Introduction
These are the notes of the second part of a course on étale cohomology taught by Johan de Jong at Columbia University in the Fall of 2009. The original note takers were Thibaut Pugin, Zachary Maddock and Min Lee. Over time we will add references to background material in the rest of the Stacks project and provide rigorous proofs of all the statements.
The trace formula
A typical course in étale cohomology would normally state and prove the proper and smooth base change theorems, purity and Poincaré duality. All of these can be found in [SGA4.5, Arcata]. Instead, we are going to study the trace formula for the frobenius, following the account of Deligne in [SGA4.5, Rapport]. We will only look at dimension 1, but using proper base change this is enough for the general case. Since all the cohomology groups considered will be étale, we drop the subscript \(_\etale\). Let us now describe the formula we are after. Let \(X\) be a finite type scheme of dimension 1 over a finite field \(k\), \(\ell\) a prime number and \(\mathcal{F}\) a constructible, flat \(\mathbf{Z}/\ell^n\mathbf{Z}\) sheaf. Then [03SK]\[\begin{equation} \sum\nolimits_{x \in X(k)} \text{Tr}(\text{Frob} | \mathcal{F}_{\bar x}) = \sum\nolimits_{i = 0}^2 (-1)^i \text{Tr}(\pi_X^* | H^i_c(X \otimes_k \bar k, \mathcal{F})) \end{equation}\] as elements of \(\mathbf{Z}/\ell^n\mathbf{Z}\). As we will see, this formulation is slightly wrong as stated. Let us nevertheless describe the symbols that occur therein.
Frobenii
In this section we will prove a “baffling” theorem. A topological analogue of the baffling theorem is the following.
Exercise
Let \(X\) be a topological space and \(g : X \to X\) a continuous map such that \(g^{-1}(U) = U\) for all opens \(U\) of \(X\). Then \(g\) induces the identity on cohomology on \(X\) (for any coefficients).
We now turn to the statement for the étale site.
Lemma
Let \(X\) be a scheme and \(g : X \to X\) a morphism. Assume that for all \(\varphi : U \to X\) étale, there is an isomorphism \[\xymatrix{ U \ar[rd]_\varphi \ar[rr]^-\sim & & {U \times_{\varphi, X, g} X} \ar[ld]^{\text{pr}_2} \\ & X }\] functorial in \(U\). Then \(g\) induces the identity on cohomology (for any sheaf).
Proof
The proof is formal and without difficulty.
Please see Varieties, Section 0CC6 for a discussion of different variants of the Frobenius morphism.
Theorem
Let \(X\) be a scheme in characteristic \(p > 0\). Then the absolute frobenius induces (by pullback) the trivial map on cohomology, i.e., for all integers \(j\geq 0\), \[F_X^* : H^j (X, \underline{\mathbf{Z}/n\mathbf{Z}}) \longrightarrow H^j (X, \underline{\mathbf{Z}/n\mathbf{Z}})\] is the identity.
This theorem is purely formal. It is a good idea, however, to review how to compute the pullback of a cohomology class. Let us simply say that in the case where cohomology agrees with Čech cohomology, it suffices to pull back (using the fiber products on a site) the Čech cocycles. The general case is quite technical, see Hypercoverings, Theorem 01H0. To prove the theorem, we merely verify that the assumption of Lemma 03SP holds for the frobenius.
Proof
We need to verify the existence of a functorial isomorphism as above. For an étale morphism \(\varphi : U \to X\), consider the diagram \[\xymatrix{ U \ar@{-->}[rd] \ar@/^1pc/[rrd]^{F_U} \ar@/_1pc/[rdd]_\varphi \\ & {U \times_{\varphi, X, F_X} X} \ar[r]_-{\text{pr}_1} \ar[d]^{\text{pr}_2} & U \ar[d]^\varphi \\ & X \ar[r]^{F_X} & X. }\] The dotted arrow is an étale morphism and a universal homeomorphism, so it is an isomorphism. See Étale Morphisms, Lemma 0EBS.
Definition
Let \(k\) be a finite field with \(q = p^f\) elements. Let \(X\) be a scheme over \(k\). The geometric frobenius of \(X\) is the morphism \(\pi_X : X \to X\) over \(\Spec(k)\) which equals \(F_X^f\).
Since \(\pi_X\) is a morphism over \(k\), we can base change it to any scheme over \(k\). In particular we can base change it to the algebraic closure \(\bar k\) and get a morphism \(\pi_X : X_{\bar k} \to X_{\bar k}\). Using \(\pi_X\) also for this base change should not be confusing as \(X_{\bar k}\) does not have a geometric frobenius of its own.
Lemma
Let \(\mathcal{F}\) be a sheaf on \(X_\etale\). Then there are canonical isomorphisms \(\pi_X^{-1} \mathcal{F} \cong \mathcal{F}\) and \(\mathcal{F} \cong {\pi_X}_*\mathcal{F}\).
This is false for the fppf site.
Proof
Let \(\varphi : U \to X\) be étale. Recall that \({\pi_X}_* \mathcal{F} (U) = \mathcal{F} (U \times_{\varphi, X, \pi_X} X)\). Since \(\pi_X = F_X^f\), it follows from the proof of Theorem 03SN that there is a functorial isomorphism \[\xymatrix{ U \ar[rd]_{\varphi} \ar[rr]_-{\gamma_U} & & U \times_{\varphi, X, \pi_X} X \ar[ld]^{\text{pr}_2} \\ & X }\] where \(\gamma_U = (\varphi, F_U^f)\). Now we define an isomorphism \[\mathcal{F} (U) \longrightarrow {\pi_X}_* \mathcal{F} (U) = \mathcal{F} (U \times_{\varphi, X, \pi_X} X)\] by taking the restriction map of \(\mathcal{F}\) along \(\gamma_U^{-1}\). The other isomorphism is analogous.
Remark
It may or may not be the case that \(F^f_U\) equals \(\pi_U\).
We continue discussion cohomology of sheaves on our scheme \(X\) over the finite field \(k\) with \(q = p^f\) elements. Fix an algebraic closure \(\bar k\) of \(k\) and write \(G_k = \text{Gal}(\bar k/k)\) for the absolute Galois group of \(k\). Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). We will define a left \(G_k\)-module structure cohomology group \(H^j (X_{\bar k}, \mathcal{F}|_{X_{\bar k}})\) as follows: if \(\sigma \in G_k\), the diagram \[\xymatrix{ X_{\bar k} \ar[rd] \ar[rr]^{\Spec(\sigma) \times \text{id}_X} & & X_{\bar k} \ar[ld] \\ & X }\] commutes. Thus we can set, for \(\xi \in H^j (X_{\bar k}, \mathcal{F}|_{X_{\bar k}})\) \[\sigma \cdot \xi := (\Spec(\sigma) \times \text{id}_X)^*\xi \in H^j(X_{\bar k}, (\Spec(\sigma) \times \text{id}_X)^{-1} \mathcal{F}|_{X_{\bar k}}) = H^j (X_{\bar k}, \mathcal{F}|_{X_{\bar k}}),\] where the last equality follows from the commutativity of the previous diagram. This endows the latter group with the structure of a \(G_k\)-module.
Lemma
In the situation above denote \(\alpha : X \to \Spec(k)\) the structure morphism. Consider the stalk \((R^j\alpha_*\mathcal{F})_{\Spec(\bar k)}\) endowed with its natural Galois action as in Étale Cohomology, Section 03QW. Then the identification \[(R^j\alpha_*\mathcal{F})_{\Spec(\bar k)} \cong H^j (X_{\bar k}, \mathcal{F}|_{X_{\bar k}})\] from Étale Cohomology, Theorem 03Q9 is an isomorphism of \(G_k\)-modules.
A similar result holds comparing \((R^j\alpha_!\mathcal{F})_{\Spec(\bar k)}\) with \(H^j_c (X_{\bar k}, \mathcal{F}|_{X_{\bar k}})\).
Proof
Omitted.
Definition
The arithmetic frobenius is the map \(\text{frob}_k : \bar k \to \bar k\), \(x \mapsto x^q\) of \(G_k\).
Theorem
Let \(\mathcal{F}\) be an abelian sheaf on \(X_\etale\). Then for all \(j\geq 0\), \(\text{frob}_k\) acts on the cohomology group \(H^j(X_{\bar k}, \mathcal{F}|_{X_{\bar k}})\) as the inverse of the map \(\pi_X^*\).
The map \(\pi_X^*\) is defined by the composition \[H^j(X_{\bar k}, \mathcal{F}|_{X_{\bar k}}) \xrightarrow{{\pi_X}_{\bar k}^*} H^j(X_{\bar k}, (\pi_X^{-1} \mathcal{F})|_{X_{\bar k}}) \cong H^j(X_{\bar k}, \mathcal{F}|_{X_{\bar k}}).\] where the last isomorphism comes from the canonical isomorphism \(\pi_X^{-1} \mathcal{F} \cong \mathcal{F}\) of Lemma 03SR.
Proof
The composition \(X_{\bar k} \xrightarrow{\Spec(\text{frob}_k)} X_{\bar k} \xrightarrow{\pi_X} X_{\bar k}\) is equal to \(F_{X_{\bar k}}^f\), hence the result follows from the baffling theorem suitably generalized to nontrivial coefficients. Note that the previous composition commutes in the sense that \(F_{X_{\bar k}}^f = \pi_X \circ \Spec(\text{frob}_k) = \Spec(\text{frob}_k) \circ \pi_X\).
Definition
If \(x \in X(k)\) is a rational point and \(\bar x : \Spec(\bar k) \to X\) the geometric point lying over \(x\), we let \(\pi_x : \mathcal{F}_{\bar x} \to \mathcal{F}_{\bar x}\) denote the action by \(\text{frob}_k^{-1}\) and call it the geometric frobenius1
We can now make a more precise statement (albeit a false one) of the trace formula (03SK). Let \(X\) be a finite type scheme of dimension 1 over a finite field \(k\), \(\ell\) a prime number and \(\mathcal{F}\) a constructible, flat \(\mathbf{Z}/\ell^n\mathbf{Z}\) sheaf. Then [03SX]\[\begin{equation} \sum\nolimits_{x \in X(k)} \text{Tr}(\pi_x | \mathcal{F}_{\bar x}) = \sum\nolimits_{i = 0}^2 (-1)^i \text{Tr}(\pi_X^* | H^i_c(X_{\bar k}, \mathcal{F})) \end{equation}\] as elements of \(\mathbf{Z}/\ell^n\mathbf{Z}\). The reason this equation is wrong is that the trace in the right-hand side does not make sense for the kind of sheaves considered. Before addressing this issue, we try to motivate the appearance of the geometric frobenius (apart from the fact that it is a natural morphism!).
Let us consider the case where \(X = \mathbf{P}^1_k\) and \(\mathcal{F} = \underline{\mathbf{Z}/\ell\mathbf{Z}}\). For any point, the Galois module \(\mathcal{F}_{\bar x}\) is trivial, hence for any morphism \(\varphi\) acting on \(\mathcal{F}_{\bar x}\), the left-hand side is \[\sum\nolimits_{x \in X(k)} \text{Tr}(\varphi | \mathcal{F}_{\bar x}) = \#\mathbf{P}^1_k(k) = q+1.\] Now \(\mathbf{P}^1_k\) is proper, so compactly supported cohomology equals standard cohomology, and so for a morphism \(\pi : \mathbf{P}^1_k \to \mathbf{P}^1_k\), the right-hand side equals \[\text{Tr}(\pi^* | H^0 (\mathbf{P}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}})) + \text{Tr}(\pi^* | H^2 (\mathbf{P}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}})).\] The Galois module \(H^0 (\mathbf{P}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}}) = \mathbf{Z}/\ell\mathbf{Z}\) is trivial, since the pullback of the identity is the identity. Hence the first trace is 1, regardless of \(\pi\). For the second trace, we need to compute the pullback \(\pi^* : H^2(\mathbf{P}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}}))\) for a map \(\pi : \mathbf{P}^1_{\bar k} \to \mathbf{P}^1_{\bar k}\). This is a good exercise and the answer is multiplication by the degree of \(\pi\) (for a proof see Étale Cohomology, Lemma 0AMB). In other words, this works as in the familiar situation of complex cohomology. In particular, if \(\pi\) is the geometric frobenius we get \[\text{Tr}(\pi_X^* | H^2 (\mathbf{P}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}})) = q\] and if \(\pi\) is the arithmetic frobenius then we get \[\text{Tr}(\text{frob}_k^* | H^2 (\mathbf{P}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}})) = q^{-1}.\] The latter option is clearly wrong.
Remark
The computation of the degrees can be done by lifting (in some obvious sense) to characteristic 0 and considering the situation with complex coefficients. This method almost never works, since lifting is in general impossible for schemes which are not projective space.
The question remains as to why we have to consider compactly supported cohomology. In fact, in view of Poincaré duality, it is not strictly necessary for smooth varieties, but it involves adding in certain powers of \(q\). For example, let us consider the case where \(X = \mathbf{A}^1_k\) and \(\mathcal{F} = \underline{\mathbf{Z}/\ell\mathbf{Z}}\). The action on stalks is again trivial, so we only need look at the action on cohomology. But then \(\pi_X^*\) acts as the identity on \(H^0(\mathbf{A}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}})\) and as multiplication by \(q\) on \(H^2_c(\mathbf{A}^1_{\bar k}, \underline{\mathbf{Z}/\ell\mathbf{Z}})\).
Traces
We now explain how to take the trace of an endomorphism of a module over a noncommutative ring. Fix a finite ring \(\Lambda\) with cardinality prime to \(p\). Typically, \(\Lambda\) is the group ring \((\mathbf{Z}/\ell^n\mathbf{Z})[G]\) for some finite group \(G\). By convention, all the \(\Lambda\)-modules considered will be left \(\Lambda\)-modules.
We introduce the following notation: We set \(\Lambda^\natural\) to be the quotient of \(\Lambda\) by its additive subgroup generated by the commutators (i.e., the elements of the form \(ab-ba\), \(a, b \in \Lambda\)). Note that \(\Lambda^\natural\) is not a ring.
For instance, the module \((\mathbf{Z}/\ell^n\mathbf{Z})[G]^\natural\) is the dual of the class functions, so \[(\mathbf{Z}/\ell^n\mathbf{Z})[G]^\natural = \bigoplus\nolimits_{\text{conjugacy classes of }G} \mathbf{Z}/\ell^n\mathbf{Z}.\] For a free \(\Lambda\)-module, we have \(\text{End}_\Lambda(\Lambda^{\oplus m}) = \text{Mat}_m(\Lambda)\). Note that since the modules are left modules, representation of endomorphism by matrices is a right action: if \(a \in \text{End}(\Lambda^{\oplus m})\) has matrix \(A\) and \(v \in \Lambda\), then \(a(v) = v A\).
Definition
The trace of the endomorphism \(a\) is the sum of the diagonal entries of a matrix representing it. This defines an additive map \(\text{Tr} : \text{End}_\Lambda(\Lambda^{\oplus m}) \to \Lambda^\natural\).
Exercise
Given maps \[\Lambda^{\oplus m} \xrightarrow{a} \Lambda^{\oplus n} \quad\text{and}\quad \Lambda^{\oplus n} \xrightarrow{b} \Lambda^{\oplus m}\] show that \(\text{Tr}(ab) = \text{Tr}(ba)\).
We extend the definition of the trace to a finite projective \(\Lambda\)-module \(P\) and an endomorphism \(\varphi\) of \(P\) as follows. Write \(P\) as the summand of a free \(\Lambda\)-module, i.e., consider maps \(P \xrightarrow{a} \Lambda^{\oplus n} \xrightarrow{b} P\) with
\(\Lambda^{\oplus n} = \Im(a) \oplus \Ker(b)\); and
\(b\circ a = \text{id}_P\).
Then we set \(\text{Tr}(\varphi) = \text{Tr}(a\varphi b)\). It is easy to check that this is well-defined, using the previous exercise.
Why derived categories?
With this definition of the trace, let us now discuss another issue with the formula as stated. Let \(C\) be a smooth projective curve over \(k\). Then there is a correspondence between finite locally constant sheaves \(\mathcal{F}\) on \(C_\etale\) whose stalks are isomorphic to \({(\mathbf{Z}/\ell^n\mathbf{Z})}^{\oplus m}\) on the one hand, and continuous representations \(\rho : \pi_1 (C, \bar c) \to \text{GL}_m(\mathbf{Z}/\ell^n\mathbf{Z}))\) (for some fixed choice of \(\bar c\)) on the other hand. We denote \(\mathcal{F}_\rho\) the sheaf corresponding to \(\rho\). Then \(H^2 (C_{\bar k}, \mathcal{F}_\rho)\) is the group of coinvariants for the action of \(\rho(\pi_1 (C, \bar c))\) on \({(\mathbf{Z}/\ell^n\mathbf{Z})}^{\oplus m}\), and there is a short exact sequence \[0 \longrightarrow \pi_1 (C_{\bar k}, \bar c) \longrightarrow \pi_1 (C, \bar c) \longrightarrow G_k \longrightarrow 0.\] For instance, let \(\mathbf{Z} = \mathbf{Z} \sigma\) act on \(\mathbf{Z}/\ell^2\mathbf{Z}\) via \(\sigma(x) = (1+\ell) x\). The coinvariants are \((\mathbf{Z}/\ell^2\mathbf{Z})_{\sigma} = \mathbf{Z}/\ell\mathbf{Z}\), which is not a flat \(\mathbf{Z}/\ell^2\mathbf{Z}\)-module. Hence we cannot take the trace of some action on \(H^2(C_{\bar k}, \mathcal{F}_\rho)\), at least not in the sense of the previous section.
In fact, our goal is to consider a trace formula for \(\ell\)-adic coefficients. But \(\mathbf{Q}_\ell = \mathbf{Z}_\ell[1/\ell]\) and \(\mathbf{Z}_\ell = \lim \mathbf{Z}/\ell^n\mathbf{Z}\), and even for a flat \(\mathbf{Z}/\ell^n\mathbf{Z}\) sheaf, the individual cohomology groups may not be flat, so we cannot compute traces. One possible remedy is consider the total derived complex \(R\Gamma(C_{\bar k}, \mathcal{F}_\rho)\) in the derived category \(D(\mathbf{Z}/\ell^n\mathbf{Z})\) and show that it is a perfect object, which means that it is quasi-isomorphic to a finite complex of finite free module. For such complexes, we can define the trace, but this will require an account of derived categories.
Derived categories
To set up notation, let \(\mathcal{A}\) be an abelian category. Let \(\text{Comp}(\mathcal{A})\) be the abelian category of complexes in \(\mathcal{A}\). Let \(K(\mathcal{A})\) be the category of complexes up to homotopy, with objects equal to complexes in \(\mathcal{A}\) and morphisms equal to homotopy classes of morphisms of complexes. This is not an abelian category. Loosely speaking, \(D(A)\) is defined to be the category obtained by inverting all quasi-isomorphisms in \(\text{Comp}(\mathcal{A})\) or, equivalently, in \(K(\mathcal{A})\). Moreover, we can define \(\text{Comp}^+(\mathcal{A}), K^+(\mathcal{A}), D^+(\mathcal{A})\) analogously using only bounded below complexes. Similarly, we can define \(\text{Comp}^-(\mathcal{A}), K^-(\mathcal{A}), D^-(\mathcal{A})\) using bounded above complexes, and we can define \(\text{Comp}^b(\mathcal{A}), K^b(\mathcal{A}), D^b(\mathcal{A})\) using bounded complexes.
