Étale cohomology — the Weil conjectures and ℓ-adic methods

Intuition Beginner

Algebraic geometry studies shapes cut out by polynomial equations. When those equations have coefficients in the complex numbers, the resulting variety is also a real geometric surface, and we can probe it with the tools of ordinary topology — count its holes, compute its singular cohomology, and so on.

But the deepest questions about varieties live over finite fields like ${\mathbb{F}}_q$, where there is no familiar surface to fall back on. A finite field has only finitely many points and no continuous notion of "nearby." Singular cohomology, which depends on analytic discs and paths, simply does not exist there.

The remedy is to invent a new kind of topology suited to algebra. An étale neighbourhood of a point is not a small disc around it (there are no discs over a finite field) but a finite algebraic extension — another sheet of the variety that lies infinitesimally close, reached by adjoining roots of equations. The French word étale suggests something stretched out flat, with no branching or folding.

With these algebraic neighbourhoods replacing discs, sheaves 04.03.01 and their cohomology transfer almost verbatim. The result, étale cohomology, is a cohomology theory that works over any field, including finite ones. It was built by Grothendieck and his school in the 1960s precisely to prove the Weil conjectures, a set of predictions about how many points a variety has over each finite field.

Visual Beginner

A scheme drawn as a surface, with an étale neighbourhood shown as a thin algebraic sheet lifting a chosen point. Where a topological disc would spread continuously, the étale neighbourhood is a finite branched-free cover.

The intuition: replace "points close to $x$" by "finite algebraic extensions of $x$," and rebuild sheaf cohomology on that notion of locality.

Worked example Beginner

The payoff of étale cohomology is that it counts points of varieties over finite fields. Consider the elliptic curve $E:y^2=x^3-x$ over ${\mathbb{F}}_5$. We count its ${\mathbb{F}}_5$-points by hand.

Step 1. For each $x\in {\mathbb{F}}_5=\{0,1,2,3,4\}$, compute $x^3-x\mathrm{mod}5$ and find the $y$ with $y^2$ equal to it.

Step 2. $x=0$ gives $0$; only $y=0$ works (one point). $x=1$ also gives $0$; one point. $x=2$ gives $8-2=6\equiv 1$; $y=1,4$ both square to $1$, so two points. $x=3$ gives $24\equiv 4$; $y=2,3$, two points. $x=4$ gives $60\equiv 0$; one point.

Step 3. Including the point at infinity, the total is $1+1+2+2+1+1=8$. So $|E({\mathbb{F}}_5)|=8$.

The Hasse bound says the count always lies close to $q+1=6$, within $2\sqrt{q}\approx 4.47$. Our count $8$ satisfies $|8-6|=2\le 4.47$. Étale cohomology is the machine that explains this bound: Frobenius acts on $H_{\acute{\mathrm{e}}\mathrm{t}}^1$ with eigenvalues of absolute value $\sqrt{q}$, and the point count is read off from those eigenvalues.

Check your understanding Beginner

Formal definition Intermediate+

Let $X$ be a scheme 04.02.01. A morphism $f:U\to X$ is étale if it is flat, locally of finite presentation, and its fibres are finite separable field extensions; equivalently (the Jacobian criterion) $f$ is the algebraic analogue of a local isomorphism 04.02.05. Étale morphisms are the building blocks of the étale topology in the same way that open immersions are the building blocks of the Zariski topology.

Sites and the étale site

A Grothendieck topology (or site) $\mathcal{C}$ is a category equipped with a notion of covering family for each object, satisfying axioms mimicking those of open covers (stability under pullback, refinement of refinements). A sheaf on $\mathcal{C}$ (of sets, abelian groups, etc.) is a presheaf $F:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Ab}$ such that for every covering family $\{U_i\to U\}$, the sequence

$$0\to F(U)\to \prod _{i}F(U_i)\rightrightarrows \prod _{i,j}F(U_i{\times }_U{U}_j)$$

is an equaliser ^[SGA4].

