, R^ia_!ℱ_j, S is also locally constant.  Since S is connected, it suffices to prove (8.5) (i), (ii) for a particular polynomial Q. We will take Q=∑_i x_i^d. This polynomial satisfies the nonsingularity condition because (d, p)=1. For this polynomial variables in the exponential sum (8.4) are separated. This corresponds to the fact that ℱ_j is the tensor product of the inverse images of similar sheaves ℱ_j^1 on the factors of dimension one 𝔸^1 of 𝔸=𝔸^n. By Kunneth's formula 
    H^*(𝔸, ℱ_j)=⊕ H^*(𝔸^1, ℱ_j^1).
 This reduces the proof of (8.5) (i), (ii) to the case when n=1 and Q is x^d.

(8.11) Let's deal with this particular case. The covering X of 𝔸 is irreducible, so for i=0, 2 
    H_c^i(𝔸, E_λ) ∼→ H_c^i(X, E_λ).
 So for i ≠ 1 and j ≠ 0 we have[In the original paper 𝔸 appears instead of X.] 
    H_c^i(X, ℱ_j)=0.
 Assertion (ii) follows from (2.8) or (2.12) and the fact that u_!ℱ_j=u_*ℱ_j. To prove (i) it remains to verify that 
    χ_c(𝔸, ℱ_j)=1-d.
 According to the Euler-Poincare formula (see expose Bourbaki 286 of February 1965, by M.Raynaud), it is equivalent to the following lemma.


Swan's conductor of ℱ_j at infinity equals d.

This statement is equivalent to the following. 


Let k be a finite field of characteristic p, y ∈ k[[x]] an element of valuation d coprime to p, L the extension of K=k((x)) generated by the roots of T^p-T=y^-1 and χ the following character on Gal(L/K) with values in ℤ/p: 
    χ(σ)=σ T-T.
 Then χ has conductor d+1. 


By extension of the residue field we may assume that k is algebraically closed rather than finite and apply: J.P.Serre, Sur les corps locaux a corps residuel algebriquement clos, Bull. Soc. Math. France, 89 (1961), p. 105-154, n^o 4.4.


999
 A. Grothendieck, Formule de Lefschetz et rationalite des fonctions L, Seminare Bourbaki, 279, December 1964 (Benjamin).
 S. Lefschetz, L'analysis situs et la geometrie algebrique (Gauthier-Villars), 1924. Reproduced in: Selected papers (Chelsea Publ. Co.).

 R. A. Rankin Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions.. II, Proc. Camb. Phil. Soc., 35 (1939), 351-372.

 A.Weil Numbers of solutions of equations in finite fields, Bull. Math. Soc., 55 (1949), p. 497-508. 

SGA, Seminaire de Geometrie Algebrique du Bois-Marie (IHES):

SGA 4, Theorie des topos et cohomologie etale des schemas (led by M. Artin, A. Grothendieck and J.-L. Verdier), Lecture notes in Math., 269, 270, 305.

SGA 5, Cohomologie l-adique et fonctions L, supervised by I'IHES.

SGA 7, Groupes de monodromie en geometrie algebrique.

 1-st part: led by A.Grothendieck, Lecture Notes in Math., 288.

2-nd part: by P.Deligne and N.Katz, Lecture Notes in Math., 340.





Manuscript received on 20 September 1973.