Remark
Notes on derived categories.
There are some set-theoretical problems when \(\mathcal{A}\) is somewhat arbitrary, which we will happily disregard.
The categories \(K(A)\) and \(D(A)\) are endowed with the structure of a triangulated category.
The categories \(\text{Comp}(\mathcal{A})\) and \(K(\mathcal{A})\) can also be defined when \(\mathcal{A}\) is an additive category.
The homology functor \(H^i : \text{Comp}(\mathcal{A}) \to \mathcal{A}\) taking a complex \(K^\bullet \mapsto H^i(K^\bullet)\) extends to functors \(H^i : K(\mathcal{A}) \to \mathcal{A}\) and \(H^i : D(\mathcal{A}) \to \mathcal{A}\).
Lemma
An object \(E\) of \(D(\mathcal{A})\) is contained in \(D^+(\mathcal{A})\) if and only if \(H^i(E) =0\) for all \(i \ll 0\). Similar statements hold for \(D^-\) and \(D^b\).
Proof
Hint: use truncation functors. See Derived Categories, Lemma 05RV.
Lemma
Morphisms between objects in the derived category.
Let \(I^\bullet \in \text{Comp}^+(\mathcal{A})\) with \(I^n\) injective for all \(n \in \mathbf{Z}\). Then \[\Hom_{D(\mathcal{A})}(K^\bullet, I^\bullet) = \Hom_{K(\mathcal{A})}(K^\bullet, I^\bullet).\]
Let \(P^\bullet \in \text{Comp}^-(\mathcal{A})\) with \(P^n\) is projective for all \(n \in \mathbf{Z}\). Then \[\Hom_{D(\mathcal{A})}(P^\bullet, K^\bullet) = \Hom_{K(\mathcal{A})}(P^\bullet, K^\bullet).\]
If \(\mathcal{A}\) has enough injectives and \(\mathcal{I} \subset \mathcal{A}\) is the additive subcategory of injectives, then \(D^+(\mathcal{A})\cong K^+(\mathcal{I})\) (as triangulated categories).
If \(\mathcal{A}\) has enough projectives and \(\mathcal{P} \subset \mathcal{A}\) is the additive subcategory of projectives, then \(D^-(\mathcal{A}) \cong K^-(\mathcal{P}).\)
Proof
Omitted.
Definition
Let \(F: \mathcal{A} \to \mathcal{B}\) be a left exact functor and assume that \(\mathcal{A}\) has enough injectives. We define the total right derived functor of \(F\) as the functor \(RF: D^+(\mathcal{A}) \to D^+(\mathcal{B})\) fitting into the diagram \[\xymatrix{ D^+(\mathcal{A}) \ar[r]^{RF} & D^+(\mathcal{B}) \\ K^+(\mathcal I) \ar[u] \ar[r]^F & K^+(\mathcal{B}). \ar[u] }\] This is possible since the left vertical arrow is invertible by the previous lemma. Similarly, let \(G: \mathcal{A} \to \mathcal{B}\) be a right exact functor and assume that \(\mathcal{A}\) has enough projectives. We define the total left derived functor of \(G\) as the functor \(LG: D^-(\mathcal{A}) \to D^-(\mathcal{B})\) fitting into the diagram \[\xymatrix{ D^-(\mathcal{A}) \ar[r]^{LG} & D^-(\mathcal{B}) \\ K^-(\mathcal{P}) \ar[u] \ar[r]^G & K^-(\mathcal{B}). \ar[u] }\] This is possible since the left vertical arrow is invertible by the previous lemma.
Remark
In these cases, it is true that \(R^iF(K^\bullet) = H^i(RF(K^\bullet))\), where the left hand side is defined to be \(i\)th homology of the complex \(F(K^\bullet)\).
Filtered derived category
It turns out we have to do it all again and build the filtered derived category also.
Definition
Let \(\mathcal{A}\) be an abelian category.
Let \(\text{Fil}(\mathcal{A})\) be the category of filtered objects \((A, F)\) of \(\mathcal{A}\), where \(F\) is a filtration of the form \[A \supset \ldots \supset F^n A \supset F^{n+1}A \supset \ldots \supset 0.\] This is an additive category.
We denote \(\text{Fil}^f(\mathcal{A})\) the full subcategory of \(\text{Fil}(\mathcal{A})\) whose objects \((A, F)\) have finite filtration. This is also an additive category.
An object \(I \in \text{Fil}^f(\mathcal{A})\) is called filtered injective (respectively projective) provided that \(\text{gr}^p(I) = \text{gr}_F^p(I) = F^pI/F^{p+1}I\) is injective (resp. projective) in \(\mathcal{A}\) for all \(p\).
The category of complexes \(\text{Comp}(\text{Fil}^f(\mathcal{A})) \supset \text{Comp}^+(\text{Fil}^f(\mathcal{A}))\) and its homotopy category \(K(\text{Fil}^f(\mathcal{A})) \supset K^+(\text{Fil}^f(\mathcal A))\) are defined as before.
A morphism \(\alpha : K^\bullet \to L^\bullet\) of complexes in \(\text{Comp}(\text{Fil}^f(\mathcal{A}))\) is called a filtered quasi-isomorphism provided that \[\text{gr}^p(\alpha): \text{gr}^p(K^\bullet) \to \text{gr}^p(L^\bullet)\] is a quasi-isomorphism for all \(p \in \mathbf{Z}\).
We define \(DF(\mathcal{A})\) (resp. \(DF^+(\mathcal{A})\)) by inverting the filtered quasi-isomorphisms in \(K(\text{Fil}^f(\mathcal{A}))\) (resp. \(K^+(\text{Fil}^f(\mathcal{A}))\)).
Lemma
If \(\mathcal{A}\) has enough injectives, then \(DF^+(\mathcal{A}) \cong K^+(\mathcal{I})\), where \(\mathcal{I}\) is the full additive subcategory of \(\text{Fil}^f(\mathcal{A})\) consisting of filtered injective objects. Similarly, if \(\mathcal{A}\) has enough projectives, then \(DF^-(\mathcal{A}) \cong K^-(\mathcal{P})\), where \(\mathcal P\) is the full additive subcategory of \(\text{Fil}^f(\mathcal{A})\) consisting of filtered projective objects.
Proof
Omitted.
Filtered derived functors
And then there are the filtered derived functors.
Definition
Let \(T: \mathcal{A} \to \mathcal{B}\) be a left exact functor and assume that \(\mathcal{A}\) has enough injectives. Define \(RT: DF^+(\mathcal{A}) \to D F^+(\mathcal{B})\) to fit in the diagram \[\xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} & DF^+(\mathcal{B}) \\ K^+(\mathcal{I}) \ar[u] \ar[r]^{T \quad} & K^+(\text{Fil}^f(\mathcal{B})). \ar[u]}\] This is well-defined by the previous lemma. Let \(G: \mathcal{A} \to \mathcal{B}\) be a right exact functor and assume that \(\mathcal{A}\) has enough projectives. Define \(LG: DF^-(\mathcal{A}) \to DF^-(\mathcal{B})\) to fit in the diagram \[\xymatrix{ DF^-(\mathcal{A}) \ar[r]^{LG} & DF^-(\mathcal{B}) \\ K^-(\mathcal{P}) \ar[u] \ar[r]^{G \quad} & K^-(\text{Fil}^f(\mathcal{B})). \ar[u]}\] Again, this is well-defined by the previous lemma. The functors \(RT\), resp. \(LG\), are called the filtered derived functor of \(T\), resp. \(G\).
Proposition
In the situation above, we have \[\text{gr}^p \circ RT = RT \circ \text{gr}^p\] where the \(RT\) on the left is the filtered derived functor while the one on the right is the total derived functor. That is, there is a commuting diagram \[\xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} \ar[d]_{\text{gr}^p} & DF^+(\mathcal{B}) \ar[d]^{\text{gr}^p}\\ D^+(\mathcal{A}) \ar[r]^{RT} & D^+(\mathcal{B}).}\]
Proof
Omitted.
Given \(K^\bullet \in DF^+(\mathcal{B})\), we get a spectral sequence \[E_1^{p, q} = H^{p+q}(\text{gr}^p K^\bullet) \Rightarrow H^{p+q}(\text{forget filt}(K^\bullet)).\]
Application of filtered complexes
Let \(\mathcal{A}\) be an abelian category with enough injectives, and \(0 \to L \to M \to N \to 0\) a short exact sequence in \(\mathcal{A}\). Consider \(\widetilde M \in \text{Fil}^f(\mathcal{A})\) to be \(M\) along with the filtration defined by \[F^1M = L, \ F^nM = M \text{ for }n \leq 0\text{, and }F^nM = 0\text{ for }n \geq 2.\] By definition, we have \[\text{forget filt}(\widetilde M) = M, \quad \text{gr}^0(\widetilde M) = N, \quad \text{gr}^1(\widetilde M) = L\] and \(\text{gr}^n(\widetilde M) = 0\) for all other \(n \neq 0, 1\). Let \(T: \mathcal{A} \to \mathcal{B}\) be a left exact functor. Assume that \(\mathcal{A}\) has enough injectives. Then \(RT(\widetilde M) \in DF^+(\mathcal{B})\) is a filtered complex with \[\text{gr}^p(RT(\widetilde M)) \stackrel{\text{qis}}{=} \left\{ \begin{matrix} 0 & \text{if} & p \neq 0, 1, \\ RT(N) & \text{if} & p = 0, \\ RT(L) & \text{if} & p = 1. \end{matrix} \right.\] and \(\text{forget filt}(RT(\widetilde M))\stackrel{\text{qis}}{ = } RT(M)\). The spectral sequence applied to \(RT(\widetilde M)\) gives \[E_1^{p, q} = R^{p+q}T(\text{gr}^p(\widetilde M)) \Rightarrow R^{p+q}T(\text{forget filt}(\widetilde M)).\] Unwinding the spectral sequence gives us the long exact sequence \[\xymatrix{ 0 \ar[r] & T(L) \ar[r] & T(M) \ar[r] & T(N) \ar@(rd, ul)[rdllllr] \\ & R^1T(L) \ar[r] & R^1T(M) \ar[r] & \ldots }\] This will be used as follows. Let \(X/k\) be a scheme of finite type. Let \(\mathcal{F}\) be a flat constructible \(\mathbf{Z}/\ell^n \mathbf{Z}\)-module. Then we want to show that the trace \[\text{Tr}( \pi_X^\ast | R\Gamma_c(X_{\bar k}, \mathcal{F})) \in \mathbf{Z}/\ell^n \mathbf{Z}\] is additive on short exact sequences. To see this, it will not be enough to work with \(R\Gamma_c(X_{\bar k}, -) \in D^+(\mathbf{Z}/\ell^n \mathbf{Z})\), but we will have to use the filtered derived category.
Perfectness
Let \(\Lambda\) be a (possibly noncommutative) ring, \(\text{Mod}_{\Lambda}\) the category of left \(\Lambda\)-modules, \(K(\Lambda) = K(\text{Mod}_\Lambda)\) its homotopy category, and \(D(\Lambda)= D(\text{Mod}_\Lambda)\) the derived category.
Definition
We denote by \(K_{perf}(\Lambda)\) the category whose objects are bounded complexes of finite projective \(\Lambda\)-modules, and whose morphisms are morphisms of complexes up to homotopy. The functor \(K_{perf}(\Lambda)\to D(\Lambda)\) is fully faithful (Derived Categories, Lemma 064B). Denote \(D_{perf}(\Lambda)\) its essential image. An object of \(D(\Lambda)\) is called perfect if it is in \(D_{perf}(\Lambda)\).
Proposition
Let \(K\in D_{perf}(\Lambda)\) and \(f\in \text{End}_{D(\Lambda)}(K)\). Then the trace \(\text{Tr}(f)\in \Lambda^\natural\) is well defined.
Proof
We will use Derived Categories, Lemma 064B without further mention in this proof. Let \(P^\bullet\) be a bounded complex of finite projective \(\Lambda\)-modules and let \(\alpha : P^\bullet \to K\) be an isomorphism in \(D(\Lambda)\). Then \(\alpha^{-1}\circ f\circ \alpha\) corresponds to a morphism of complexes \(f^\bullet : P^\bullet \to P^\bullet\) well defined up to homotopy. Set \[\text{Tr}(f) = \sum_i (-1)^i \text{Tr}(f^i : P^i \to P^i) \in \Lambda^\natural.\] Given \(P^\bullet\) and \(\alpha\), this is independent of the choice of \(f^\bullet\). Namely, any other choice is of the form \(\tilde{f}^\bullet = f^\bullet + dh +hd\) for some \(h^i : P^i \to P^{i-1}(i\in \mathbf{Z})\). But \[\begin{eqnarray*} \text{Tr}(dh) & = & \sum_i (-1)^i \text{Tr}(P^i\xrightarrow{dh} P^i) \\ & = & \sum_i (-1)^i \text{Tr}(P^{i-1}\xrightarrow{hd} P^{i-1}) \\ & = & -\sum_i (-1)^{i-1}\text{Tr}(P^{i-1}\xrightarrow{hd} P^{i-1}) \\ & = & - \text{Tr}(hd) \end{eqnarray*}\] and so \(\sum_i (-1)^i \text{Tr} ((dh+hd)|_{P^i})=0\). Furthermore, this is independent of the choice of \((P^\bullet , \alpha)\): suppose \((Q^\bullet, \beta)\) is another choice. The compositions \[Q^\bullet \xrightarrow{\beta} K \xrightarrow{\alpha^{-1}} P^\bullet \quad\text{and}\quad P^\bullet \xrightarrow{\alpha} K \xrightarrow{\beta^{-1}} Q^\bullet\] are representable by morphisms of complexes \(\gamma_1^\bullet\) and \(\gamma_2^\bullet\) respectively, such that \(\gamma_1^\bullet \circ \gamma_2^\bullet\) is homotopic to the identity. Thus, the morphism of complexes \(\gamma_2^\bullet\circ f^\bullet\circ \gamma_1^\bullet : Q^\bullet\to Q^\bullet\) represents the morphism \(\beta^{-1}\circ f\circ\beta\) in \(D(\Lambda)\). Now \[\begin{eqnarray*} \text{Tr}(\gamma_2^\bullet\circ f^\bullet\circ\gamma_1^\bullet|_{Q^\bullet}) & = & \text{Tr}(\gamma_1^\bullet \circ\gamma_2^\bullet \circ f^\bullet|_{P^\bullet})\\ & = & \text{Tr}(f^\bullet|_{P^\bullet}) \end{eqnarray*}\] by the fact that \(\gamma_1^\bullet \circ \gamma_2^\bullet\) is homotopic to the identity and the independence of the choice of \(f^\bullet\) we saw above.
Filtrations and perfect complexes
We now present a filtered version of the category of perfect complexes. An object \((M, F)\) of \(\text{Fil}^f(\text{Mod}_\Lambda)\) is called filtered finite projective if for all \(p\), \(\text{gr}^p_F (M)\) is finite and projective. We then consider the homotopy category \(KF_{\text{perf}}(\Lambda)\) of bounded complexes of filtered finite projective objects of \(\text{Fil}^f(\text{Mod}_\Lambda)\). We have a diagram of categories \[\begin{matrix} KF(\Lambda) & \supset & KF_{\text{perf}}(\Lambda)\\ \downarrow & & \downarrow\\ DF(\Lambda) & \supset & DF_{\text{perf}}(\Lambda) \end{matrix}\] where the vertical functor on the right is fully faithful and the category \(DF_{\text{perf}}(\Lambda)\) is its essential image, as before.
Lemma
Let \(K\in DF_{\text{perf}}(\Lambda)\) and \(f\in \text{End}_{DF}(K)\). Then \[\text{Tr}(f|_K) = \sum\nolimits_{p\in \mathbf{Z}} \text{Tr}(f|_{\text{gr}^p K}).\]
Proof
By Proposition 03TI, we may assume we have a bounded complex \(P^\bullet\) of filtered finite projectives of \(\text{Fil}^f(\text{Mod}_\Lambda)\) and a map \(f^\bullet : P^\bullet\to P^\bullet\) in \(\text{Comp}(\text{Fil}^f(\text{Mod}_\Lambda))\). So the lemma follows from the following result, which proof is left to the reader.
Lemma
Let \(P \in \text{Fil}^f(\text{Mod}_\Lambda)\) be filtered finite projective, and \(f : P \to P\) an endomorphism in \(\text{Fil}^f(\text{Mod}_\Lambda)\). Then \[\text{Tr}(f|_P) = \sum\nolimits_p \text{Tr}(f|_{\text{gr}^p(P)}).\]
Proof
Omitted.
Characterizing perfect objects
For the commutative case see More on Algebra, Sections 064N, 0651, and 0656.
Definition
Let \(\Lambda\) be a (possibly noncommutative) ring. An object \(K\in D(\Lambda)\) has finite \(\text{Tor}\)-dimension if there exist \(a, b \in \mathbf{Z}\) such that for any right \(\Lambda\)-module \(N\), we have \(H^i(N \otimes_{\Lambda}^\mathbf{L} K) = 0\) for all \(i \not \in [a, b]\).
This in particular means that \(K \in D^b(\Lambda)\) as we see by taking \(N = \Lambda\).
Lemma
Let \(\Lambda\) be a left Noetherian ring and \(K\in D(\Lambda)\). Then \(K\) is perfect if and only if the two following conditions hold:
\(K\) has finite \(\text{Tor}\)-dimension, and
for all \(i \in \mathbf{Z}\), \(H^i(K)\) is a finite \(\Lambda\)-module.
Proof
See More on Algebra, Lemma 0658 for the proof in the commutative case.
The reader is strongly urged to try and prove this. The proof relies on the fact that a finite module on a finitely left-presented ring is flat if and only if it is projective.
Remark
A variant of this lemma is to consider a Noetherian scheme \(X\) and the category \(D_{perf}(\mathcal{O}_X)\) of complexes which are locally quasi-isomorphic to a finite complex of finite locally free \(\mathcal{O}_X\)-modules. Objects \(K\) of \(D_{perf}(\mathcal{O}_X)\) can be characterized by having coherent cohomology sheaves and bounded tor dimension.
Cohomology of nice complexes
The following is a special case of a more general result about compactly supported cohomology of objects of \(D_{ctf}(X, \Lambda)\).
Proposition
Let \(X\) be a projective curve over a field \(k\), \(\Lambda\) a finite ring and \(K\in D_{ctf}(X, \Lambda)\). Then \(R\Gamma(X_{\bar k}, K)\in D_{perf}(\Lambda)\).