The small étale site $X_{\acute{\mathrm{e}}\mathrm{t}}$ of $X$ has objects the étale morphisms $U\to X$ and morphisms the $X$-maps; a covering family of $U\to X$ is a family $\{U_i\to U\}$ of étale morphisms that is jointly surjective. A sheaf on $X_{\acute{\mathrm{e}}\mathrm{t}}$ is an étale sheaf. The category ${\mathbf{Sh}}_{\mathbf{Ab}}(X_{\acute{\mathrm{e}}\mathrm{t}})$ is a Grothendieck abelian category with enough injectives, so we may form derived functors.

Étale cohomology and ℓ-adic sheaves

The étale cohomology of $X$ with coefficients in an étale sheaf $\mathcal{F}$ is

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(X;\mathcal{F}):=R^i\Gamma (X_{\acute{\mathrm{e}}\mathrm{t}},\mathcal{F}),$$

the $i$-th right derived functor of the global-sections functor on the étale site 04.03.01. This is sheaf cohomology transplanted from the Zariski site to the étale site.

Fix a prime $\ell \ne \mathrm{char}(k)$. For each $n\ge 1$ we have the finite constant sheaf $\mathbb{Z}/{\ell }^n\mathbb{Z}$ on $X_{\acute{\mathrm{e}}\mathrm{t}}$. The ℓ-adic cohomology is the inverse limit

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(X;{\mathbb{Z}}_{\ell }):=\underset{n}{\underleftarrow{\lim }}{H}_{\acute{\mathrm{e}}\mathrm{t}}^i(X;\mathbb{Z}/{\ell }^n\mathbb{Z}),H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{\bar{k}};{\mathbb{Q}}_{\ell }):=H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{\bar{k}};{\mathbb{Z}}_{\ell }){\otimes }_{{\mathbb{Z}}_{\ell }}{\mathbb{Q}}_{\ell }.$$

An ℓ-adic sheaf is a compatible inverse system $({\mathcal{F}}_n{)}_n$ of sheaves of $\mathbb{Z}/{\ell }^n\mathbb{Z}$-modules; the category of smooth ${\mathbb{Q}}_{\ell }$-sheaves is the bridge to Galois representations 21.05.01.

The étale fundamental group

For a geometric point $\bar{x}\to X$, the étale fundamental group ${\pi }_1^{\acute{\mathrm{e}}\mathrm{t}}(X,\bar{x})$ is the automorphism group of the fibre functor

$$F_{\bar{x}}:{\mathbf{FEt}}_X\to \mathbf{Sets},(Y\to X)\mapsto Y{\times }_X\bar{x},$$

on the category of finite étale covers of $X$. When $X=\mathrm{Spec}(k)$ this recovers the absolute Galois group $\mathrm{Gal}(\bar{k}/k)$ 01.04.01; when $X$ is a smooth complex variety it recovers the profinite completion of the topological fundamental group.

Key theorem with proof Intermediate+

Theorem (Grothendieck–Lefschetz trace formula). Let $X_0$ be a smooth projective variety over ${\mathbb{F}}_q$, let $X=X_0{\otimes }_{{\mathbb{F}}_q}{\bar{\mathbb{F}}}_q$, and fix $\ell \ne \mathrm{char}({\mathbb{F}}_q)$. Let $F:X\to X$ be the geometric Frobenius. Then the number of ${\mathbb{F}}_q$-rational points of $X_0$ is the alternating sum of traces of Frobenius on ℓ-adic cohomology:

$$|X_0({\mathbb{F}}_q)|=\sum _{i=0}^{2\dim X_0}(-1{)}^i\mathrm{Tr}(F\mid H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{{\bar{\mathbb{F}}}_q};{\mathbb{Q}}_{\ell })).$$

Proof sketch. This is the Lefschetz fixed-point formula applied to the Frobenius correspondence. Let ${\Gamma }_F\subset X\times X$ be the graph of Frobenius and $\Delta \subset X\times X$ the diagonal; both are correspondences on $X$. Classical Lefschetz computes the fixed points of an endomorphism as the intersection number ${\Gamma }_F\cdot \Delta$; Grothendieck lifts this to the ℓ-adic setting by constructing a cohomological correspondence ${\Gamma }_F^{\ast }$ on $H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(X\times X;{\mathbb{Q}}_{\ell })$.

