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Weil cohomology in practice

Kiran S. Kedlaya

Chapter 16 Étale fundamental groups

Original lecture date: November 26, 2019.
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In this lecture, we’ll continue the discussion of the étale fundamental group, giving the formulation of Deligne’s “Weil II” theorem.
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Readings 16.0.1.

We follow the setup of [31]. While there is a whole parallel \(p\)-adic setup, we did not have time to say much it in these lectures (besides Remark 16.1.7 below); for more, see [77].
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Section 16.1

Remark 16.1.1.

Let’s recall our setup from last time. Let \(X\) be a smooth connected scheme over a finite field \(k\) of characteristic \(p\text{.}\) (We didn’t require smoothness last time, but it will be convenient later to add this hypothesis.) Let \(\ell\) be a prime nonzero in \(k\text{.}\) We “defined” the notion of a lisse \(\overline{\QQ}_\ell\)-sheaf \(\calF\) in terms of an associated continuous representation \(\Lambda_\calF\colon\pi_1(X,x)\rightarrow \GL(r,\overline{\QQ}_\ell)\text{,}\) where \(x\) is a geometric point of \(X\text{.}\) For any closed point \(y\in X\text{,}\) we have a well-defined conjugacy class of elements \(\Frob_y\in \pi_1(X,x)\text{.}\)
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Definition 16.1.2.

Suppose now that \(X\) is geometrically irreducible. The geometric profinite fundamental group of \(X\) is defined as \(\pi_1(X_{\overline{k}})\text{,}\) where the basepoint is omitted.
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The geometric fundamental group is related to the actual fundamental group by functoriality for the morphism \(X_{\overline{k}}\rightarrow X\rightarrow k\text{.}\) This sequence of maps behaves a bit like a homotopy fiber sequence, in that we obtain the following exact sequence.
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Proof.

Remark 16.1.4.

In lieu of saying more about the proof of Proposition 16.1.3, we point out a philosophical observation: the group \(\pi_1(X_{\overline{k}})\) has “many” representations, but \(\pi_1(X)\) has “few” representations.
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To wit, the exact sequence gives rise to a map \(G_k\rightarrow\Out(\pi_1(X_{\overline{k}}))\) to the group of outer automorphisms of \(\pi_1(X_{\overline{k}})\) (which makes sense without regard for the basepoint). The group \(\Out(\pi_1(X_{\overline{k}}))\) acts on the set of isomorphism classes of continuous \(\overline{\QQ}_\ell\)-representations of \(\pi_1(X_{\overline{k}})\text{,}\) and the class of any representation that extends to \(\pi_1(X_{\overline{k}})\) must be a fixed point for this action. However, “most” classes are not fixed points.
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Example 16.1.5.

Let \(X\) be the scheme obtained from a smooth, projective, geometrically irreducible curve of genus \(g\) over \(k\) by removing a zero-dimensional closed subscheme of length \(m\) over \(k\text{.}\) While the group \(\pi_1(X_{\overline{k}})\) is somewhat difficult to define, Grothendieck defined a quotient of it, the tame profinite fundamental group \(\pi_1^{\tame}(X_{\overline{k}})\text{,}\) which is much easier to compute: it is a certain profinite completion of the free group generated by \(2g+m\) letters \(a_1,\dots,a_g,b_1,\dots,b_g,c_1,\dots,c_m\) modulo the relation \([a_1,b_1]\cdots[a_g,b_g]c_1\cdots c_m\text{,}\) where the brackets denote commutators. (That is, we take the profinite completion of the ordinary fundamental group of a genus-\(g\) Riemann surface with \(m\) punctures.)
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It is quite easy to write down continuous \(\overline{\QQ_\ell}\)-representations of \(\pi_1^{\tame}(X_{\overline{k}})\text{:}\) this just comes down to writing down systems of matrices corresponding to the generators, with a bit of care to ensure that the resulting map extends to the profinite completion. However, most of these maps will not be preserved by the outer action of \(G_k\text{.}\)
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Remark 16.1.6.

Proposition 16.1.3 remains true for an arbitrary field \(k\text{.}\) In the case \(k= \QQ\text{,}\) the resulting “mysterious” action of \(G_k\) on \(\pi_1(X_{\overline{k}})\) gives rise to a “mysterious” action on finite covers of \(\PP^1\) branched over \(\{0,1,\infty\}\) (since these covers are rigid, they always give rise to curves defined over number fields) and in turn to a “mysterious” action on Grothendieck’s dessins d’enfants.
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Remark 16.1.7.