Proof
The first step is to show:
The cohomology of \(R\Gamma(X_{\bar k}, K)\) is bounded.
Consider the spectral sequence \[H^i(X_{\bar k}, \underline H^j(K)) \Rightarrow H^{i+j} (R\Gamma(X_{\bar k}, K)).\] Since \(K\) is bounded and \(\Lambda\) is finite, the sheaves \(\underline H^j(K)\) are torsion. Moreover, \(X_{\bar k}\) has finite cohomological dimension, so the left-hand side is nonzero for finitely many \(i\) and \(j\) only. Therefore, so is the right-hand side.
The cohomology groups \(H^{i+j} (R\Gamma(X_{\bar k}, K))\) are finite.
Since the sheaves \(\underline H^j(K)\) are constructible, the groups \(H^i(X_{\bar k}, \underline H^j(K))\) are finite (Étale Cohomology, Section 03SB) so it follows by the spectral sequence again.
\(R\Gamma(X_{\bar k}, K)\) has finite \(\text{Tor}\)-dimension.
Let \(N\) be a right \(\Lambda\)-module (in fact, since \(\Lambda\) is finite, it suffices to assume that \(N\) is finite). By the projection formula (change of module), \[N \otimes^\mathbf{L}_\Lambda R \Gamma(X_{\bar k}, K) = R\Gamma(X_{\bar k}, N \otimes^\mathbf{L}_\Lambda K).\] Therefore, \[H^i (N \otimes^\mathbf{L}_\Lambda R\Gamma(X_{\bar k}, K)) = H^i(R\Gamma(X_{\bar k}, N \otimes_{\Lambda}^\mathbf{L} K)).\] Now consider the spectral sequence \[H^i (X_{\bar k}, \underline H^j (N \otimes_{\Lambda}^\mathbf{L} K)) \Rightarrow H^{i+j}(R\Gamma(X_{\bar k}, N \otimes_{\Lambda}^\mathbf{L} K)).\] Since \(K\) has finite \(\text{Tor}\)-dimension, \(\underline H^j (N \otimes_{\Lambda}^\mathbf{L} K)\) vanishes universally for \(j\) small enough, and the left-hand side vanishes whenever \(i < 0\). Therefore \(R\Gamma(X_{\bar k}, K)\) has finite \(\text{Tor}\)-dimension, as claimed. So it is a perfect complex by Lemma 03TO.
Lefschetz numbers
The fact that the total cohomology of a constructible complex of finite tor dimension is a perfect complex is the key technical reason why cohomology behaves well, and allows us to define rigorously the traces occurring in the trace formula.
Definition
Let \(\Lambda\) be a finite ring, \(X\) a projective curve over a finite field \(k\) and \(K \in D_{ctf}(X, \Lambda)\) (for instance \(K = \underline\Lambda\)). There is a canonical map \(c_K : \pi_X^{-1}K \to K\), and its base change \(c_K|_{X_{\bar k}}\) induces an action denoted \(\pi_X^*\) on the perfect complex \(R\Gamma(X_{\bar k}, K|_{X_{\bar k}})\). The global Lefschetz number of \(K\) is the trace \(\text{Tr}(\pi_X^* |_{R\Gamma(X_{\bar k}, K)})\) of that action. It is an element of \(\Lambda^\natural\).
Definition
With \(\Lambda, X, k, K\) as in Definition 03TX. Since \(K\in D_{ctf}(X, \Lambda)\), for any geometric point \(\bar x\) of \(X\), the complex \(K_{\bar x}\) is a perfect complex (in \(D_{perf}(\Lambda)\)). As we have seen in Section 03SL, the Frobenius \(\pi_X\) acts on \(K_{\bar x}\). The local Lefschetz number of \(K\) is the sum \[\sum\nolimits_{x\in X(k)} \text{Tr}(\pi_x |_{K_{\overline{x}}})\] which is again an element of \(\Lambda^\natural\).
At last, we can formulate precisely the trace formula.
Theorem
Let \(X\) be a projective curve over a finite field \(k\), \(\Lambda\) a finite ring and \(K \in D_{ctf}(X, \Lambda)\). Then the global and local Lefschetz numbers of \(K\) are equal, i.e., [03U0]\[\begin{equation} \text{Tr}(\pi^*_X |_{R\Gamma(X_{\bar k}, K)}) = \sum\nolimits_{x\in X(k)} \text{Tr}(\pi_X |_{K_{\bar x}}) \end{equation}\] in \(\Lambda^\natural\).
Proof
See discussion below.
We will use, rather than prove, the trace formula. Nevertheless, we will give quite a few details of the proof of the theorem as given in [SGA4.5] (some of the things that are not adequately explained are listed in Section 03V5).
We only stated the formula for curves, and in some weak sense it is a consequence of the following result.
Theorem
Let \(C\) be a nonsingular projective curve over an algebraically closed field \(k\), and \(\varphi : C \to C\) a \(k\)-endomorphism of \(C\) distinct from the identity. Let \(V(\varphi) = \Delta_C \cdot \Gamma_\varphi\), where \(\Delta_C\) is the diagonal, \(\Gamma_\varphi\) is the graph of \(\varphi\), and the intersection number is taken on \(C \times C\). Let \(J = \underline{\Picardfunctor}^0_{C/k}\) be the jacobian of \(C\) and denote \(\varphi^* : J \to J\) the action induced by \(\varphi\) by taking pullbacks. Then \[V(\varphi) = 1 - \text{Tr}_J(\varphi^*) + \deg \varphi.\]
Proof
The number \(V(\varphi)\) is the number of fixed points of \(\varphi\), it is equal to \[V(\varphi) = \sum\nolimits_{c \in |C| : \varphi(c) = c} m_{\text{Fix}(\varphi)} (c)\] where \(m_{\text{Fix}(\varphi)} (c)\) is the multiplicity of \(c\) as a fixed point of \(\varphi\), namely the order or vanishing of the image of a local uniformizer under \(\varphi - \text{id}_C\). Proofs of this theorem can be found in [Lang] and [Weil].
Example
Let \(C = E\) be an elliptic curve and \(\varphi = [n]\) be multiplication by \(n\). Then \(\varphi^* = \varphi^t\) is multiplication by \(n\) on the jacobian, so it has trace \(2n\) and degree \(n^2\). On the other hand, the fixed points of \(\varphi\) are the points \(p \in E\) such that \(n p = p\), which is the \((n-1)\)-torsion, which has cardinality \((n-1)^2\). So the theorem reads \[(n-1)^2 = 1 - 2n + n^2.\]
Jacobians. We now discuss without proofs the correspondence between a curve and its jacobian which is used in Weil’s proof. Let \(C\) be a nonsingular projective curve over an algebraically closed field \(k\) and choose a base point \(c_0 \in C(k)\). Denote by \(A^1(C \times C)\) (or \(\Pic(C \times C)\), or \(\text{CaCl}(C \times C)\)) the abelian group of codimension 1 divisors of \(C \times C\). Then \[A^1(C \times C) = \text{pr}_1^* (A^1(C)) \oplus \text{pr}_2^* (A^1(C)) \oplus R\] where \[R = \{ Z \in A^1(C \times C) \ | \ Z|_{C \times \{c_0\}} \sim_\text{rat} 0 \text{ and } Z|_{\{c_0\} \times C} \sim_\text{rat} 0 \}.\] In other words, \(R\) is the subgroup of line bundles which pull back to the trivial one under either projection. Then there is a canonical isomorphism of abelian groups \(R \cong \text{End}(J)\) which maps a divisor \(Z\) in \(R\) to the endomorphism \[\begin{matrix} J & \to & J \\ \left[ \mathcal{O}_C(D) \right] & \mapsto & (\text{pr}_1 |_Z)_* (\text{pr}_2 |_Z)^* (D). \end{matrix}\] The aforementioned correspondence is the following. We denote by \(\sigma\) the automorphism of \(C \times C\) that switches the factors. \[\begin{matrix} \hline & \\ \text{End}(J) & R \\ & \\ \hline & \\ \text{composition of }\alpha, \beta & {\text{pr}_{13}}_* ({\text{pr}_{12}}^*(\alpha) \circ {\text{pr}_{23}}^*(\beta)) \\ & \\ \text{id}_J & \Delta_C - \{c_0\} \times C - C \times \{c_0\} \\ & \\ \varphi^* & \Gamma_\varphi - C \times \{\varphi(c_0)\} - \sum_{\varphi(c) = c_0} \{c\} \times C \\ & \\ { \begin{matrix} \text{the trace form} \\ \alpha, \beta \mapsto \text{Tr}(\alpha \beta) \end{matrix} } & \alpha, \beta \mapsto - \int_{C \times C} \alpha . \sigma^*\beta \\ & \\ { \begin{matrix} \text{the Rosati involution} \\ \alpha \mapsto \alpha^\dagger \end{matrix} } & \alpha \mapsto \sigma^*\alpha \\ & \\ { \begin{matrix} \text{positivity of Rosati} \\ \text{Tr}(\alpha\alpha^\dagger) > 0 \end{matrix} } & { \begin{matrix} \text{Hodge index theorem on }C \times C \\ - \int_{C \times C} \alpha \sigma^*\alpha > 0. \end{matrix} } \\ & \\ \hline \end{matrix}\] In fact, in light of the Kunneth formula, the subgroup \(R\) corresponds to the \(1, 1\) hodge classes in \(H^1(C)\otimes H^1(C)\).
Weil’s proof. Using this correspondence, we can prove the trace formula. We have \[\begin{eqnarray*} V(\varphi) & = & \int_{C \times C} \Gamma_\varphi.\Delta \\ & = & \int_{C \times C} \Gamma_\varphi. \left(\Delta_C - \{c_0\} \times C - C \times \{c_0\}\right) + \int_{C \times C} \Gamma_\varphi. \left(\{c_0\} \times C + C \times \{c_0\}\right). \end{eqnarray*}\] Now, on the one hand \[\int_{C \times C} \Gamma_\varphi. \left(\{c_0\} \times C + C \times \{c_0\}\right) = 1 + \deg \varphi\] and on the other hand, since \(R\) is the orthogonal of the ample divisor \(\{c_0\} \times C + C \times \{c_0\}\), \[\begin{eqnarray*} & & \int_{C \times C} \Gamma_\varphi. \left(\Delta_C - \{c_0\} \times C - C \times \{c_0\}\right) \\ & = & \int_{C \times C} \left(\Gamma_\varphi - C \times \{\varphi(c_0)\} - \sum_{\varphi(c) = c_0} \{c\} \times C \right). \left(\Delta_C - \{c_0\} \times C - C \times \{c_0\}\right) \\ & = & - \text{Tr}_J (\varphi^* \circ \text{id}_J). \end{eqnarray*}\] Recapitulating, we have \[V(\varphi) = 1 - \text{Tr}_J (\varphi^*) + \deg \varphi\] which is the trace formula.
Lemma
Consider the situation of Theorem 03U1 and let \(\ell\) be a prime number invertible in \(k\). Then \[\sum\nolimits_{i = 0}^2 (-1)^i \text{Tr}(\varphi^* |_{H^i (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})}) = V(\varphi) \mod \ell^n.\]
Proof
Observe first that the assumption makes sense because \(H^i(C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})\) is a free \(\mathbf{Z}/\ell^n \mathbf{Z}\)-module for all \(i\). The trace of \(\varphi^*\) on the 0th degree cohomology is 1. The choice of a primitive \(\ell^n\)th root of unity in \(k\) gives an isomorphism \[H^i(C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}}) \cong H^i(C, \mu_{\ell^n})\] compatibly with the action of the geometric Frobenius. On the other hand, \(H^1(C, \mu_{\ell^n}) = J[\ell^n]\). Therefore, \[\begin{eqnarray*} \text{Tr}(\varphi^* |_{H^1 (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})})) & = & \text{Tr}_J (\varphi^*) \mod \ell^n \\ & = & \text{Tr}_{\mathbf{Z}/\ell^n \mathbf{Z}} (\varphi^* : J[\ell^n] \to J[\ell^n]). \end{eqnarray*}\] Moreover, \(H^2(C, \mu_{\ell^n}) = \Pic(C)/\ell^n\Pic(C) \cong \mathbf{Z}/\ell^n \mathbf{Z}\) where \(\varphi^*\) is multiplication by \(\deg \varphi\). Hence \[\text{Tr} (\varphi^*|_{H^2 (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})}) = \deg \varphi.\] Thus we have \[\sum_{i = 0}^2 (-1)^i \text{Tr}(\varphi^* |_{H^i (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})}) = 1 - \text{Tr}_J(\varphi^*) + \deg \varphi \mod \ell^n\] and the corollary follows from Theorem 03U1.
An alternative way to prove this corollary is to show that \[X \mapsto H^* (X, \mathbf{Q}_\ell) = \mathbf{Q}_\ell \otimes \lim_n H^*(X, \mathbf{Z}/\ell^n\mathbf{Z})\] defines a Weil cohomology theory on smooth projective varieties over \(k\). Then the trace formula \[V(\varphi) = \sum_{i = 0}^2 (-1)^i \text{Tr}(\varphi^* |_{H^i(C, \mathbf{Q}_\ell)})\] is a formal consequence of the axioms (it’s an exercise in linear algebra, the proof is the same as in the topological case).
Preliminaries and sorites
Notation: We fix the notation for this section. We denote by \(A\) a commutative ring, \(\Lambda\) a (possibly noncommutative) ring with a ring map \(A\to \Lambda\) which image lies in the center of \(\Lambda\). We let \(G\) be a finite group, \(\Gamma\) a monoid extension of \(G\) by \(\mathbf{N}\), meaning that there is an exact sequence \[1\to G\to \tilde\Gamma\to \mathbf{Z}\to 1\] and \(\Gamma\) consists of those elements of \(\tilde\Gamma\) which image is nonnegative. Finally, we let \(P\) be an \(A[\Gamma]\)-module which is finite and projective as an \(A[G]\)-module, and \(M\) a \(\Lambda[\Gamma]\)-module which is finite and projective as a \(\Lambda\)-module.
Our goal is to compute the trace of \(1 \in \mathbf{N}\) acting over \(\Lambda\) on the coinvariants of \(G\) on \(P \otimes_A M\), that is, the number \[\text{Tr}_{\Lambda}\left(1; \left(P \otimes_A M\right)_G\right) \in \Lambda^\natural.\] The element \(1\in \mathbf{N}\) will correspond to the Frobenius.
Lemma
Let \(e\in G\) denote the neutral element. The map \[\begin{matrix} \Lambda[G] & \longrightarrow & \Lambda^{\natural}\\ \sum \lambda_g\cdot g & \longmapsto & \lambda_e \end{matrix}\] factors through \(\Lambda[G]^\natural\). We denote \(\varepsilon : \Lambda[G]^\natural\to \Lambda^\natural\) the induced map.
Proof
We have to show the map annihilates commutators. One has \[\left(\sum\lambda_g g\right)\left(\sum\mu_g g\right)-\left(\sum \mu_g g\right)\left(\sum\lambda_g g\right) = \sum_g\left(\sum_{g_1g_2=g} \lambda_{g_1}\mu_{g_2}-\mu_{g_1}\lambda_{g_2}\right)g\] The coefficient of \(e\) is \[\sum_g\left(\lambda_g\mu_{g^{-1}}-\mu_g\lambda_{g^{-1}}\right) = \sum_g\left(\lambda_g\mu_{g^{-1}}-\mu_{g^{-1}}\lambda_g\right)\] which is a sum of commutators, hence it zero in \(\Lambda^\natural\).
Definition
Let \(f : P\to P\) be an endomorphism of a finite projective \(\Lambda[G]\)-module \(P\). We define \[\text{Tr}_{\Lambda}^G(f; P) := \varepsilon\left(\text{Tr}_{\Lambda[G]}(f; P)\right)\] to be the \(G\)-trace of \(f\) on \(P\).
Lemma
Let \(f : P\to P\) be an endomorphism of the finite projective \(\Lambda[G]\)-module \(P\). Then \[\text{Tr}_{\Lambda}(f; P) = \# G \cdot \text{Tr}_\Lambda^G(f; P).\]
Proof
By additivity, reduce to the case \(P = \Lambda[G]\). In that case, \(f\) is given by right multiplication by some element \(\sum\lambda_g\cdot g\) of \(\Lambda[G]\). In the basis \((g)_{g \in G}\), the matrix of \(f\) has coefficient \(\lambda_{g_2^{-1}g_1}\) in the \((g_1, g_2)\) position. In particular, all diagonal coefficients are \(\lambda_e\), and there are \(\# G\) such coefficients.
Lemma
The map \(A\to \Lambda\) defines an \(A\)-module structure on \(\Lambda^\natural\).
Proof
This is clear.
Lemma
Let \(P\) be a finite projective \(A[G]\)-module and \(M\) a \(\Lambda[G]\)-module, finite projective as a \(\Lambda\)-module. Then \(P \otimes_A M\) is a finite projective \(\Lambda[G]\)-module, for the structure induced by the diagonal action of \(G\).
Note that \(P \otimes_A M\) is naturally a \(\Lambda\)-module since \(M\) is. Explicitly, together with the diagonal action this reads \[\left(\sum\lambda_g g\right)\left(p \otimes m\right) = \sum g p \otimes \lambda_g g m.\]
Proof
For any \(\Lambda[G]\)-module \(N\) one has \[\Hom_{\Lambda[G]}\left(P \otimes_A M, N\right)= \Hom_{A[G]}\left(P, \Hom_{\Lambda}(M, N)\right)\] where the \(G\)-action on \(\Hom_{\Lambda}(M, N)\) is given by \((g\cdot \varphi)(m) = g \varphi (g^{-1} m)\). Now it suffices to observe that the right-hand side is a composition of exact functors, because of the projectivity of \(P\) and \(M\).
Lemma
With assumptions as in Lemma 03U9, let \(u\in \text{End}_{A[G]}(P)\) and \(v\in \text{End}_{\Lambda[G]}(M)\). Then \[\text{Tr}_\Lambda^G \left(u \otimes v; P \otimes_A M\right) = \text{Tr}_A^G(u; P)\cdot \text{Tr}_\Lambda(v;M).\]
Proof
Reduce to the case \(P=A[G]\). In that case, \(u\) is right multiplication by some element \(a = \sum a_gg\) of \(A[G]\), which we write \(u = R_a\). There is an isomorphism of \(\Lambda[G]\)-modules \[\begin{matrix} \varphi : & A[G]\otimes_A M & \cong & \left(A[G]\otimes_A M\right)'\\ & g \otimes m & \longmapsto & g \otimes g^{-1}m \end{matrix}\] where \(\left(A[G]\otimes_A M\right)'\) has the module structure given by the left \(G\)-action, together with the \(\Lambda\)-linearity on \(M\). This transport of structure changes \(u \otimes v\) into \(\sum_ga_gR_g \otimes g^{-1}v\). In other words, \[\varphi \circ (u \otimes v) \circ \varphi^{-1} = \sum_ga_gR_g \otimes g^{-1}v.\] Working out explicitly both sides of the equation, we have to show \[\text{Tr}_\Lambda^G\left(\sum_g a_gR_g \otimes g^{-1}v\right) = a_e\cdot \text{Tr}_\Lambda(v; M).\] This is done by showing that \[\text{Tr}_\Lambda^G\left(a_gR_g \otimes g^{-1}v\right) = \left\{ \begin{matrix} 0 & \text{ if } g\neq e\\ a_e\text{Tr}_\Lambda\left(v; M\right) & \text{ if }g = e \end{matrix} \right.\] by reducing to \(M=\Lambda\).