The Künneth formula decomposes $H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(X\times X;{\mathbb{Q}}_{\ell })\cong H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(X;{\mathbb{Q}}_{\ell })\otimes H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(X;{\mathbb{Q}}_{\ell })$, and under this decomposition the action of the correspondence ${\Gamma }_F^{\ast }$ on each summand $H^i\otimes H^i$ is $F$ on the left factor against the dual of $F$ on the right, producing the trace $\mathrm{Tr}(F\mid H^i)$. Taking the alternating sum over $i$ and applying the Lefschetz fixed-point theorem (Frobenius fixed points are precisely the ${\mathbb{F}}_q$-rational points) yields the formula. The heavy input — proved later by Deligne — is that $H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{{\bar{\mathbb{F}}}_q};{\mathbb{Q}}_{\ell })$ is finite-dimensional and Frobenius acts with eigenvalues controlled by the Weil conjectures. $\square$

For the elliptic curve of the worked example, $\dim X_0=1$, so only $H^0$, $H^1$, $H^2$ appear. Frobenius acts as the identity on $H^0\cong {\mathbb{Q}}_{\ell }$ (trace $1$) and on $H^2\cong {\mathbb{Q}}_{\ell }(-1)$ (trace $q$), and the two eigenvalues $\alpha ,\beta$ on $H^1$ have $|\alpha |=|\beta |=\sqrt{q}$, giving $|E({\mathbb{F}}_q)|=1-(\alpha +\beta )+q$. For $q=5$ this reads $8=1-(\alpha +\beta )+5$, so $\alpha +\beta =-2$, consistent with $|\alpha +\beta |\le 2\sqrt{5}$.

Bridge. The trace formula builds toward 21.05.01 (ℓ-adic Galois representations), where the eigenvalues of Frobenius on $H_{\acute{\mathrm{e}}\mathrm{t}}^i$ become the central objects of arithmetic — each variety over ${\mathbb{F}}_q$ produces a Galois representation — and appears again in 04.02.05 (smooth, étale and unramified morphisms), where the local structure that makes the étale site well-defined is established. The foundational reason the formula works is that ℓ-adic cohomology carries a canonical Frobenius action while remaining a finite-dimensional vector space, and the bridge is that counting points over ${\mathbb{F}}_q$ reduces to linear algebra on a cohomology group; putting these together, this is exactly the pattern by which a geometric counting problem becomes a spectral problem in the Grothendieck reformulation of arithmetic geometry.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — recorded in the unit metadata. Mathlib has the categorical infrastructure for sites and Grothendieck topologies (Mathlib.CategoryTheory.Sites.Sheaf, Mathlib.CategoryTheory.GrothendieckTopology) and a growing library on étale and finite étale morphisms of schemes, but no construction of the small étale site of a scheme, no ℓ-adic sheaf pro-category, and no derived-functor cohomology on a site in the form needed to state the trace formula. A serious formal attack would need to build the cohomology-of-sites layer first; until then the Weil conjectures are beyond Lean's reach.

Advanced results Master

The four Weil conjectures (1955), conjectured by Weil and proved over two decades by Dwork, Grothendieck, and Deligne, govern the zeta function

$$Z(X_0,t)=\exp \left(\sum _{n\ge 1}|X_0({\mathbb{F}}_{q^n})|\frac{t^n}{n}\right)$$

of a smooth projective variety $X_0$ over ${\mathbb{F}}_q$. Dwork (1960) proved rationality of $Z$ by $p$-adic analysis. Grothendieck (1965) re-proved rationality and added the functional equation by constructing ℓ-adic cohomology and the trace formula ^[SGA4]. Deligne (1974) proved the deepest part — the Riemann hypothesis for varieties over finite fields ^{[Deligne1974]}:

Every eigenvalue of the geometric Frobenius on $H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{{\bar{\mathbb{F}}}_q};{\mathbb{Q}}_{\ell })$ has complex absolute value $q^{i/2}$.