Although we have not included in these lectures a detailed account, there is a parallel \(p\)-adic construction of “lisse sheaves” to which much of the following discussion carries over. Let us briefly indicate the analogue of Proposition 16.1.3 in this setup.
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The analogue of a continuous \(\ell\)-adic representation of \(\pi_1(X_{\overline{k}})\) is an overconvergent isocrystal. For \(X\) affine, such an object can be described as a finite projective module (“vector bundle”) over a dagger lift \(A^\dagger\) equipped with an integrable \(K\)-linear connection, where \(K\) again denotes the fraction field of the ring of Witt vectors \(W(k)\text{.}\) (An additional condition must be imposed to ensure that this definition is independent of the choice of the dagger lift.)
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The analogue of a continuous \(\ell\)-adic representation of \(\pi_1(X)\) is an overconvergent \(F\)-isocrystal. Such an object consists of an overconvergent isocrystal plus an isomorphism with its Frobenius pullback.
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As in the étale case, it is relatively easy to manufacture overconvergent isocrystals from the definition, but most of these will not admit a compatible Frobenius action.
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Definition 16.1.8.

Fix now a lisse \(\overline{\QQ}_\ell\)-sheaf \(\calF\text{.}\) Let \(H^i(X_{\overline{k}},\calF)\) and \(H^i_c(X_{\overline{k}},\calF)\) denote étale cohomology with coefficients in \(\calF\) and étale cohomology with compact support with coefficients in \(\calF\) respectively. We will refer to these for short as \(H^i\) and \(H^i_c\text{.}\)
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It turns out that \(H^i\) and \(H^i_c\) are finite-dimensional \(\overline{\QQ}_\ell\) vector spaces on which \(G_k\) acts continuously; that is, there are lisse \(\overline{\QQ}_\ell\)-sheaves over the point \(\Spec(k)\text{.}\) These are special cases of higher direct images \(R^i f_* \calF, R^i f_! \calF\) for \(f: X \to S\) a smooth morphism, but in general these land in a large category than the lisse \(\overline{\QQ}_\ell\)-sheaves (namely, the category of constructible \(\overline{\QQ}_\ell\)-sheaves). When \(f\) is smooth proper, they are indeed lisse \(\overline{\QQ}_\ell\)-sheaves.
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To formulate Poincaré duality for étale cohomology, before stating it, we need three more constructions.
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  • The forgetting supports map \(H^i_c \to H^i\text{,}\) which is an isomorphism when \(X\) is proper.
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  • The trace map: when \(X\) is of dimension \(n\text{,}\) there is a \(G_k\)-equivariant map
    \begin{equation*} H^{2n}_c(X_{\overline{k}})\twoheadrightarrow \overline{\QQ}_l(-n) \end{equation*}
    where \(\overline{\QQ}_\ell(-n)\) is the \((-n)\)-th Tate twist of \(\overline{\QQ}_\ell\text{.}\) This map is an isomorphism if \(X\) is (smooth and) geometrically irreducible.
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  • The cup product pairing:
    \begin{equation*} H^i_c(X_{\overline{k}},\calF)\times H^{2n-i}(X_{\overline{k}},\calF^\vee)\rightarrow H^{2n}_c(X_{\overline{k}},\calF\otimes \calF^\vee)\rightarrow H^{2n}_c(X_{\overline{k}},\overline{\QQ}_l)\twoheadrightarrow \overline{\QQ}_l(-n). \end{equation*}
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With this pairing defined, we can state Poincaré duality in étale cohomology.
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We also have a Lefschetz trace formula for étale cohomology. In the following formulation, it does not even require \(X\) to be smooth over \(k\text{.}\)
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This implies a factorization of the \(L\)-function of a now-familiar form.
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In order to state Weil II, we need to add a little more notation in order to measure archimedean absolute values.
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Definition 16.1.12.

Fix an algebraic (but in no way topological!) embedding \(\iota\colon \overline{\QQ}_\ell \rightarrow \CC\text{,}\) and let \({|x|}_\iota=|\iota(x)|\) be the induced absolute value on \(\overline{\QQ}_l\text{.}\)
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We say that a lisse \(\overline{\QQ}_\ell\)-sheaf \(\calF\) is \(\iota\)-pure of weight \(w\in\RR\) (resp. \(\iota\)-mixed of weight \(\leq w\), \(\iota\)-mixed of weight \(\geq w\)) if for all finite extensions \(k^\prime /k\) and all \(x\in X(k^\prime)\text{,}\) all the eigenvalues of \(\Lambda_\calF(\Frob_x)\) have \(\iota\)-absolute value equal to (resp. greater than or equal to, less than or equal to) \((\# k^\prime)^{w/2}\text{.}\)
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Remark 16.1.13.

The construction of an embedding \(\iota\) as above is not at all effective: it depends on the axiom of choice. However, while one cannot easily run the proof of Weil II without making such an artificial choice, in practice one only ever applies such an embedding to algebraic numbers, for which it is much less exotic: it amounts to choosing a place of \(\overline{\QQ}\) above \(\ell\text{.}\)
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In its simplest form, “Weil II” is the following statement.
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Remark 16.1.15.

By Poincaré duality, if \(U\) is proper, then we get that \(H^1_c(U_{\overline{k}})\) is \(\ell\)-pure of weight \(w+1\text{.}\)
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