Notation: Consider the monoid extension \(1 \to G\to \Gamma\to \mathbf{N} \to 1\) and let \(\gamma\in \Gamma\). Then we write \(Z_\gamma = \{g\in G | g\gamma = \gamma g\}\).
Lemma
Let \(P\) be a \(\Lambda[\Gamma]\)-module, finite and projective as a \(\Lambda[G]\)-module, and \(\gamma \in \Gamma\). Then \[\text{Tr}_{\Lambda}(\gamma, P) = \# Z_\gamma \cdot \text{Tr}_\Lambda^{Z_\gamma}\left(\gamma, P\right).\]
Proof
This follows readily from Lemma 03U7.
Lemma
Let \(P\) be an \(A[\Gamma]\)-module, finite projective as \(A[G]\)-module. Let \(M\) be a \(\Lambda[\Gamma]\)-module, finite projective as a \(\Lambda\)-module. Then \[\text{Tr}_{\Lambda}^{Z_\gamma}(\gamma, P \otimes_A M) = \text{Tr}_A^{Z_\gamma}(\gamma, P)\cdot \text{Tr}_\Lambda(\gamma, M).\]
Proof
This follows directly from Lemma 03UA.
Lemma
Let \(P\) be a \(\Lambda[\Gamma]\)-module, finite projective as \(\Lambda[G]\)-module. Then the coinvariants \(P_G = \Lambda \otimes_{\Lambda[G]} P\) form a finite projective \(\Lambda\)-module, endowed with an action of \(\Gamma/G = \mathbf{N}\). Moreover, we have \[\text{Tr}_\Lambda(1; P_G) = \sum\nolimits'_{\gamma \mapsto 1} \text{Tr}_\Lambda^{Z_\gamma}(\gamma, P)\] where \(\sum_{\gamma\mapsto 1}'\) means taking the sum over the \(G\)-conjugacy classes in \(\Gamma\).
Proof
We first prove this after multiplying by \(\# G\). \[\# G\cdot \text{Tr}_\Lambda(1; P_G) = \text{Tr}_\Lambda(\sum\nolimits_{\gamma\mapsto 1} \gamma, P_G) = \text{Tr}_\Lambda(\sum\nolimits_{\gamma\mapsto 1} \gamma, P)\] where the second equality follows by considering the commutative triangle \[\xymatrix{ P^G \ar[rd]_a & & P_G \ar[ll]^c \\ & P \ar[ur]_b }\] where \(a\) is the canonical inclusion, \(b\) the canonical surjection and \(c = \sum_{\gamma \mapsto 1} \gamma\). Then we have \[(\sum\nolimits_{\gamma \mapsto 1} \gamma) |_P = a \circ c \circ b \quad\text{and}\quad (\sum\nolimits_{\gamma \mapsto 1} \gamma) |_{P_G} = b \circ a \circ c\] hence they have the same trace. We then have \[\# G\cdot \text{Tr}_\Lambda(1; P_G) = {\sum_{\gamma\mapsto 1}}' \frac{\# G}{\# Z_\gamma}\text{Tr}_\Lambda(\gamma, P) = \# G{\sum_{\gamma\mapsto 1}}' \text{Tr}_\Lambda^{Z_\gamma}(\gamma, P).\] To finish the proof, reduce to case \(\Lambda\) torsion-free by some universality argument. See [SGA4.5] for details.
Remark
Let us try to illustrate the content of the formula of Lemma 03UC. Suppose that \(\Lambda\), viewed as a trivial \(\Gamma\)-module, admits a finite resolution \(0\to P_r\to \ldots \to P_1 \to P_0\to \Lambda\to 0\) by some \(\Lambda[\Gamma]\)-modules \(P_i\) which are finite and projective as \(\Lambda[G]\)-modules. In that case \[H_*\left(\left(P_\bullet\right)_G\right) = \text{Tor}_*^{\Lambda[G]}\left(\Lambda, \Lambda\right) = H_*(G, \Lambda)\] and \[\text{Tr}_\Lambda^{Z_\gamma}\left(\gamma, P_\bullet\right) =\frac{1}{\# Z_\gamma}\text{Tr}_\Lambda(\gamma, P_\bullet)=\frac{1}{\# Z_\gamma}\text{Tr}(\gamma, \Lambda) = \frac{1}{\# Z_\gamma}.\] Therefore, Lemma 03UC says \[\text{Tr}_\Lambda (1 , P_G) = \text{Tr}\left(1 |_{H_*(G, \Lambda)}\right) = {\sum_{\gamma\mapsto 1}}'\frac{1}{\# Z_\gamma}.\] This can be interpreted as a point count on the stack \(BG\). If \(\Lambda = \mathbf{F}_\ell\) with \(\ell\) prime to \(\# G\), then \(H_*(G, \Lambda)\) is \(\mathbf{F}_\ell\) in degree 0 (and \(0\) in other degrees) and the formula reads \[1 = \sum\nolimits_{ \frac{\sigma\text{-conjugacy}}{\text{classes}\langle\gamma\rangle} } \frac{1}{\# Z_\gamma} \mod \ell.\] This is in some sense a “trivial” trace formula for \(G\). Later we will see that (03U0) can in some cases be viewed as a highly nontrivial trace formula for a certain type of group, see Section 03VO.
Proof of the trace formula
Theorem
Let \(k\) be a finite field and \(X\) a finite type, separated scheme of dimension at most 1 over \(k\). Let \(\Lambda\) be a finite ring whose cardinality is prime to that of \(k\), and \(K\in D_{ctf}(X, \Lambda)\). Then [03UH]\[\begin{equation} \text{Tr}(\pi_X^* |_{R\Gamma_c(X_{\bar k}, K)}) = \sum\nolimits_{x\in X(k)} \text{Tr}(\pi_x |_{K_{\bar x}}) \end{equation}\] in \(\Lambda^{\natural}\).
Please see Remark 03UI for some remarks on the statement. Notation: For short, we write \[T'(X, K) = \sum\nolimits_{x\in X(k)} \text{Tr}(\pi_x |_{K_{\bar x}})\] for the right-hand side of (03UH) and \[T''(X, K) =\text{Tr}(\pi_x^* |_{R\Gamma_c(X_{\bar k}, K)})\] for the left-hand side.
Proof
The proof proceeds in a number of steps.
Step 1. Let \(j : \mathcal{U}\hookrightarrow X\) be an open immersion with complement \(Y = X - \mathcal{U}\) and \(i : Y \hookrightarrow X\). Then \(T''(X, K) = T''(\mathcal{U}, j^{-1} K)+ T''(Y, i^{-1}K)\) and \(T'(X, K) = T'(\mathcal{U}, j^{-1} K)+ T'(Y, i^{-1}K)\).
This is clear for \(T'\). For \(T''\) use the exact sequence \[0\to j_!j^{-1} K \to K \to i_* i^{-1} K \to 0\] to get a filtration on \(K\). This gives rise to an object \(\widetilde K \in DF(X, \Lambda)\) whose graded pieces are \(j_!j^{-1}K\) and \(i_*i^{-1}K\), both of which lie in \(D_{ctf}(X, \Lambda)\). Then, by filtered derived abstract nonsense (INSERT REFERENCE), \(R\Gamma_c(X_{\bar k}, K)\in DF_{perf}(\Lambda)\), and it comes equipped with \(\pi_x^*\) in \(DF_{perf}(\Lambda)\). By the discussion of traces on filtered complexes (INSERT REFERENCE) we get \[\begin{eqnarray*} \text{Tr}(\pi_X^* |_{R\Gamma_c(X_{\bar k}, K)}) & = & \text{Tr}(\pi_X^* |_{R\Gamma_c(X_{\bar k}, j_!j^{-1}K)}) + \text{Tr}(\pi_X^* |_{R\Gamma_c(X_{\bar k}, i_*i^{-1}K)}) \\ & = & T''(U, i^{-1}K) + T''(Y, i^{-1}K). \end{eqnarray*}\]
Step 2. The theorem holds if \(\dim X\leq 0\).
Indeed, in that case \[R\Gamma_c(X_{\bar k}, K) = R\Gamma(X_{\bar k}, K) = \Gamma(X_{\bar k}, K) = \bigoplus\nolimits_{\bar x \in X_{\bar k}} K_{\bar x}\] which comes equipped with the endormophism \(\pi_X^*\). Since the fixed points of \(\pi_X : X_{\bar k}\to X_{\bar k}\) are exactly the points \(\bar x \in X_{\bar k}\) which lie over a \(k\)-rational point \(x \in X(k)\) we get \[\text{Tr}\big(\pi_X^*|_{R\Gamma_c(X_{\bar k}, K)}\big) = \sum\nolimits_{x \in X(k)} \text{Tr}(\pi_x|_{K_{\bar x}}).\] as desired.
Step 3.
It suffices to prove the equality \(T'(\mathcal{U}, \mathcal{F}) = T''(\mathcal{U}, \mathcal{F})\) in the case where
\(\mathcal{U}\) is a smooth irreducible affine curve over \(k\),
\(\mathcal{U}(k) = \emptyset\),
\(K=\mathcal{F}\) is a finite locally constant sheaf of \(\Lambda\)-modules on \(\mathcal{U}\) whose stalk(s) are finite projective \(\Lambda\)-modules, and
\(\Lambda\) is killed by a power of a prime \(\ell\) and \(\ell \in k^*\).
Indeed, because of Step 2, we can throw out any finite set of points. But we have only finitely many rational points, so we may assume there are none2. We may assume that \(\mathcal{U}\) is smooth irreducible and affine by passing to irreducible components and throwing away the bad points if necessary. The assumptions of \(\mathcal{F}\) come from unwinding the definition of \(D_{ctf}(X, \Lambda)\) and those on \(\Lambda\) from considering its primary decomposition.
For the remainder of the proof, we consider the situation \[\xymatrix{ \mathcal{V} \ar[d]_f \ar[r] & Y \ar[d]^{\bar f} \\ \mathcal{U} \ar[r] & X }\] where \(\mathcal{U}\) is as above, \(f\) is a finite étale Galois covering, \(\mathcal{V}\) is connected and the horizontal arrows are projective completions. Denoting \(G=\text{Aut}(\mathcal{V}|\mathcal{U})\), we also assume (as we may) that \(f^{-1}\mathcal{F} =\underline M\) is constant, where the module \(M = \Gamma(\mathcal{V}, f^{-1}\mathcal{F})\) is a \(\Lambda[G]\)-module which is finite and projective over \(\Lambda\). This corresponds to the trivial monoid extension \[1\to G\to \Gamma = G \times \mathbf{N}\to \mathbf{N}\to 1.\] In that context, using the reductions above, we need to show that \(T''(\mathcal{U}, \mathcal{F}) = 0\).
Step 4. There is a natural action of \(G\) on \(f_*f^{-1}\mathcal{F}\) and the trace map \(f_*f^{-1}\mathcal{F}\to \mathcal{F}\) defines an isomorphism \[(f_*f^{-1}\mathcal{F})\otimes_{\Lambda[G]} \Lambda = (f_*f^{-1}\mathcal{F})_G \cong \mathcal{F}.\]
To prove this, simply unwind everything at a geometric point.
Step 5. Let \(A = \mathbf{Z}/\ell^n \mathbf{Z}\) with \(n\gg 0\). Then \(f_*f^{-1}\mathcal{F} \cong (f_*\underline A) \otimes_{\underline A} \underline M\) with diagonal \(G\)-action.
Step 6. There is a canonical isomorphism \((f_*\underline A \otimes_{\underline A} \underline M) \otimes_{\Lambda[G]} \underline \Lambda \cong \mathcal{F}\).
In fact, this is a derived tensor product, because of the projectivity assumption on \(\mathcal{F}\).
Step 7. There is a canonical isomorphism \[R\Gamma_c(\mathcal{U}_{\bar k}, \mathcal{F}) = (R\Gamma_c(\mathcal{U}_{\bar k}, f_*A)\otimes_A^\mathbf{L} M)\otimes_{\Lambda[G]}^\mathbf{L} \Lambda,\] compatible with the action of \(\pi^*_\mathcal{U}\).
This comes from the universal coefficient theorem, i.e., the fact that \(R\Gamma_c\) commutes with \(\otimes^\mathbf{L}\), and the flatness of \(\mathcal{F}\) as a \(\Lambda\)-module.
We have \[\begin{eqnarray*} \text{Tr}( \pi_\mathcal{U}^* |_{R\Gamma_c(\mathcal{U}_{\bar k}, \mathcal{F})}) & = & {\sum_{g \in G}}' \text{Tr}_{\Lambda}^{Z_g} \left( (g, \pi_\mathcal{U}^*) |_{R\Gamma_c(\mathcal{U}_{\bar k}, f_*A)\otimes_A^\mathbf{L} M} \right) \\ & = & {\sum_{g\in G}}' \text{Tr}_A^{Z_g} ( (g, \pi_\mathcal{U}^*) |_{R\Gamma_c(\mathcal{U}_{\bar k}, f_*A)} ) \cdot \text{Tr}_\Lambda(g|_M) \end{eqnarray*}\] where \(\Gamma\) acts on \(R\Gamma_c(\mathcal{U}_{\bar k}, \mathcal{F})\) by \(G\) and \((e, 1)\) acts via \(\pi_\mathcal{U}^*\). So the monoidal extension is given by \(\Gamma = G \times \mathbf{N} \to \mathbf{N}\), \(\gamma \mapsto 1\). The first equality follows from Lemma 03UD and the second from Lemma 03UC.
Step 8. It suffices to show that \(\text{Tr}_A^{Z_g}((g, \pi_\mathcal{U}^*) |_{R\Gamma_c(\mathcal{U}_{\bar k}, f_*A)}) \in A\) maps to zero in \(\Lambda\).
Recall that \[\begin{eqnarray*} \# Z_g \cdot \text{Tr}_A^{Z_g}((g, \pi_\mathcal{U}^*) |_{R\Gamma_c(\mathcal{U}_{\bar k}, f_*A)}) & = & \text{Tr}_A((g, \pi_\mathcal{U}^*) |_{R\Gamma_c(\mathcal{U}_{\bar k}, f_*A)})\\ & = & \text{Tr}_A((g^{-1}\pi_\mathcal{V})^* |_{R\Gamma_c(\mathcal{V}_{\bar k}, A)}). \end{eqnarray*}\] The first equality is Lemma 03UB, the second is the Leray spectral sequence, using the finiteness of \(f\) and the fact that we are only taking traces over \(A\). Now since \(A=\mathbf{Z}/\ell^n\mathbf{Z}\) with \(n \gg 0\) and \(\# Z_g = \ell^a\) for some (fixed) \(a\), it suffices to show the following result.
Step 9. We have \(\text{Tr}_A((g^{-1}\pi_\mathcal{V})^* |_{R\Gamma_c(\mathcal{V}, A)}) = 0\) in \(A\).
By additivity again, we have \[\begin{eqnarray*} & \text{Tr}_A((g^{-1}\pi_\mathcal{V})^* |_{R\Gamma_c(\mathcal{V}_{\bar k} A)}) + \text{Tr}_A((g^{-1}\pi_\mathcal{V})^* |_{R\Gamma_c(Y-\mathcal {V})_{\bar k}, A)}) \\ & = \text{Tr}_A((g^{-1}\pi_Y)^* |_{R\Gamma(Y_{\bar k}, A)}) \end{eqnarray*}\] The latter trace is the number of fixed points of \(g^{-1}\pi_Y\) on \(Y\), by Weil’s trace formula Theorem 03U1. Moreover, by the 0-dimensional case already proven in step 2, \[\text{Tr}_A((g^{-1}\pi_\mathcal{V})^*|_{R\Gamma_c(Y-\mathcal{V})_{\bar k}, A)})\] is the number of fixed points of \(g^{-1}\pi_Y\) on \((Y-\mathcal{V})_{\bar k}\). Therefore, \[\text{Tr}_A((g^{-1}\pi_\mathcal{V})^* |_{R\Gamma_c(\mathcal{V}_{\bar k}, A)})\] is the number of fixed points of \(g^{-1}\pi_Y\) on \(\mathcal{V}_{\bar k}\). But there are no such points: if \(\bar y\in Y_{\bar k}\) is fixed under \(g^{-1}\pi_Y\), then \(\bar f(\bar y) \in X_{\bar k}\) is fixed under \(\pi_X\). But \(\mathcal{U}\) has no \(k\)-rational point, so we must have \(\bar f(\bar y)\in (X-\mathcal{U})_{\bar k}\) and so \(\bar y\notin \mathcal{V}_{\bar k}\), a contradiction. This finishes the proof.
Remark
Remarks on Theorem 03UG.
This formula holds in any dimension. By a dévissage lemma (which uses proper base change etc.) it reduces to the current statement – in that generality.
The complex \(R\Gamma_c(X_{\bar k}, K)\) is defined by choosing an open immersion \(j : X \hookrightarrow \bar X\) with \(\bar X\) projective over \(k\) of dimension at most 1 and setting \[R\Gamma_c(X_{\bar k}, K) := R\Gamma(\bar X_{\bar k}, j_!K).\] This is independent of the choice of \(\bar X\) follows from (insert reference here). We define \(H^i_c(X_{\bar k}, K)\) to be the \(i\)th cohomology group of \(R\Gamma_c(X_{\bar k}, K)\).
Remark
Even though all we did are reductions and mostly algebra, the trace formula Theorem 03UG is much stronger than Weil’s geometric trace formula (Theorem 03U1) because it applies to coefficient systems (sheaves), not merely constant coefficients.
Applications
OK, having indicated the proof of the trace formula, let’s try to use it for something.
On l-adic sheaves
Definition
Let \(X\) be a Noetherian scheme. A \(\mathbf{Z}_\ell\)-sheaf on \(X\), or simply an \(\ell\)-adic sheaf \(\mathcal{F}\) is an inverse system \(\left\{\mathcal{F}_n\right\}_{n\geq 1}\) where
\(\mathcal{F}_n\) is a constructible \(\mathbf{Z}/\ell^n\mathbf{Z}\)-module on \(X_\etale\), and
the transition maps \(\mathcal{F}_{n+1}\to \mathcal{F}_n\) induce isomorphisms \(\mathcal{F}_{n+1} \otimes_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} \cong \mathcal{F}_n\).
We say that \(\mathcal{F}\) is lisse if each \(\mathcal{F}_n\) is locally constant. A morphism of such is merely a morphism of inverse systems.