Poincaré duality (ℓ-adic). For $X_0/{\mathbb{F}}_q$ smooth projective of pure dimension $d$, there is a perfect pairing

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(X_{{\bar{\mathbb{F}}}_q};{\mathbb{Q}}_{\ell })\times H_{\acute{\mathrm{e}}\mathrm{t}}^{2d-i}(X_{{\bar{\mathbb{F}}}_q};{\mathbb{Q}}_{\ell }(d))\to {\mathbb{Q}}_{\ell },$$

giving the functional equation for $Z(X_0,t)$. The Tate twist ${\mathbb{Q}}_{\ell }(1)$ is the dual of the ℓ-adic cyclotomic character and encodes the "weight" $2$ of $H^2$.

Smooth and proper base change. If $f:X\to S$ is smooth and proper and $\mathcal{F}$ is a lisse ${\mathbb{Q}}_{\ell }$-sheaf on $X$, then $R^i{f}_{\ast }\mathcal{F}$ is lisse on $S$ and its formation commutes with base change. Geometrically: the Betti numbers of the fibres of a smooth proper family are locally constant. This is the ℓ-adic substitute for the Ehresmann fibration theorem of differential geometry.

Künneth formula. For smooth projective $X,Y$ over an algebraically closed field,

$$H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(X\times Y;{\mathbb{Q}}_{\ell })\cong H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(X;{\mathbb{Q}}_{\ell })\otimes H_{\acute{\mathrm{e}}\mathrm{t}}^{\bullet }(Y;{\mathbb{Q}}_{\ell }),$$

exactly as for singular cohomology. Combined with the cycle class map, this drives the standard conjectures on algebraic cycles, which remain open.

Comparison with singular cohomology (Artin). If $X$ is a smooth variety over $\mathbb{C}$, the canonical map

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(X;\mathbb{Z}/n\mathbb{Z})\to H_{\mathrm{sing}}^i(X^{\mathrm{an}};\mathbb{Z}/n\mathbb{Z})$$

is an isomorphism for every $n$. Étale cohomology is thus a genuine extension of singular cohomology to positive characteristic, not merely an analogy 04.03.01.

Weights and Deligne's Weil II. In the 1980 paper La conjecture de Weil II Deligne went further, defining a theory of weights for ℓ-adic sheaves: a sheaf is pure of weight $w$ if every eigenvalue of Frobenius at every closed point has absolute value $(\#\kappa (x){)}^{w/2}$. The Weil II theorem states that the higher direct images of a pure sheaf under a smooth morphism are pure, and that the local monodromy theorem holds. This is the engine behind the proof of the Kazhdan–Lusztig conjectures, the Deligne–Lusztig theory of finite reductive groups, and modern instances of the Langlands program 21.05.01.

Synthesis. Étale cohomology is the right cohomology for algebraic geometry in positive characteristic: it generalises singular cohomology by replacing open discs with finite étale neighbourhoods, the foundational reason being that the étale site carries enough finite covering data to support a derived-functor cohomology even when no analytic topology exists. The trace formula builds toward 21.05.01 where Frobenius eigenvalues become Galois representations, the central insight that a point-counting problem becomes a spectral problem, and the pattern appears again in 04.02.05 where the local étale structure is defined. Putting these together, this is exactly the bridge between topology and arithmetic: Poincaré duality, the Künneth formula, and base change lift verbatim, while Deligne's theorem that Frobenius eigenvalues have absolute value $q^{i/2}$ is the Riemann hypothesis in the arithmetic setting. The bridge is that the same sheaf-cohomology formalism of 04.03.01, transported from the Zariski site to the étale site, recovers singular cohomology over $\mathbb{C}$ and proves the Weil conjectures over ${\mathbb{F}}_q$ in one stroke.

Full proof set Master

Proposition (Exactness of the Kummer sequence in the étale topology). Let $X$ be a scheme and $n$ an integer invertible on $X$ (that is, $n$ is prime to the residue characteristics of every point). Then the sequence of étale sheaves

$$0\to {\mu }_n\to {\mathbb{G}}_m\stackrel{x\mapsto x^n}{\to }{\mathbb{G}}_m\to 0$$

is exact, where ${\mu }_n$ is the sheaf of $n$-th roots of unity and the right-hand map raises sections to the $n$-th power.

Proof. We check exactness on stalks at every geometric point $\bar{x}\to X$. The strict Henselisation ${\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}$ is a strictly Henselian local ring; its units fit into

$$0\to {\mu }_n({\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}})\to ({\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}{)}^{\times }\stackrel{x\mapsto x^n}{\to }({\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}{)}^{\times }.$$

The kernel of $x\mapsto x^n$ is by definition ${\mu }_n$, which establishes exactness at the first two terms.

It remains to show $x\mapsto x^n$ is surjective on the strictly Henselian stalk. Let $u\in ({\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}{)}^{\times }$. The polynomial $T^n-u\in {\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}[T]$ has derivative $n{T}^{n-1}$, which is invertible at any root $T_0$ with $T_0^n=u$ because $T_0$ is a unit and $n$ is invertible on $X$. Hence $T^n-u$ is a separable polynomial, and the algebra ${\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}[T]/(T^n-u)$ is finite étale over ${\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}$. By strict Henselianity, any factorisation of $T^n-u$ over the residue field lifts to a root in ${\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}$ itself; in particular, since $\bar{u}$ has an $n$-th root in the separably closed residue field, $u$ has an $n$-th root in $({\mathcal{O}}_{X,\bar{x}}^{\mathrm{sh}}{)}^{\times }$. This proves surjectivity, hence exactness on stalks, hence exactness as a sequence of étale sheaves. $\square$

Corollary. The Kummer exact sequence yields a long exact sequence in étale cohomology,

$$\cdots \to H_{\acute{\mathrm{e}}\mathrm{t}}^0(X;{\mathbb{G}}_m)\stackrel{x\mapsto x^n}{\to }{H}_{\acute{\mathrm{e}}\mathrm{t}}^0(X;{\mathbb{G}}_m)\to H_{\acute{\mathrm{e}}\mathrm{t}}^1(X;{\mu }_n)\to H_{\acute{\mathrm{e}}\mathrm{t}}^1(X;{\mathbb{G}}_m)\stackrel{n}{\to }{H}_{\acute{\mathrm{e}}\mathrm{t}}^1(X;{\mathbb{G}}_m)\to \cdots$$

Using Hilbert's theorem 90 ($H_{\acute{\mathrm{e}}\mathrm{t}}^1(X;{\mathbb{G}}_m)=\mathrm{Pic}(X)$ for a regular scheme), this identifies $H_{\acute{\mathrm{e}}\mathrm{t}}^1(X;{\mu }_n)\cong \mathrm{Pic}(X)[n]$ and feeds the descent-theoretic construction of the Brauer group $H_{\acute{\mathrm{e}}\mathrm{t}}^2(X;{\mathbb{G}}_m)$. This single exact sequence is the most-used computational input in concrete étale-cohomology calculations.

Connections Master

• Sheaf cohomology 04.03.01. Étale cohomology is literally the sheaf cohomology of 04.03.01 transplanted from the Zariski site to the étale site: same derived-functor definition of $\Gamma$, same long exact sequences, same Čech comparison. The only change is the underlying site. Connection type: direct-application-of.

• Schemes 04.02.01 and smooth/étale morphisms 04.02.05. The étale site is built out of étale morphisms, whose local structure (flat, finite presentation, separable fibres) is established in 04.02.05. Without the scheme theory of 04.02.01 there is no étale site to speak of. Connection type: depends-on.

• Fields and Galois theory 01.04.01. For $X=\mathrm{Spec}(k)$ the étale site classifies finite separable extensions of $k$ and the étale fundamental group is the absolute Galois group $\mathrm{Gal}(\bar{k}/k)$. Étale cohomology of $\mathrm{Spec}(k)$ is Galois cohomology. Connection type: specialises-to.

• ℓ-adic Galois representations 21.05.01. The ℓ-adic cohomology of a variety over a number field or finite field carries a continuous action of the absolute Galois group, producing the Galois representations at the heart of the Langlands program and of modularity. Connection type: feeds-into.

• Crystalline and prismatic cohomology. For the complementary prime $\ell =p=\mathrm{char}(k)$, where étale cohomology fails (Exercise 6), the correct substitute is crystalline (Berthelot) or prismatic (Bhatt–Scholze) cohomology. These complete the cohomological picture in mixed characteristic. Connection type: complementary-theory.