Lemma
Let \(\{\mathcal{G}_n\}_{n\geq 1}\) be an inverse system of constructible \(\mathbf{Z}/\ell^n\mathbf{Z}\)-modules. Suppose that for all \(k\geq 1\), the maps \[\mathcal{G}_{n+1}/\ell^k \mathcal{G}_{n+1}\to \mathcal{G}_n /\ell^k \mathcal{G}_n\] are isomorphisms for all \(n\gg 0\) (where the bound possibly depends on \(k\)). In other words, assume that the system \(\{\mathcal{G}_n/\ell^k\mathcal{G}_n\}_{n\geq 1}\) is eventually constant, and call \(\mathcal{F}_k\) the corresponding sheaf. Then the system \(\left\{\mathcal{F}_k\right\}_{k\geq 1}\) forms a \(\mathbf{Z}_\ell\)-sheaf on \(X\).
Proof
The proof is obvious.
Lemma
The category of \(\mathbf{Z}_\ell\)-sheaves on \(X\) is abelian.
Proof
Let \(\Phi = \left\{\varphi_n\right\}_{n\geq 1} : \left\{\mathcal{F}_n\right\} \to \left\{\mathcal{G}_n\right\}\) be a morphism of \(\mathbf{Z}_\ell\)-sheaves. Set \[\Coker(\Phi) = \left\{ \Coker\left(\mathcal{F}_n \xrightarrow{\varphi_n} \mathcal{G}_n\right) \right\}_{n\geq 1}\] and \(\Ker(\Phi)\) is the result of Lemma 03UN applied to the inverse system \[\left\{ \bigcap_{m\geq n} \Im\left(\Ker(\varphi_m) \to \Ker(\varphi_n)\right) \right\}_{n \geq 1}.\] That this defines an abelian category is left to the reader.
Example
Let \(X=\Spec(\mathbf{C})\) and \(\Phi : \mathbf{Z}_\ell\to \mathbf{Z}_\ell\) be multiplication by \(\ell\). More precisely, \[\Phi = \left\{ \mathbf{Z}/\ell^n\mathbf{Z} \xrightarrow{\ell} \mathbf{Z}/\ell^n\mathbf{Z}\right\}_{n \geq 1}.\] To compute the kernel, we consider the inverse system \[\ldots\to \mathbf{Z}/\ell\mathbf{Z}\xrightarrow{0} \mathbf{Z}/\ell\mathbf{Z}\xrightarrow{0}\mathbf{Z}/\ell\mathbf{Z}.\] Since the images are always zero, \(\Ker(\Phi)\) is zero as a system.
Remark
If \(\mathcal{F} = \left\{\mathcal{F}_n\right\}_{n\geq 1}\) is a \(\mathbf{Z}_\ell\)-sheaf on \(X\) and \(\bar x\) is a geometric point then \(M_n = \left\{\mathcal{F}_{n, \bar x}\right\}\) is an inverse system of finite \(\mathbf{Z}/\ell^n\mathbf{Z}\)-modules such that \(M_{n+1}\to M_n\) is surjective and \(M_n = M_{n+1}/\ell^n M_{n+1}\). It follows that \[M = \lim_n M_n = \lim \mathcal{F}_{n, \bar x}\] is a finite \(\mathbf{Z}_\ell\)-module. This follows from Algebra, Lemmas 09B8 and 031D and the fact that \(M/\ell M = M_1\) is finite over \(\mathbf{F}_\ell\). Therefore, \(M\cong \mathbf{Z}_\ell^{\oplus r} \oplus \oplus_{i = 1}^t \mathbf{Z}_\ell/\ell^{e_i}\mathbf{Z}_\ell\) for some \(r, t\geq 0\), \(e_i\geq 1\). The module \(M = \mathcal{F}_{\bar x}\) is called the stalk of \(\mathcal{F}\) at \(\bar x\).
Definition
A \(\mathbf{Z}_\ell\)-sheaf \(\mathcal{F}\) is torsion if \(\ell^n : \mathcal{F} \to \mathcal{F}\) is the zero map for some \(n\). The abelian category of \(\mathbf{Q}_\ell\)-sheaves on \(X\) is the quotient of the abelian category of \(\mathbf{Z}_\ell\)-sheaves by the Serre subcategory of torsion sheaves. In other words, its objects are \(\mathbf{Z}_\ell\)-sheaves on \(X\), and if \(\mathcal{F}, \mathcal{G}\) are two such, then \[\Hom_{\mathbf{Q}_\ell} \left(\mathcal{F}, \mathcal{G} \right) = \Hom_{\mathbf{Z}_\ell} \left(\mathcal{F}, \mathcal{G}\right) \otimes_{\mathbf{Z}_\ell} \mathbf{Q}_\ell.\] We denote by \(\mathcal{F} \mapsto \mathcal{F} \otimes \mathbf{Q}_\ell\) the quotient functor (right adjoint to the inclusion). If \(\mathcal{F} = \mathcal{F}' \otimes \mathbf{Q}_\ell\) where \(\mathcal{F}'\) is a \(\mathbf{Z}_\ell\)-sheaf and \(\bar x\) is a geometric point, then the stalk of \(\mathcal{F}\) at \(\bar x\) is \(\mathcal{F}_{\bar x} = \mathcal{F}'_{\bar x} \otimes \mathbf{Q}_\ell\).
Remark
Since a \(\mathbf{Z}_\ell\)-sheaf is only defined on a Noetherian scheme, it is torsion if and only if its stalks are torsion.
Definition
If \(X\) is a separated scheme of finite type over an algebraically closed field \(k\) and \(\mathcal{F} = \left\{\mathcal{F}_n\right\}_{n\geq 1}\) is a \(\mathbf{Z}_\ell\)-sheaf on \(X\), then we define \[H^i(X, \mathcal{F}) := \lim_n H^i(X, \mathcal{F}_n) \quad\text{and}\quad H_c^i(X, \mathcal{F}) := \lim_n H_c^i(X, \mathcal{F}_n).\] If \(\mathcal{F} = \mathcal{F}'\otimes \mathbf{Q}_\ell\) for a \(\mathbf{Z}_\ell\)-sheaf \(\mathcal{F}'\) then we set \[H_c^i(X , \mathcal{F}) := H_c^i(X, \mathcal{F}')\otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell.\] We call these the \(\ell\)-adic cohomology of \(X\) with coefficients \(\mathcal{F}\).
L-functions
Definition
Let \(X\) be a scheme of finite type over a finite field \(k\). Let \(\Lambda\) be a finite ring of order prime to the characteristic of \(k\) and \(\mathcal{F}\) a constructible flat \(\Lambda\)-module on \(X_\etale\). Then we set \[L(X, \mathcal{F}) := \prod\nolimits_{x\in |X|} \det(1 - \pi_x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \Lambda [[ T ]]\] where \(|X|\) is the set of closed points of \(X\), \(\deg x = [\kappa(x): k]\) and \(\bar x\) is a geometric point lying over \(x\). This definition clearly generalizes to the case where \(\mathcal{F}\) is replaced by a \(K \in D_{ctf}(X, \Lambda)\). We call this the \(L\)-function of \(\mathcal{F}\).
Remark
Intuitively, \(T\) should be thought of as \(T = t^f\) where \(p^f = \# k\). The definitions are then independent of the size of the ground field.
Definition
Now assume that \(\mathcal{F}\) is a \(\mathbf{Q}_\ell\)-sheaf on \(X\). In this case we define \[L(X, \mathcal{F}) := \prod\nolimits_{x \in |X|} \det(1 - \pi_x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \mathbf{Q}_\ell[[T]].\] Note that this product converges since there are finitely many points of a given degree. We call this the \(L\)-function of \(\mathcal{F}\).
Cohomological interpretation
This is how Grothendieck interpreted the \(L\)-function.
Theorem
Let \(X\) be a scheme of finite type over a finite field \(k\). Let \(\Lambda\) be a finite ring of order prime to the characteristic of \(k\) and \(\mathcal{F}\) a constructible flat \(\Lambda\)-module on \(X_\etale\). Then \[L(X, \mathcal{F}) = \det(1 - \pi_X^*\ T |_{R\Gamma_c(X_{\bar k}, \mathcal{F})})^{-1} \in \Lambda[[T]].\]
Proof
Omitted.
Thus far, we don’t even know whether each cohomology group \(H^i_c(X_{\bar k}, \mathcal{F})\) is free.
Theorem
Let \(X\) be a scheme of finite type over a finite field \(k\), and \(\mathcal{F}\) a \(\mathbf{Q}_\ell\)-sheaf on \(X\). Then \[L(X, \mathcal{F}) = \prod\nolimits_i \det(1 - \pi_X^*T |_{H_c^i(X_{\bar k} , \mathcal{F})})^{(-1)^{i + 1}} \in \mathbf{Q}_\ell[[T]].\]
Proof
This is sketched below.
Remark
Since we have only developed some theory of traces and not of determinants, Theorem 03UZ is harder to prove than Theorem 03V0. We will only prove the latter, for the former see [SGA4.5]. Observe also that there is no version of this theorem more general for \(\mathbf{Z}_\ell\) coefficients since there is no \(\ell\)-torsion.
We reduce the proof of Theorem 03V0 to a trace formula. Since \(\mathbf{Q}_\ell\) has characteristic 0, it suffices to prove the equality after taking logarithmic derivatives. More precisely, we apply \(T\frac{d}{dT} \log\) to both sides. We have on the one hand \[\begin{align*} T \frac{d}{dT} \log L(X, \mathcal{F}) & = T\frac{d}{dT} \log \prod_{x \in |X|} \det(1 - \pi_x^* T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \\ & = \sum_{x \in |X|} T \frac{d}{dT} \log ( \det(1 - \pi_x^* T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1}) \\ & = \sum_{x \in |X|} \deg x \sum_{n \geq 1} \text{Tr}((\pi_x^n)^* |_{\mathcal{F}_{\bar x}}) T^{n\deg x} \end{align*}\] where the last equality results from the formula \[T\frac{d}{dT}\log\left(\det\left(1-fT|_M\right)^{-1}\right) = \sum_{n\geq 1} \text{Tr}(f^n|_M)T^n\] which holds for any commutative ring \(\Lambda\) and any endomorphism \(f\) of a finite projective \(\Lambda\)-module \(M\). On the other hand, we have \[\begin{align*} & T\frac{d}{dT} \log\left( \prod\nolimits_i \det(1-\pi_X^*T |_{H_c^i\left(X_{\bar k} , \mathcal{F}\right)})^{(-1)^{i+1}} \right) \\ & = \sum\nolimits_i (-1)^i \sum\nolimits_{n \geq 1} \text{Tr}\left((\pi_X^n)^* |_{H_c^i(X_{\bar k},\mathcal{F})}\right) T^n \end{align*}\] by the same formula again. Now, comparing powers of \(T\) and using the Mobius inversion formula, we see that Theorem 03V0 is a consequence of the following equality \[\sum_{d | n} d \sum_{x \in |X| \atop \deg x = d} \text{Tr}((\pi_X^{n/d})^* |_{\mathcal{F}_{\bar x}}) = \sum_i (-1)^i \text{Tr}((\pi^n_X)^* |_{H^i_c(X_{\bar k}, \mathcal{F})}).\] Writing \(k_n\) for the degree \(n\) extension of \(k\), \(X_n = X \times_{\Spec k} \Spec(k_n)\) and \(_n\mathcal{F} = \mathcal{F}|_{X_n}\), this boils down to \[\sum_{x \in X_n(k_n)} \text{Tr}(\pi_X^* |_{_n\mathcal{F}_{\bar x}}) = \sum_i (-1)^i \text{Tr}((\pi^n_X)^* |_{H^i_c({(X_n)}_{\bar k}, _n\mathcal{F})})\] which is a consequence of Theorem 03V2.
Theorem
Let \(X/k\) be as above, let \(\Lambda\) be a finite ring with \(\#\Lambda \in k^*\) and \(K\in D_{ctf}(X, \Lambda)\). Then \(R\Gamma_c(X_{\bar k}, K)\in D_{perf}(\Lambda)\) and \[\sum_{x\in X(k)}\text{Tr}\left(\pi_x |_{K_{\bar x}}\right) = \text{Tr}\left(\pi_X^* |_{R\Gamma_c(X_{\bar k}, K )}\right).\]
Proof
Note that we have already proved this (REFERENCE) when \(\dim X \leq 1\). The general case follows easily from that case together with the proper base change theorem.
Theorem
Let \(X\) be a separated scheme of finite type over a finite field \(k\) and \(\mathcal{F}\) be a \(\mathbf{Q}_\ell\)-sheaf on \(X\). Then \(\dim_{\mathbf{Q}_\ell}H_c^i(X_{\bar k}, \mathcal{F})\) is finite for all \(i\), and is nonzero for \(0\leq i \leq 2 \dim X\) only. Furthermore, we have \[\sum_{x\in X(k)} \text{Tr}\left(\pi_x |_{\mathcal{F}_{\bar x}}\right) = \sum_i (-1)^i\text{Tr}\left(\pi_X^* |_{H_c^i(X_{\bar k}, \mathcal{F})}\right).\]
Proof
We explain how to deduce this from Theorem 03V3. We first use some étale cohomology arguments to reduce the proof to an algebraic statement which we subsequently prove.
Let \(\mathcal{F}\) be as in the theorem. We can write \(\mathcal{F}\) as \(\mathcal{F}'\otimes \mathbf{Q}_\ell\) where \(\mathcal{F}' = \left\{\mathcal{F}'_n\right\}\) is a \(\mathbf{Z}_\ell\)-sheaf without torsion, i.e., \(\ell : \mathcal{F}'\to \mathcal{F}'\) has trivial kernel in the category of \(\mathbf{Z}_\ell\)-sheaves. Then each \(\mathcal{F}_n'\) is a flat constructible \(\mathbf{Z}/\ell^n\mathbf{Z}\)-module on \(X_\etale\), so \(\mathcal{F}'_n \in D_{ctf}(X, \mathbf{Z}/\ell^n\mathbf{Z})\) and \(\mathcal{F}_{n+1}' \otimes^{\mathbf{L}}_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} = \mathcal{F}_n'\). Note that the last equality holds also for standard (non-derived) tensor product, since \(\mathcal{F}'_n\) is flat (it is the same equality). Therefore,
the complex \(K_n = R\Gamma_c\left(X_{\bar k}, \mathcal{F}_n'\right)\) is perfect, and it is endowed with an endomorphism \(\pi_n : K_n\to K_n\) in \(D(\mathbf{Z}/\ell^n\mathbf{Z})\),
there are identifications \[K_{n+1} \otimes^{\mathbf{L}}_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} = K_n\] in \(D_{perf}(\mathbf{Z}/\ell^n\mathbf{Z})\), compatible with the endomorphisms \(\pi_{n+1}\) and \(\pi_n\) (see [SGA4.5, Rapport 4.12]),
the equality \(\text{Tr}\left(\pi_X^* |_{K_n}\right) = \sum_{x\in X(k)} \text{Tr}\left(\pi_x |_{(\mathcal{F}'_n)_{\bar x}}\right)\) holds, and
for each \(x\in X(k)\), the elements \(\text{Tr}(\pi_x |_{\mathcal{F}'_{n, \bar x}}) \in \mathbf{Z}/\ell^n\mathbf{Z}\) form an element of \(\mathbf{Z}_\ell\) which is equal to \(\text{Tr}(\pi_x |_{\mathcal{F}_{\bar x}}) \in \mathbf{Q}_\ell\).
It thus suffices to prove the following algebra lemma.
Lemma
Suppose we have \(K_n\in D_{perf}(\mathbf{Z}/\ell^n\mathbf{Z})\), \(\pi_n : K_n\to K_n\) and isomorphisms \(\varphi_n : K_{n+1} \otimes^\mathbf{L}_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} \to K_n\) compatible with \(\pi_{n+1}\) and \(\pi_n\). Then
the elements \(t_n = \text{Tr}(\pi_n |_{K_n})\in \mathbf{Z}/\ell^n\mathbf{Z}\) form an element \(t_\infty = \{t_n\}\) of \(\mathbf{Z}_\ell\),
the \(\mathbf{Z}_\ell\)-module \(H_\infty^i = \lim_n H^i(k_n)\) is finite and is nonzero for finitely many \(i\) only, and
the operators \(H^i(\pi_n): H^i(K_n)\to H^i(K_n)\) are compatible and define \(\pi_\infty^i : H_\infty^i\to H_\infty^i\) satisfying \[\sum (-1)^i \text{Tr}( \pi_\infty^i |_{H_\infty^i \otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell}) = t_\infty.\]
Proof
Since \(\mathbf{Z}/\ell^n\mathbf{Z}\) is a local ring and \(K_n\) is perfect, each \(K_n\) can be represented by a finite complex \(K_n^\bullet\) of finite free \(\mathbf{Z}/\ell^n \mathbf{Z}\)-modules such that the map \(K_n^p \to K_n^{p+1}\) has image contained in \(\ell K_n^{p+1}\). It is a fact that such a complex is unique up to isomorphism. Moreover \(\pi_n\) can be represented by a morphism of complexes \(\pi_n^\bullet : K_n^\bullet\to K_n^\bullet\) (which is unique up to homotopy). By the same token the isomorphism \(\varphi_n : K_{n+1} \otimes_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}}^{\mathbf{L}} \mathbf{Z}/\ell^n\mathbf{Z}\to K_n\) is represented by a map of complexes \[\varphi_n^\bullet : K_{n+1}^\bullet \otimes_{\mathbf{Z}/\ell^{n+1}\mathbf{Z}} \mathbf{Z}/\ell^n\mathbf{Z} \to K_n^\bullet.\] In fact, \(\varphi_n^\bullet\) is an isomorphism of complexes, thus we see that
there exist \(a, b\in \mathbf{Z}\) independent of \(n\) such that \(K_n^i = 0\) for all \(i\notin[a, b]\), and
the rank of \(K_n^i\) is independent of \(n\).
Therefore, the module \(K_\infty^i = \lim_n \{K_n^i, \varphi_n^i\}\) is a finite free \(\mathbf{Z}_\ell\)-module and \(K_\infty^\bullet\) is a finite complex of finite free \(\mathbf{Z}_\ell\)-modules. By induction on the number of nonzero terms, one can prove that \(H^i\left(K_\infty^\bullet\right) = \lim_n H^i\left(K_n^\bullet\right)\) (this is not true for unbounded complexes). We conclude that \(H_\infty^i = H^i\left(K_\infty^\bullet\right)\) is a finite \(\mathbf{Z}_\ell\)-module. This proves ii. To prove the remainder of the lemma, we need to overcome the possible noncommutativity of the diagrams \[\xymatrix{ {K_{n+1}^\bullet} \ar[d]_{\pi_{n+1}^\bullet} \ar[r]^{\varphi_n^\bullet} & {K_n^\bullet} \ar[d]^{\pi_n^\bullet} \\ {K_{n+1}^\bullet} \ar[r]_{\varphi_n^\bullet} & {K_n^\bullet.} }\] However, this diagram does commute in the derived category, hence it commutes up to homotopy. We inductively replace \(\pi_n^\bullet\) for \(n\geq 2\) by homotopic maps of complexes making these diagrams commute. Namely, if \(h^i : K_{n+1}^i \to K_n^{i-1}\) is a homotopy, i.e., \[\pi_n^\bullet \circ \varphi_n^\bullet - \varphi_n^\bullet \circ \pi_{n + 1}^\bullet = dh + hd,\] then we choose \(\tilde h^i : K_{n+1}^i\to K_{n+1}^{i-1}\) lifting \(h^i\). This is possible because \(K_{n+1}^i\) free and \(K_{n+1}^{i-1}\to K_n^{i-1}\) is surjective. Then replace \(\pi_n^\bullet\) by \(\tilde\pi_n^\bullet\) defined by \[\tilde\pi_{n+1}^\bullet = \pi_{n+1}^\bullet + d\tilde h+\tilde hd.\] With this choice of \(\{\pi_n^\bullet\}\), the above diagrams commute, and the maps fit together to define an endomorphism \(\pi_\infty^\bullet = \lim_n\pi_n^\bullet\) of \(K_\infty^\bullet\). Then part i is clear: the elements \(t_n = \sum(-1)^i \text{Tr}\left(\pi_n^i |_{K_n^i}\right)\) fit into an element \(t_\infty\) of \(\mathbf{Z}_\ell\). Moreover \[\begin{align*} t_\infty & = \sum (-1)^i \text{Tr}_{\mathbf{Z}_\ell}(\pi_\infty^i |_{K_\infty^i}) \\ & = \sum (-1)^i \text{Tr}_{\mathbf{Q}_\ell}( \pi_\infty^i |_{K_\infty^i \otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell}) \\ & = \sum (-1)^i \text{Tr}( \pi_\infty |_{H^i(K_\infty^\bullet \otimes \mathbf{Q}_\ell)}) \end{align*}\] where the last equality follows from the fact that \(\mathbf{Q}_\ell\) is a field, so the complex \(K_\infty^\bullet \otimes \mathbf{Q}_\ell\) is quasi-isomorphic to its cohomology \(H^i(K_\infty^\bullet \otimes \mathbf{Q}_\ell)\). The latter is also equal to \(H^i(K_\infty^\bullet)\otimes_{\mathbf{Z}}\mathbf{Q}_\ell = H_\infty^i \otimes \mathbf{Q}_\ell\), which finishes the proof of the lemma, and also that of Theorem 03V2.
List of things which we should add above
What did we skip the proof of in the lectures so far:
curves and their Jacobians,
proper base change theorem,
inadequate discussion of \(R\Gamma_c\),
more generally, given \(f : X \to S\) finite type, separated \(S\) quasi-projective, discussion of \(Rf_!\) on étale sheaves.
discussion of \(\otimes^{\mathbf{L}}\)
discussion of why \(R\Gamma_c\) commutes with \(\otimes^{\mathbf{L}}\)
Examples of L-functions
We use Theorem 03V0 for curves to give examples of \(L\)-functions
Constant sheaves
Let \(k\) be a finite field, \(X\) a smooth, geometrically irreducible curve over \(k\) and \(\mathcal{F} = \underline{\mathbf{Q}_\ell}\) the constant sheaf. If \(\bar x\) is a geometric point of \(X\), the Galois module \(\mathcal{F}_{\bar x} = \mathbf{Q}_\ell\) is trivial, so \[\det(1-\pi_x^*\ T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} = \frac{1}{1-T^{\deg x}}.\] Applying Theorem 03V0, we get \[\begin{align*} L(X, \mathcal{F}) & = \prod_{i = 0}^2 \det(1 - \pi_X^*T |_{H_c^i(X_{\bar k}, \mathbf{Q}_\ell)})^{(-1)^{i+1}} \\ & = \frac{\det(1 - \pi_X^*T |_{H_c^1(X_{\bar k}, \mathbf{Q}_\ell)})}{ \det(1 - \pi_X^*T |_{H_c^0(X_{\bar k}, \mathbf{Q}_\ell)}) \cdot \det(1 - \pi_X^*T |_{H_c^2(X_{\bar k}, \mathbf{Q}_\ell)})}. \end{align*}\] To compute the latter, we distinguish two cases.
Projective case. Assume that \(X\) is projective, so \(H_c^i(X_{\bar k}, \mathbf{Q}_\ell) = H^i(X_{\bar k}, \mathbf{Q}_\ell)\), and we have \[H^i(X_{\bar k}, \mathbf{Q}_\ell) = \left\{ \begin{matrix} \mathbf{Q}_\ell & \pi_X^* = 1 & \text{if }i = 0, \\ \mathbf{Q}_\ell^{2g} & \pi_X^* = ? & \text{if }i = 1, \\ \mathbf{Q}_\ell & \pi_X^* = q & \text{if }i = 2. \end{matrix} \right.\] The identification of the action of \(\pi_X^*\) on \(H^2\) comes from Étale Cohomology, Lemma 0AMB and the fact that the degree of \(\pi_X\) is \(q = \#(k)\). We do not know much about the action of \(\pi_X^*\) on the degree 1 cohomology. Let us call \(\alpha_1, \ldots, \alpha_{2g}\) its eigenvalues in \(\bar{\mathbf{Q}}_\ell\). Putting everything together, Theorem 03V0 yields the equality \[\prod\nolimits_{x \in |X|} \frac{1}{1 - T^{\deg x}} = \frac{\det(1- \pi_X^* T|_{H^1(X_{\bar k}, \mathbf{Q}_\ell)})}{(1-T)(1-qT)} = \frac{(1 - \alpha_1 T) \ldots (1 - \alpha_{2g}T)}{(1-T)(1-qT)}\] from which we deduce the following result.
Lemma
Let \(X\) be a smooth, projective, geometrically irreducible curve over a finite field \(k\). Then
the \(L\)-function \(L(X, \mathbf{Q}_\ell)\) is a rational function,
the eigenvalues \(\alpha_1, \ldots, \alpha_{2g}\) of \(\pi_X^*\) on \(H^1(X_{\bar k}, \mathbf{Q}_\ell)\) are algebraic integers independent of \(\ell\),
the number of rational points of \(X\) on \(k_n\), where \([k_n : k] = n\), is \[\# X(k_n) = 1 - \sum\nolimits_{i = 1}^{2g}\alpha_i^n + q^n,\]
for each \(i\), \(|\alpha_i| < q\).
Proof
Part (3) is Theorem 03V2 applied to \(\mathcal{F} = \underline{\mathbf{Q}_\ell}\) on \(X \otimes k_n\). For part (4), use the following result.
Exercise
Let \(\alpha_1, \ldots, \alpha_n \in \mathbf{C}\). Then for any conic sector containing the positive real axis of the form \(C_\varepsilon = \{ z \in \mathbf{C} \ | \ |\arg z| < \varepsilon \}\) with \(\varepsilon > 0\), there exists an integer \(k \geq 1\) such that \(\alpha_1^k, \ldots, \alpha_n^k \in C_\varepsilon\).
Then prove that \(|\alpha_i| \leq q\) for all \(i\). Then, use elementary considerations on complex numbers to prove (as in the proof of the prime number theorem) that \(|\alpha_i| < q\). In fact, the Riemann hypothesis says that for all \(|\alpha_i| = \sqrt{q}\) for all \(i\). We will come back to this later.
Affine case. Assume now that \(X\) is affine, say \(X= \bar X-\left\{x_1, \ldots, x_n\right\}\) where \(j : X \hookrightarrow \bar X\) is a projective nonsingular completion. Then \(H_c^0(X_{\bar k}, \mathbf{Q}_\ell) = 0\) and \(H_c^2(X_{\bar k}, \mathbf{Q}_\ell) = H^2(\bar X_{\bar k}, \mathbf{Q}_\ell)\) so Theorem 03V0 reads \[L(X, \mathbf{Q}_\ell) = \prod_{x \in |X|}\frac{1}{1 - T^{\deg x}} = \frac{\det(1-\pi_X^*T |_{H_c^1(X_{\bar k}, \mathbf{Q}_\ell)})}{1 - qT}.\] On the other hand, the previous case gives \[\begin{eqnarray*} L(X, \mathbf{Q}_\ell) & = & L(\bar X, \mathbf{Q}_\ell)\prod_{i = 1}^n\left(1-T^{\deg x_i}\right) \\ & = & \frac{\prod_{i = 1}^n(1-T^{\deg x_i})\prod_{j = 1}^{2g}(1-\alpha_jT)}{(1-T)(1-qT)}. \end{eqnarray*}\] Therefore, we see that \(\dim H_c^1(X_{\bar k}, \mathbf{Q}_\ell) = 2g+\sum_{i = 1}^n \deg(x_i)-1\), and the eigenvalues \(\alpha_1, \ldots, \alpha_{2g}\) of \(\pi_{\bar X}^*\) acting on the degree 1 cohomology are roots of unity. More precisely, each \(x_i\) gives a complete set of \(\deg(x_i)\)th roots of unity, and one occurrence of 1 is omitted. To see this directly using coherent sheaves, consider the short exact sequence on \(\bar X\) \[0\to j_!\mathbf{Q}_\ell\to \mathbf{Q}_\ell\to\bigoplus_{i = 1}^n \mathbf{Q}_{\ell, x_i}\to 0.\] The long exact cohomology sequence reads \[0\to \mathbf{Q}_\ell \to \bigoplus_{i = 1}^n \mathbf{Q}_\ell^{\oplus \deg x_i} \to H_c^1(X_{\bar k}, \mathbf{Q}_\ell) \to H_c^1(\bar X_{\bar k}, \mathbf{Q}_\ell)\to 0\] where the action of Frobenius on \(\bigoplus_{i = 1}^n \mathbf{Q}_\ell^{\oplus \deg x_i}\) is by cyclic permutation of each term; and \(H_c^2(X_{\bar k}, \mathbf{Q}_\ell) = H_c^2(\bar X_{\bar k}, \mathbf{Q}_\ell)\).
The Legendre family
Let \(k\) be a finite field of odd characteristic, \(X = \Spec(k[\lambda, \frac{1}{\lambda(\lambda - 1)}])\), and consider the family of elliptic curves \(f : E \to X\) on \(\mathbf{P}^2_X\) whose affine equation is \(y^2 = x(x - 1)(x - \lambda)\). We set \(\mathcal{F} = Rf_*^1\mathbf{Q}_\ell = \left\{R^1f_*\mathbf{Z}/\ell^n\mathbf{Z}\right\}_{n\geq 1} \otimes \mathbf{Q}_\ell\). In this situation, the following is true
for each \(n \geq 1\), the sheaf \(R^1f_*(\mathbf{Z}/\ell^n\mathbf{Z})\) is finite locally constant – in fact, it is free of rank 2 over \(\mathbf{Z}/\ell^n\mathbf{Z}\),
the system \(\{R^1f_*\mathbf{Z}/\ell^n\mathbf{Z}\}_{n\geq 1}\) is a lisse \(\ell\)-adic sheaf, and
for all \(x\in |X|\), \(\det(1 - \pi_x\ T^{\deg x} |_{\mathcal{F}_{\bar x}}) = (1 - \alpha_x T^{\deg x})(1 - \beta_x T^{\deg x})\) where \(\alpha_x, \beta_x\) are the eigenvalues of the geometric frobenius of \(E_x\) acting on \(H^1(E_{\bar x}, \mathbf{Q}_\ell)\).
Note that \(E_x\) is only defined over \(\kappa(x)\) and not over \(k\). The proof of these facts uses the proper base change theorem and the local acyclicity of smooth morphisms. For details, see [SGA4.5]. It follows that \[L(E/X) := L(X, \mathcal{F}) = \prod_{x\in |X|} \frac{1}{(1-\alpha_xT^{\deg x})(1-\beta_xT^{\deg x })} .\] Applying Theorem 03V0 we get \[L(E/X) = \prod_{i = 0}^2 \det\left(1 - \pi_X^*T |_{H_c^i(X_{\bar k}, \mathcal{F})}\right)^{(-1)^{i+1}},\] and we see in particular that this is a rational function. Furthermore, it is relatively easy to show that \(H_c^0(X_{\bar k}, \mathcal{F}) = H_c^2(X_{\bar k}, \mathcal{F}) = 0\), so we merely have \[L(E/X) = \det(1 - \pi_X^*T |_{H_c^1(X, \mathcal{F})}).\] To compute this determinant explicitly, consider the Leray spectral sequence for the proper morphism \(f : E \to X\) over \(\mathbf{Q}_\ell\), namely \[H_c^i(X_{\bar k}, R^jf_*\mathbf{Q}_\ell) \Rightarrow H_c^{i+j}(E_{\bar k}, \mathbf{Q}_\ell)\] which degenerates. We have \(f_*\mathbf{Q}_\ell = \mathbf{Q}_\ell\) and \(R^1f_*\mathbf{Q}_\ell = \mathcal{F}\). The sheaf \(R^2f_*\mathbf{Q}_\ell = \mathbf{Q}_\ell(-1)\) is the Tate twist of \(\mathbf{Q}_\ell\), i.e., it is the sheaf \(\mathbf{Q}_\ell\) where the Galois action is given by multiplication by \(\#\kappa(x)\) on the stalk at \(\bar x\). It follows that, for all \(n\geq 1\), \[\begin{align*} \# E(k_n) & = \sum\nolimits_i (-1)^i \text{Tr}({\pi_E^n}^* |_{H_c^i(E_{\bar k}, \mathbf{Q}_\ell)}) \\ & = \sum\nolimits_{i, j} (-1)^{i+j} \text{Tr}({\pi^n_X}^* |_{H_c^i(X_{\bar k}, R^jf_*\mathbf{Q}_\ell)}) \\ & = (q^n - 2) + \text{Tr}({\pi_X^n}^* |_{H_c^1(X_{\bar k}, \mathcal{F})}) + q^n(q^n - 2) \\ & = q^{2n} - q^n - 2 + \text{Tr}({\pi_X^n}^* |_{H_c^1(X_{\bar k}, \mathcal{F})}) \end{align*}\] where the first equality follows from Theorem 03V2, the second one from the Leray spectral sequence and the third one by writing down the higher direct images of \(\mathbf{Q}_\ell\) under \(f\). Alternatively, we could write \[\# E(k_n) = \sum_{x \in X(k_n)} \# E_x(k_n)\] and use the trace formula for each curve. We can also find the number of \(k_n\)-rational points simply by counting. The zero section contributes \(q^n -2\) points (we omit the points where \(\lambda = 0, 1\)) hence \[\# E(k_n) = q^n - 2 + \# \{y^2 = x(x - 1)(x - \lambda), \lambda\neq 0, 1\}.\] Now we have \[\begin{matrix} \# \{y^2 = x(x - 1)(x - \lambda),\ \lambda\neq 0, 1\} \\ \\ \quad = \# \{y^2 = x(x - 1)(x - \lambda)\text{ in }\mathbf{A}^3\} - \# \{y^2 = x^2(x - 1)\} - \# \{y^2 = x(x - 1)^2\}\\ \\ \quad = \# \{\lambda = \frac{-y^2}{x(x - 1)} + x,\ x\neq 0, 1\} + \# \{y^2 = x(x - 1)(x - \lambda), x = 0, 1\} - 2(q^n - \varepsilon_n) \\ \\ \quad = q^n(q^n - 2)+2q^n - 2(q^n - \varepsilon_n)\\ \\ \quad = q^{2n}-2q^n+2\varepsilon_n \end{matrix}\] where \(\varepsilon_n = 1\) if \(-1\) is a square in \(k_n\), 0 otherwise, i.e., \[\varepsilon_n = \frac{1}{2}\left(1+\left(\frac{-1}{k_n}\right)\right) = \frac{1}{2}\left(1+(-1)^{\frac{q^n - 1}{2}}\right).\] Thus \(\# E(k_n) = q^{2n} - q^n - 2+ 2\varepsilon_n\). Comparing with the previous formula, we find \[\text{Tr}({\pi_X^n}^* |_{H_c^1(X_{\bar k}, \mathcal{F})}) = 2 \varepsilon_n = 1 + (-1)^{\frac{q^n - 1}{2}},\] which implies, by elementary algebra of complex numbers, that if \(-1\) is a square in \(k_n^*\), then \(\dim H_c^1(X_{\bar k}, \mathcal{F}) = 2\) and the eigenvalues are \(1\) and \(1\). Therefore, in that case we have \[L(E/X) = (1 - T)^2.\]
Exponential sums
A standard problem in number theory is to evaluate sums of the form \[S_{a, b}(p) = \sum_{x\in \mathbf{F}_p - \left\{0, 1\right\}} e^{\frac{2\pi ix^a(x - 1)^b}{p}}.\] In our context, this can be interpreted as a cohomological sum as follows. Consider the base scheme \(S = \Spec(\mathbf{F}_p[x, \frac{1}{x(x - 1)}])\) and the affine curve \(f : X \to \mathbf{P}^1-\{0, 1, \infty\}\) over \(S\) given by the equation \(y^{p - 1} = x^a(x - 1)^b\). This is a finite étale Galois cover with group \(\mathbf{F}_p^*\) and there is a splitting \[f_*(\bar{\mathbf{Q}}_\ell^*) = \bigoplus_{\chi : \mathbf{F}_p^*\to \bar{\mathbf{Q}}_\ell^*} \mathcal{F}_\chi\] where \(\chi\) varies over the characters of \(\mathbf{F}_p^*\) and \(\mathcal{F}_\chi\) is a rank 1 lisse \(\mathbf{Q}_\ell\)-sheaf on which \(\mathbf{F}_p^*\) acts via \(\chi\) on stalks. We get a corresponding decomposition \[H_c^1(X_{\bar k}, \mathbf{Q}_\ell) = \bigoplus_\chi H^1(\mathbf{P}_{\bar k}^1-\{0, 1, \infty\}, \mathcal{F}_\chi)\] and the cohomological interpretation of the exponential sum is given by the trace formula applied to \(\mathcal{F}_\chi\) over \(\mathbf{P}^1 - \{0, 1, \infty\}\) for some suitable \(\chi\). It reads \[S_{a, b}(p) = -\text{Tr}(\pi_X^* |_{H^1(\mathbf{P}_{\bar k}^1-\{0, 1, \infty\}, \mathcal{F}_\chi)}).\] The general yoga of Weil suggests that there should be some cancellation in the sum. Applying (roughly) the Riemann-Hurwitz formula, we see that \[2g_X-2 \approx -2 (p-1) + 3(p-2) \approx p\] so \(g_X\approx p/2\), which also suggests that the \(\chi\)-pieces are small.
Trace formula in terms of fundamental groups
In the following sections we reformulate the trace formula completely in terms of the fundamental group of a curve, except if the curve happens to be \(\mathbf{P}^1\).