Throughlines. Étale cohomology is the convergence point of the algebraic-geometry strand and the number-theory strand: it is sheaf cohomology on a Grothendieck site, it reduces to Galois cohomology on a point, and its Frobenius action produces the Galois representations that drive arithmetic. The pattern established here — build a cohomology theory suited to the arithmetic setting, read off point counts and L-functions from a trace formula — recurs in rigid cohomology for non-proper varieties, in crystalline cohomology for the equal-characteristic $p$ setting, and in the Langlands program's passage from geometry to automorphic forms.

Historical & philosophical context Master

The Weil conjectures were formulated by André Weil in 1949 from his study of the Riemann hypothesis for curves over finite fields and his insight that the zeta function of a variety over ${\mathbb{F}}_q$ should behave like the Riemann zeta function. Weil conjectured four properties — rationality, a functional equation, an analogue of the Riemann hypothesis on the zeros, and a connection with Betti numbers — and observed that any "good" cohomology theory for varieties in positive characteristic would imply them at once, by the same Lefschetz fixed-point argument that works in topology.

That good cohomology theory did not exist. Alexander Grothendieck, in the Séminaire de Géométrie Algébrique du Bois-Marie (SGA), built it from scratch across the 1960s. SGA 4 (with Artin and Verdier) developed the machinery of sites, topologies, and sheaf cohomology on a Grothendieck topology ^{[SGA4 Artin–Grothendieck–Verdier 1972–73]}; SGA 1 developed the étale fundamental group as a Galois theory of schemes ^{[SGA1 Grothendieck 1971]}; SGA 5 and SGA 7 handled the trace formula and monodromy. Grothendieck proved three of the four Weil conjectures by 1965.

The fourth — the Riemann hypothesis, that Frobenius eigenvalues on $H^i$ have absolute value $q^{i/2}$ — resisted for another decade and was finally proved by Pierre Deligne in La conjecture de Weil I ^{[Deligne1974]}, using an ingenious reduction to the rank-one case and a theorem of Kazhdan–Margulis on the eigenvalues of Frobenius. Deligne's 1980 Weil II extended the theory to weights of ℓ-adic sheaves and became a cornerstone of modern arithmetic geometry.

Philosophically, étale cohomology is the decisive example of Grothendieck's method: enlarge the ambient category until the problem becomes formal. By replacing topological spaces with sites — categories carrying a notion of covering — Grothendieck made sheaf cohomology available wherever a "local" structure exists, whether analytic, algebraic, or arithmetic. This philosophical move, more than any single theorem, defines modern algebraic geometry.

Bibliography Master

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  author = {Milne, James S.},
  title = {Étale Cohomology},
  publisher = {Princeton University Press},
  year = {1980},
  series = {Princeton Mathematical Series, 33},
}

@article{DeligneWeilI,
  author = {Deligne, Pierre},
  title = {La conjecture de Weil I},
  journal = {Publications Mathématiques de l'IHÉS},
  volume = {43},
  pages = {273--307},
  year = {1974},
}

@article{DeligneWeilII,
  author = {Deligne, Pierre},
  title = {La conjecture de Weil II},
  journal = {Publications Mathématiques de l'IHÉS},
  volume = {52},
  pages = {137--252},
  year = {1980},
}

@book{SGA4,
  author = {Artin, Michael and Grothendieck, Alexander and Verdier, Jean-Louis},
  title = {Théorie des topos et cohomologie étale des schémas (SGA 4)},
  publisher = {Springer, Lecture Notes in Mathematics 269, 270, 305},
  year = {1972--1973},
}

@book{SGA1,
  author = {Grothendieck, Alexander},
  title = {Revêtements étales et groupe fondamental (SGA 1)},
  publisher = {Springer, Lecture Notes in Mathematics 224},
  year = {1971},
}

@book{FreitagKiehl,
  author = {Freitag, Eberhard and Kiehl, Reinhardt},
  title = {Étale Cohomology and the Weil Conjecture},
  publisher = {Springer},
  year = {1988},
}

@book{Tamme,
  author = {Tamme, Günter},
  title = {Introduction to Étale Cohomology},
  publisher = {Springer},
  year = {1994},
}