Fundamental groups
This material is discussed in more detail in the chapter on fundamental groups. See Fundamental Groups, Section 0BQ7. Let \(X\) be a connected scheme and let \(\overline{x}\to X\) be a geometric point. Consider the functor \[\begin{matrix} F_{\overline{x}}: & \text{ finite \'etale } \atop \text{ schemes over } X & \longrightarrow & \text{ finite sets} \\ & Y/X & \longmapsto & F_{\overline{x}}(Y) = \left\{\text{ geom points }\overline y \atop \text{ of } Y \text{ lying over }\overline{x}\right\} = Y_{\overline{x}} \end{matrix}\] Set \[\pi_1(X, \overline{x}) = Aut(F_{\overline{x}}) = \text{ set of automorphisms of the functor }F_{\overline{x}}\] Note that for every finite étale \(Y \to X\) there is an action \[\pi_1(X, \overline{x}) \times F_{\overline{x}}(Y) \to F_{\overline{x}}(Y)\]
Definition
A subgroup of the form \(\text{Stab}(\overline y\in F_{\overline{x}}(Y))\subset \pi_1(X, \overline{x})\) is called open.
Theorem
Let \(X\) be a connected scheme.
There is a topology on \(\pi_1(X, \overline{x})\) such that the open subgroups form a fundamental system of open nbhds of \(e\in \pi_1(X, \overline x)\).
With topology of (1) the group \(\pi_1(X, \overline{x})\) is a profinite group.
The functor \[\begin{matrix} \text{ schemes finite } \atop \text{ \'etale over }X & \to & \text{ finite discrete continuous } \atop \pi_1(X, \overline{x})\text{-sets}\\ Y / X& \mapsto & F_{\overline{x}}(Y) \text{ with its natural action} \end{matrix}\] is an equivalence of categories.
Proof
See [SGA1].
Proposition
Let \(X\) be an integral normal Noetherian scheme. Let \(\overline y\to X\) be an algebraic geometric point lying over the generic point \(\eta\in X\). Then \[\pi_x(X, \overline \eta) = Gal(M/\kappa(\eta))\] (\(\kappa(\eta)\), function field of \(X\)) where \[\kappa(\overline \eta)\supset M\supset \kappa(\eta) = k(X)\] is the max sub-extension such that for every finite sub extension \(M\supset L\supset \kappa(\eta)\) the normalization of \(X\) in \(L\) is finite étale over \(X\).
Proof
Omitted.
Change of base point. For any \(\overline{x}_1, \overline{x}_2\) geom. points of \(X\) there exists an isom. of fibre functions \[\mathcal{F}_{\overline{x}_1} \cong \mathcal{F}_{\overline{x}_2}\] (This is a path from \(\overline{x}_1\) to \(\overline{x}_2\).) Conjugation by this path gives isom \[\pi_1(X, \overline{x}_1) \cong \pi_1(X, \overline{x}_2)\] well defined up to inner actions.
Functoriality. For any morphism \(X_1\to X_2\) of connected schemes any \(\overline{x}\in X_1\) there is a canonical map \[\pi_1(X_1, \overline{x}) \to \pi_1(X_2, \overline{x})\] (Why? because the fibre functor ...)
Base field. Let \(X\) be a variety over a field \(k\). Then we get \[\pi_1(X, \overline{x}) \to \pi_1(Spec(k), \overline{x}) =^{\text{prop}} Gal(k^{\text{sep}}/k)\] This map is surjective if and only if \(X\) is geometrically connected over \(k\). So in the geometrically connected case we get s.e.s. of profinite groups \[1 \to \pi_1(X_{\overline{k}}, \overline{x}) \to \pi_1(X, \overline{x}) \to Gal(k^{\text{sep}}/k) \to 1\] (\(\pi_1(X_{\overline{k}}, \overline{x})\): geometric fundamental group of \(X\), \(\pi_1(X, \overline{x})\): arithmetic fundamental group of \(X\))
Comparison. If \(X\) is a variety over \(\mathbf{C}\) then \[\pi_1(X, \overline{x}) = \text{ profinite completion of } \pi_1(X(\mathbf{C})(\text{ usual topology}), x)\] (have \(x\in X(\mathbf{C})\))
Frobenii. \(X\) variety over \(k\), \(\# k < \infty\). For any \(x \in X\) closed point, let \[F_x\in \pi_1(x, \overline{x}) = \text{Gal}(\kappa(x)^{\text{sep}}/\kappa(x))\] be the geometric frobenius. Let \(\overline\eta\) be an alg. geom. gen. pt. Then \[\pi_1(X, \overline\eta) \leftarrow^{\cong} \pi_1(X, \overline{x}) {\text{functoriality} \atop \leftarrow} \pi_1(x, \overline{x})\]
Easy fact: \[\begin{matrix} \pi_1(X, \overline \eta) & \to^{\deg} \pi_1(\Spec(k), \overline \eta) * & = Gal(k^{sep}/k) \\ & & || \\ & & \widehat{\mathbf{Z}}\cdot F_{\Spec(k)} \\ F_x & \mapsto & \deg(x)\cdot F_{\Spec(k)} \end{matrix}\] Recall: \(\deg(x) = [\kappa(x):k]\)
Fundamental groups and lisse sheaves. Let \(X\) be a connected scheme, \(\overline{x}\) geom. pt. There are equivalences of categories \[\begin{matrix} (\Lambda\text{ finite ring}) & \text{fin. loc. const. sheaves of } \atop \Lambda\text{-modules of }X_\etale & \leftrightarrow & \text{ finite (discrete) }\Lambda\text{-modules} \atop \text{ with continuous }\pi_1(X, \overline{x})\text{-action}\\ \\ (\ell\text{ a prime}) & \text{ lisse }\ell\text{-adic} \atop \text{ sheaves} & \leftrightarrow & \text{finitely generated }\mathbf{Z}_\ell \text{-modules }M\text{ with continuous} \atop \pi_1(X, \overline{x})\text{-action where we use } \ell\text{-adic topology on }M \end{matrix}\] In particular lisse \(\mathbf{Q}_l\)-sheaves correspond to continuous homomorphisms \[\pi_1(X, \overline{x}) \to \text{GL}_r(\mathbf{Q}_l), \quad r\geq 0\]
Notation: A module with action \((M, \rho)\) corresponds to the sheaf \(\mathcal{F}_\rho\).
Trace formulas. \(X\) variety over \(k\), \(\# k < \infty\).
\(\Lambda\) finite ring \((\# \Lambda, \# k)=1\) \[\rho : \pi_1(X, \overline{x})\to \text{GL}_r(\Lambda)\] continuous. For every \(n\geq 1\) we have \[\sum_{d|n}d\left( \sum_{x\in |X|, \atop \deg(x)=d} \text{Tr}(\rho(F_x^{n/d}))\right) = \text{Tr}\left( (\pi_x^n)^* |_{R\Gamma_c(X_{\overline{k}}, \mathcal{F}_\rho)}\right)\]
\(l\neq char(k)\) prime, \(\rho : \pi_1(X, \overline{x})\to \text{GL}_r(\mathbf{Q}_l)\). For any \(n\geq 1\) \[\sum_{d|n} d\left( \sum_{x\in |X| \atop \deg(x)=d} \text{Tr} \left( \rho(F_x^{n/d}) \right) \right) = \sum_{i = 0}^{2\dim X} (-1)^i \text{Tr}\left( \pi_X^* |_{H_c^i(X_{\overline{k}}, \mathcal{F}_\rho)}\right)\]
Weil conjectures. (Deligne-Weil I, 1974) \(X\) smooth proj. over \(k\), \(\# k = q\), then the eigenvalues of \(\pi_X^*\) on \(H^i(X_{\overline{k}}, \mathbf{Q}_l)\) are algebraic integers \(\alpha\) with \(|\alpha|=q^{1/2}\).
Deligne’s conjectures. (almost completely proved by Lafforgue + \(\ldots\)) Let \(X\) be a normal variety over \(k\) finite \[\rho : \pi_1(X, \overline{x}) \longrightarrow \text{GL}_r(\mathbf{Q}_l)\] continuous. Assume: \(\rho\) irreducible \(\det(\rho)\) of finite order. Then
there exists a number field \(E\) such that for all \(x\in |X|\)(closed points) the char. poly of \(\rho(F_x)\) has coefficients in \(E\).
for any \(x\in |X|\) the eigenvalues \(\alpha_{x, i}\), \(i = 1, \ldots, r\) of \(\rho(F_x)\) have complex absolute value \(1\). (these are algebraic numbers not necessary integers)
for every finite place \(\lambda\)( not dividing \(p\)), of \(E\) (maybe after enlarging \(E\) a bit) there exists \[\rho\lambda : \pi_1(X, \overline{x}) \to \text{GL}_r(E_\lambda)\] compatible with \(\rho\). (some char. polys of \(F_x\)’s)
Theorem
For a sheaf \(\mathcal{F}_\rho\) with \(\rho\) satisfying the conclusions of the conjecture above then the eigenvalues of \(\pi_X^*\) on \(H_c^i(X_{\overline{k}}, \mathcal{F}_{\rho})\) are algebraic numbers \(\alpha\) with absolute values \[|\alpha|=q^{w/2}, \text{ for }w\in \mathbf{Z},\ w\leq i\] Moreover, if \(X\) smooth and proj. then \(w = i\).
Proof
See [WeilII].
Profinite groups, cohomology and homology
Let \(G\) be a profinite group.
Cohomology. Consider the category of discrete modules with continuous \(G\)-action. This category has enough injectives and we can define \[H^i(G, M) = R^iH^0(G, M) = R^i(M\mapsto M^G)\] Also there is a derived version \(RH^0(G, -)\).
Homology. Consider the category of compact abelian groups with continuous \(G\)-action. This category has enough projectives and we can define \[H_i(G, M) = L_iH_0(G, M)=L_i(M\mapsto M_G)\] and there is also a derived version.
Trivial duality. The functor \(M\mapsto M^\wedge = \Hom_{cont}(M, S^1)\) exchanges the categories above and \[H^i(G, M)^\wedge = H_i(G, M^\wedge)\] Moreover, this functor maps torsion discrete \(G\)-modules to profinite continuous \(G\)-modules and vice versa, and if \(M\) is either a discrete or profinite continuous \(G\)-module, then \(M^\wedge = \Hom(M, \mathbf{Q}/\mathbf{Z})\).
Notes on Homology.
If we look at \(\Lambda\)-modules for a finite ring \(\Lambda\) then we can identify \[H_i(G, M)=Tor_i^{\Lambda[[G]]}(M, \Lambda)\] where \(\Lambda[[G]]\) is the limit of the group algebras of the finite quotients of \(G\).
If \(G\) is a normal subgroup of \(\Gamma\), and \(\Gamma\) is also profinite then
\(H^0(G, -)\): discrete \(\Gamma\)-module\(\to\) discrete \(\Gamma/G\)-modules
\(H_0(G, -)\): compact \(\Gamma\)-modules \(\to\) compact \(\Gamma/G\)-modules
and hence the profinite group \(\Gamma/G\) acts on the cohomology groups of \(G\) with values in a \(\Gamma\)-module. In other words, there are derived functors \[RH^0(G, -) : D^{+}(\text{discrete }\Gamma\text{-modules}) \longrightarrow D^{+}(\text{discrete }\Gamma/G\text{-modules})\] and similarly for \(LH_0(G, -)\).
Cohomology of curves, revisited
Let \(k\) be a field, \(X\) be geometrically connected, smooth curve over \(k\). We have the fundamental short exact sequence \[1 \to \pi_1(X_{\overline{k}}, \overline \eta) \to \pi_1(X, \overline\eta) \to \text{Gal}(k^{^{sep}}/k) \to 1\] If \(\Lambda\) is a finite ring with \(\#\Lambda\in k^*\) and \(M\) a finite \(\Lambda\)-module, and we are given \[\rho : \pi_1(X, \overline\eta) \to \text{Aut}_{\Lambda}(M)\] continuous, then \(\mathcal{F}_\rho\) denotes the associated sheaf on \(X_\etale\).
Lemma
There is a canonical isomorphism \[H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)=(M)_{\pi_1(X_{\overline{k}}, \overline\eta)}(-1)\] as \(\text{Gal}(k^{^{sep}}/k)\)-modules.
Here the subscript \({}_{\pi_1(X_{\overline{k}}, \overline\eta)}\) indicates co-invariants, and \((-1)\) indicates the Tate twist i.e., \(\sigma\in \text{Gal}(k^{^{sep}}/k)\) acts via \[\chi_{cycl}(\sigma)^{-1}.\sigma\text{ on RHS}\] where \[\chi_{cycl} : \text{Gal}(k^{^{sep}}/k) \to \prod\nolimits_{l\neq char(k)}\mathbf{Z}_l^*\] is the cyclotomic character.
Reformulation (Deligne, Weil II, page 338). For any finite locally constant sheaf \(\mathcal{F}\) on \(X\) there is a maximal quotient \(\mathcal{F}\to \mathcal{F}''\) with \(\mathcal{F}''/X_{\overline{k}}\) a constant sheaf, hence \[\mathcal{F}'' = (X\to \Spec(k))^{-1}F''\] where \(F''\) is a sheaf \(\Spec(k)\), i.e., a \(\text{Gal}(k^{^{sep}}/k)\)-module. Then \[H_c^2(X_{\overline{k}}, \mathcal{F})\to H_c^2(X_{\overline{k}}, \mathcal{F}'')\to F''(-1)\] is an isomorphism.
Proof
Let \(Y\to^{\varphi}X\) be the finite étale Galois covering corresponding to \(\Ker(\rho) \subset \pi_1(X, \overline\eta)\). So \[\text{Aut}(Y/X)=Ind(\rho)\] is Galois group. Then \(\varphi^*\mathcal{F}_\rho =\underline M_Y\) and \[\varphi_*\varphi^*\mathcal{F}_\rho\to \mathcal{F}_\rho\] which gives \[\begin{align*} & H_c^2(X_{\overline{k}}, \varphi_*\varphi^*\mathcal{F}_\rho) \to H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)\\ & =H_c^2(Y_{\overline{k}}, \varphi^*\mathcal{F}_\rho)\\ & =H_c^2(Y_{\overline{k}}, \underline M) = \oplus_{\text{irred. comp. of } \atop Y_{\overline{k}}}M \end{align*}\] \[\Im(\rho) \to H_c^2(Y_{\overline{k}}, \underline M) = \oplus_{\text{irred. comp. of } \atop Y_{\overline{k}}} M \to_{\Im(\rho) \text{equivalent}} H_c^2(X_{\overline{k}}, \mathcal{F}_{\rho}) \to^{\text{trivial } \Im(\rho) \atop \text{action}}\] irreducible curve \(C/\overline{k}\), \(H_c^2(C, \underline M)=M\).
Since \[{\text{set of irreducible } \atop \text{components of }Y_k} = \frac{Im(\rho)}{Im(\rho|_{\pi_1(X_{\overline{k}}, \overline \eta)})}\] We conclude that \(H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)\) is a quotient of \(M_{\pi_1(X_{\overline{k}}, \overline \eta)}\). On the other hand, there is a surjection \[\mathcal{F}_\rho\to \mathcal{F}'' = {\text{ sheaf on } X\text{ associated to } \atop (M)_{\pi_1(X_{\overline{k}}, \overline \eta)}\leftarrow\pi_1(X, \overline \eta)}\] \[H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)\to M_{\pi_1(X_{\overline{k}}, \overline\eta)}\] The twist in Galois action comes from the fact that \(H_c^2(X_{\overline{k}}, \mu_n)=^{\text{can}} \mathbf{Z}/n\mathbf{Z}\).
Remark
Thus we conclude that if \(X\) is also projective then we have functorially in the representation \(\rho\) the identifications \[H^0(X_{\overline{k}}, \mathcal{F}_\rho) = M^{\pi_1(X_{\overline{k}}, \overline\eta)}\] and \[H_c^2(X_{\overline{k}}, \mathcal{F}_\rho) = M_{\pi_1(X_{\overline{k}}, \overline \eta)}(-1)\] Of course if \(X\) is not projective, then \(H^0_c(X_{\overline{k}}, \mathcal{F}_\rho) = 0\).
Proposition
Let \(X/k\) as before but \(X_{\overline{k}}\neq \mathbf{P}^1_{\overline{k}}\) The functors \((M, \rho)\mapsto H_c^{2-i}(X_{\overline{k}}, \mathcal{F}_\rho)\) are the left derived functor of \((M, \rho)\mapsto H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)\) so \[H_c^{2-i}(X_{\overline{k}}, \mathcal{F}_\rho) = H_i(\pi_1(X_{\overline{k}}, \overline \eta), M)(-1)\] Moreover, there is a derived version, namely \[R\Gamma_c(X_{\overline{k}}, \mathcal{F}_\rho) = LH_0(\pi_1(X_{\overline{k}}, \overline \eta), M(-1)) = M(-1) \otimes_{\Lambda[[\pi_1(X_{\overline{k}}, \overline \eta)]]}^\mathbf{L} \Lambda\] in \(D(\Lambda[[\widehat{\mathbf{Z}}]])\). Similarly, the functors \((M, \rho)\mapsto H^i(X_{\overline{k}}, \mathcal{F}_\rho)\) are the right derived functor of \((M, \rho)\mapsto M^{\pi_1(X_{\overline{k}}, \overline \eta)}\) so \[H^i(X_{\overline{k}}, \mathcal{F}_\rho) = H^i(\pi_1(X_{\overline{k}}, \overline \eta), M)\] Moreover, in this case there is a derived version too.
Proof
(Idea) Show both sides are universal \(\delta\)-functors.
Remark
By the proposition and Trivial duality then you get \[H^{2-i}_c(X_{\overline{k}}, \mathcal{F}_\rho) \times H^i(X_{\overline{k}}, \mathcal{F}_\rho^\wedge(1)) \to \mathbf{Q}/\mathbf{Z}\] a perfect pairing. If \(X\) is projective then this is Poincare duality.
Abstract trace formula
Suppose given an extension of profinite groups, \[1 \to G \to \Gamma \xrightarrow{\deg} \widehat{\mathbf{Z}} \to 1\] We say \(\Gamma\) has an abstract trace formula if and only if there exist
an integer \(q\geq 1\), and
for every \(d\geq 1\) a finite set \(S_d\) and for each \(x\in S_d\) a conjugacy class \(F_x \in \Gamma\) with \(\deg(F_x) = d\)
such that the following hold
for all \(\ell\) not dividing \(q\) have \(\text{cd}_\ell(G)<\infty\), and
for all finite rings \(\Lambda\) with \(q\in \Lambda^*\), for all finite projective \(\Lambda\)-modules \(M\) with continuous \(\Gamma\)-action, for all \(n > 0\) we have \[\sum\nolimits_{d|n}d \left( \sum\nolimits_{x \in S_d} \text{Tr}( F_x^{n/d} |_M) \right) = q^n \text{Tr}(F^n|_{M \otimes_{\Lambda[[G]]}^{\mathbf{L}}\Lambda})\] in \(\Lambda^\natural\).
Here \(M \otimes_{\Lambda[[G]]}^{\mathbf{L}}\Lambda = LH_0(G, M)\) denotes derived homology, and \(F=1\) in \(\Gamma/G = \widehat{\mathbf{Z}}\).
Remark
Here are some observations concerning this notion.
If modeling projective curves then we can use cohomology and we don’t need factor \(q^n\).
The only examples I know are \(\Gamma = \pi_1(X, \overline \eta)\) where \(X\) is smooth, geometrically irreducible and \(K(\pi, 1)\) over finite field. In this case \(q = (\# k)^{\dim X}\). Modulo the proposition, we proved this for curves in this course.
Given the integer \(q\) then the sets \(S_d\) are uniquely determined. (You can multiple \(q\) by an integer \(m\) and then replace \(S_d\) by \(m^d\) copies of \(S_d\) without changing the formula.)
Example
Fix an integer \(q\geq 1\) \[\begin{matrix} 1 & \to & G = \widehat{\mathbf{Z}}^{(q)} & \to & \Gamma & \to & \widehat{\mathbf{Z}} & \to & 1 \\ & & = \prod_{l\not \mid q} \mathbf{Z}_l & & F & \mapsto & 1 \end{matrix}\] with \(FxF^{-1} = ux\), \(u \in (\widehat{\mathbf{Z}}^{(q)})^*\). Just using the trivial modules \(\mathbf{Z}/m\mathbf{Z}\) we see \[q^n - (qu)^n \equiv \sum\nolimits_{d|n} d\# S_d\] in \(\mathbf{Z}/m\mathbf{Z}\) for all \((m, q)=1\) (up to \(u \to u^{-1}\)) this implies \(qu = a\in \mathbf{Z}\) and \(|a| < q\). The special case \(a = 1\) does occur with \[\Gamma = \pi_1^t(\mathbf{G}_{m, \mathbf{F}_p}, \overline \eta), \quad \# S_1 = q - 1, \quad\text{and}\quad \# S_2 = \frac{(q^2-1)-(q-1)}{2}\]
Automorphic forms and sheaves
References: See especially the amazing papers [D1], [D2] and [D0] by Drinfeld.
Unramified cusp forms. Let \(k\) be a finite field of characteristic \(p\). Let \(X\) geometrically irreducible projective smooth curve over \(k\). Set \(K = k(X)\) equal to the function field of \(X\). Let \(v\) be a place of \(K\) which is the same thing as a closed point \(x\in X\). Let \(K_v\) be the completion of \(K\) at \(v\), which is the same thing as the fraction field of the completion of the local ring of \(X\) at \(x\). Denote \(O_v\subset K_v\) the ring of integers. We further set \[O = \prod\nolimits_v O_v \subset \mathbf{A} = \prod_v' K_v\] and we let \(\Lambda\) be any ring with \(p\) invertible in \(\Lambda\).
Definition
An unramified cusp form on \(\text{GL}_2(\mathbf{A})\) with values in \(\Lambda\)3 is a function \[f : \text{GL}_2(\mathbf{A}) \to \Lambda\] such that
\(f(x\gamma) = f(x)\) for all \(x\in \text{GL}_2(\mathbf{A})\) and all \(\gamma\in \text{GL}_2(K)\)
\(f(ux) = f(x)\) for all \(x\in \text{GL}_2(\mathbf{A})\) and all \(u\in \text{GL}_2(O)\)
for all \(x\in \text{GL}_2(\mathbf{A})\), \[\int_{\mathbf{A} \mod K} f \left(x \left( \begin{matrix} 1 & z \\ 0 & 1 \end{matrix} \right) \right) dz = 0\] see [dJ-conjecture, Section 4.1] for an explanation of how to make sense out of this for a general ring \(\Lambda\) in which \(p\) is invertible.
Hecke Operators. For \(v\) a place of \(K\) and \(f\) an unramified cusp form we set \[T_v(f)(x) = \int_{g\in M_v}f(g^{-1}x)dg,\] and \[U_v(f)(x) = f\left( \left( \begin{matrix} \pi_v^{-1} & 0 \\ 0 & \pi_v^{-1} \end{matrix} \right)x\right)\] Notations used: here \(\pi_v \in O_v\) is a uniformizer \[M_v = \left\{ h\in Mat(2\times 2, O_v) | \det h = \pi_vO_v^*\right\}\] and \(dg =\) is the Haar measure on \(\text{GL}_2(K_v)\) with \(\int_{\text{GL}_2(O_v)} dg = 1\). Explicitly we have \[T_v(f)(x) = f\left( \left( \begin{matrix} \pi_v^{-1}& 0 \\ 0 & 1 \end{matrix} \right) x\right) + \sum_{i = 1}^{q_v} f\left(\left( \begin{matrix} 1 & 0 \\ -\pi_v^{-1}\lambda_i & \pi_v^{-1} \end{matrix} \right) x\right)\] with \(\lambda_i\in O_v\) a set of representatives of \(O_v/(\pi_v)=\kappa_v\), \(q_v = \#\kappa_v\).
Eigenforms. An eigenform \(f\) is an unramified cusp form such that some value of \(f\) is a unit and \(T_vf = t_vf\) and \(U_vf = u_vf\) for some (uniquely determined) \(t_v, u_v \in \Lambda\).
Theorem
Given an eigenform \(f\) with values in \(\overline{\mathbf{Q}}_l\) and eigenvalues \(u_v\in \overline{\mathbf{Z}}_l^*\) then there exists \[\rho : \pi_1(X)\to \text{GL}_2(E)\] continuous, absolutely irreducible where \(E\) is a finite extension of \(\mathbf{Q}_\ell\) contained in \(\overline{\mathbf{Q}}_l\) such that \(t_v = \text{Tr}(\rho(F_v))\), and \(u_v = q_v^{-1}\det\left(\rho(F_v)\right)\) for all places \(v\).
Proof
See [D0].
Theorem
Suppose \(\mathbf{Q}_l \subset E\) finite, and \[\rho : \pi_1(X)\to \text{GL}_2(E)\] absolutely irreducible, continuous. Then there exists an eigenform \(f\) with values in \(\overline{\mathbf{Q}}_l\) whose eigenvalues \(t_v\), \(u_v\) satisfy the equalities \(t_v = \text{Tr}(\rho(F_v))\) and \(u_v = q_v^{-1}\det(\rho(F_v))\).
Proof
See [D1].
Remark
We now have, thanks to Lafforgue and many other mathematicians, complete theorems like this two above for \(\text{GL}_n\) and allowing ramification! In other words, the full global Langlands correspondence for \(\text{GL}_n\) is known for function fields of curves over finite fields. At the same time this does not mean there aren’t a lot of interesting questions left to answer about the fundamental groups of curves over finite fields, as we shall see below.
Central character. If \(f\) is an eigenform then \[\begin{matrix} \chi_f : & O^*\backslash \mathbf{A}^*/K^* & \to & \Lambda^* \\ & (1, \ldots, \pi_v, 1, \ldots, 1) & \mapsto & u_v^{-1} \end{matrix}\] is called the central character. If corresponds to the determinant of \(\rho\) via normalizations as above. Set \[C(\Lambda) = \left\{ {\text{unr. cusp forms } f \text{ with coefficients in }\Lambda} \atop {\text{ such that } U_v f = \varphi_v^{-1}f\forall v} \right\}\]
Proposition
If \(\Lambda\) is Noetherian then \(C(\Lambda)\) is a finitely generated \(\Lambda\)-module. Moreover, if \(\Lambda\) is a field with prime subfield \(\mathbf{F} \subset \Lambda\) then \[C(\Lambda)=(C(\mathbf{F}))\otimes_{\mathbf{F}}\Lambda\] compatibly with \(T_v\) acting.
Proof
This proposition trivially implies the following lemma.
Lemma
Algebraicity of eigenvalues. If \(\Lambda\) is a field then the eigenvalues \(t_v\) for \(f\in C(\Lambda)\) are algebraic over the prime subfield \(\mathbf{F} \subset \Lambda\).
Proof
Follows from Proposition 03VW.
Combining all of the above we can do the following very useful trick.
Lemma
Switching \(l\). Let \(E\) be a number field. Start with \[\rho : \pi_1(X)\to SL_2(E_\lambda)\] absolutely irreducible continuous, where \(\lambda\) is a place of \(E\) not lying above \(p\). Then for any second place \(\lambda'\) of \(E\) not lying above \(p\) there exists a finite extension \(E'_{\lambda'}\) and a absolutely irreducible continuous representation \[\rho': \pi_1(X)\to SL_2(E'_{\lambda'})\] which is compatible with \(\rho\) in the sense that the characteristic polynomials of all Frobenii are the same.
Note how this is an instance of Deligne’s conjecture!
Proof
To prove the switching lemma use Theorem 03VU to obtain \(f\in C(\overline{\mathbf{Q}}_l)\) eigenform ass. to \(\rho\). Next, use Proposition 03VW to see that we may choose \(f\in C(E')\) with \(E \subset E'\) finite. Next we may complete \(E'\) to see that we get \(f\in C(E'_{\lambda'})\) eigenform with \(E'_{\lambda'}\) a finite extension of \(E_{\lambda'}\). And finally we use Theorem 03VT to obtain \(\rho': \pi_1(X) \to SL_2(E_{\lambda'}')\) abs. irred. and continuous after perhaps enlarging \(E'_{\lambda'}\) a bit again.
Speculation: If for a (topological) ring \(\Lambda\) we have \[\left( {\rho : \pi_1(X)\to SL_2(\Lambda) \atop \text{ abs irred}} \right) \leftrightarrow \text{ eigen forms in } C(\Lambda)\] then all eigenvalues of \(\rho(F_v)\) algebraic (won’t work in an easy way if \(\Lambda\) is a finite ring. Based on the speculation that the Langlands correspondence works more generally than just over fields one arrives at the following conjecture.
Conjecture. (See [dJ-conjecture]) For any continuous \[\rho : \pi_1(X)\to \text{GL}_n(\mathbf{F}_l[[t]])\] we have \(\# \rho(\pi_1(X_{\overline{k}}))<\infty\).
A rephrasing in the language of sheaves: "For any lisse sheaf of \(\overline{\mathbf{F}_l((t))}\)-modules the geom monodromy is finite."
Theorem
The Conjecture holds if \(n\leq 2\).
Proof
See [dJ-conjecture].
Theorem
Conjecture holds if \(l > 2n\) modulo some unproven things.
Proof
See [Gaitsgory].
It turns out the conjecture is useful for something. See work of Drinfeld on Kashiwara’s conjectures. But there is also the much more down to earth application as follows.
Theorem
(See [dJ-conjecture, Theorem 3.5]) Suppose \[\rho_0: \pi_1(X)\to \text{GL}_n(\mathbf{F}_l)\] is a continuous, \(l\neq p\). Assume
Conj. holds for \(X\),
\(\rho_0 |_{\pi_1(X_{\overline{k}})}\) abs. irred., and
\(l\) does not divide \(n\).
Then the universal deformation ring \(R_{\text{univ}}\) of \(\rho_0\) is finite flat over \(\mathbf{Z}_l\).
Explanation: There is a representation \(\rho_{\text{univ}}: \pi_1(X)\to \text{GL}_n(R_{\text{univ}})\) (Univ. Defo ring) \(R_{\text{univ}}\) loc. complete, residue field \(\mathbf{F}_l\) and \((R_{\text{univ}}\to \mathbf{F}_l)\circ\rho_{\text{univ}}\cong\rho_0\). And given any \(R\to \mathbf{F}_l\), \(R\) local complete and \(\rho : \pi_1(X)\to \text{GL}_n(R)\) then there exists \(\psi : R_{\text{univ}}\to R\) such that \(\psi\circ\rho_{\text{univ}}\cong \rho\). The theorem says that the morphism \[\Spec(R_{\text{univ}}) \longrightarrow \Spec(\mathbf{Z}_l)\] is finite and flat. In particular, such a \(\rho_0\) lifts to a \(\rho : \pi_1(X) \to \text{GL}_n(\overline{\mathbf{Q}}_l)\).
Notes:
The theorem on deformations is easy.
Any result towards the conjecture seems hard.
It would be interesting to have more conjectures on \(\pi_1(X)\)!
Counting points
Let \(X\) be a smooth, geometrically irreducible, projective curve over \(k\) and \(q = \# k\). The trace formula gives: there exists algebraic integers \(w_1, \ldots, w_{2g}\) such that \[\# X(k_n) = q^n - \sum\nolimits_{i = 1}^{2g_X} w_i^n + 1.\] If \(\sigma\in \text{Aut}(X)\) then for all \(i\), there exists \(j\) such that \(\sigma(w_i)=w_j\).
Riemann-Hypothesis. For all \(i\) we have \(|\omega_i| = \sqrt{q}\).
This was formulated by Emil Artin, in 1924, for hyperelliptic curves. Proved by Weil 1940. Weil gave two proofs
using intersection theory on \(X \times X\), using the Hodge index theorem, and
using the Jacobian of \(X\).
There is another proof whose initial idea is due to Stephanov, and which was given by Bombieri: it uses the function field \(k(X)\) and its Frobenius operator (1969). The starting point is that given \(f\in k(X)\) one observes that \(f^q - f\) is a rational function which vanishes in all the \(\mathbf{F}_q\)-rational points of \(X\), and that one can try to use this idea to give an upper bound for the number of points.
Precise form of Chebotarev
As a first application let us prove a precise form of Chebotarev for a finite étale Galois covering of curves. Let \(\varphi : Y \to X\) be a finite étale Galois covering with group \(G\). This corresponds to a homomorphism \[\pi_1(X) \longrightarrow G = \text{Aut}(Y/X)\] Assume \(Y_{\overline{k}} =\) irreducible. If \(C\subset G\) is a conjugacy class then for all \(n > 0\), we have \[| \# \{x \in X(k_n) \mid F_x \in C\} - \frac{\# C}{\# G} \cdot \# X(k_n) | \leq (\# C)(2g - 2) \sqrt{q^n}\] (Warning: Please check the coefficient \(\# C\) on the right hand side carefully before using.)
Proof
Write \[\varphi_*(\overline{\mathbf{Q}_l}) = \oplus_{\pi \in \widehat{G}} \mathcal{F}_{\pi}\] where \(\widehat{G}\) is the set of isomorphism classes of irred representations of \(G\) over \(\overline{\mathbf{Q}}_l\). For \(\pi \in \widehat{G}\) let \(\chi_{\pi}: G \to \overline{\mathbf{Q}}_l\) be the character of \(\pi\). Then \[H^*(Y_{\overline{k}}, \overline{\mathbf{Q}}_l) = \oplus_{\pi\in \widehat{G}} H^*(Y_{\overline{k}}, \overline{\mathbf{Q}}_l)_\pi =_{(\varphi\text{ finite })} \oplus_{\pi\in \widehat{G}} H^*(X_{\overline{k}}, \mathcal{F}_\pi)\] If \(\pi\neq 1\) then we have \[H^0(X_{\overline{k}}, \mathcal{F}_\pi) = H^2(X_{\overline{k}}, \mathcal{F}_\pi) = 0,\quad \dim H^1(X_{\overline{k}}, \mathcal{F}_\pi) = (2g_X - 2)d_\pi^2\] (can get this from trace formula for acting on ...) and we see that \[|\sum_{x \in X(k_n)} \chi_\pi(\mathcal{F}_x)| \leq (2g_X - 2) d_\pi^2\sqrt{q^n}\] Write \(1_C = \sum_\pi a_\pi \chi_\pi\), then \(a_\pi = \langle 1_C, \chi_\pi\rangle\), and \(a_1 = \langle 1_C, \chi_1\rangle = \frac{\# C}{\# G}\) where \[\langle f, h\rangle = \frac{1}{\# G}\sum_{g \in G} f(g)\overline{h(g)}\] Thus we have the relation \[\frac{\# C}{\# G} = ||1_C||^2 = \sum|a_\pi|^2\] Final step: \[\begin{align*} \#\left\{x \in X(k_n) \mid F_x \in C\right\} & = \sum_{x \in X(k_n)} 1_C(x) \\ & = \sum_{x \in X(k_n)} \sum_\pi a_\pi \chi_\pi(F_x) \\ & = \underbrace{\frac{\# C}{\# G} \# X(k_n)}_{ \text{term for }\pi = 1} + \underbrace{\sum_{\pi\neq 1}a_\pi\sum_{x\in X(k_n)}\chi_\pi(F_x)}_{ \text{ error term (to be bounded by }E)} \end{align*}\] We can bound the error term by \[\begin{align*} |E| & \leq \sum_{\pi \in \widehat{G}, \atop \pi \neq 1} |a_\pi| (2g - 2) d_\pi^2 \sqrt{q^n} \\ & \leq \sum_{\pi \neq 1} \frac{\# C}{\# G} (2g_X - 2) d_\pi^3 \sqrt{q^n} \end{align*}\] By Weil’s conjecture, \(\# X(k_n)\sim q^n\).
How many primes decompose completely?
This section gives a second application of the Riemann Hypothesis for curves over a finite field. For number theorists it may be nice to look at the paper by Ihara, entitled “How many primes decompose completely in an infinite unramified Galois extension of a global field?”, see [Ihara]. Consider the fundamental exact sequence \[1 \to \pi_1(X_{\overline{k}}) \to \pi_1(X) \xrightarrow{\deg} \widehat{\mathbf{Z}} \to 1\]
Proposition
There exists a finite set \(x_1, \ldots, x_n\) of closed points of \(X\) such that set of all frobenius elements corresponding to these points topologically generate \(\pi_1(X)\).
Another way to state this is: There exist \(x_1, \ldots, x_n\in |X|\) such that the smallest normal closed subgroup \(\Gamma\) of \(\pi_1(X)\) containing \(1\) frobenius element for each \(x_i\) is all of \(\pi_1(X)\). i.e., \(\Gamma = \pi_1(X)\).
Proof
Pick \(N\gg 0\) and let \[\{x_1, \ldots, x_n\} = {\text{ set of all closed points of} \atop X \text{ of degree} \leq N\text{ over } k}\] Let \(\Gamma\subset \pi_1(X)\) be as in the variant statement for these points. Assume \(\Gamma \neq \pi_1(X)\). Then we can pick a normal open subgroup \(U\) of \(\pi_1(X)\) containing \(\Gamma\) with \(U \neq \pi_1(X)\). By R.H. for \(X\) our set of points will have some \(x_{i_1}\) of degree \(N\), some \(x_{i_2}\) of degree \(N - 1\). This shows \(\deg : \Gamma \to \widehat{\mathbf{Z}}\) is surjective and so the same holds for \(U\). This exactly means if \(Y \to X\) is the finite étale Galois covering corresponding to \(U\), then \(Y_{\overline{k}}\) irreducible. Set \(G = \text{Aut}(Y/X)\). Picture \[Y \to^G X,\quad G = \pi_1(X)/U\] By construction all points of \(X\) of degree \(\leq N\), split completely in \(Y\). So, in particular \[\# Y(k_N)\geq (\# G)\# X(k_N)\] Use R.H. on both sides. So you get \[q^N+1+2g_Yq^{N/2}\geq \# G\# X(k_N)\geq \# G(q^N+1-2g_Xq^{N/2})\] Since \(2g_Y-2 = (\# G)(2g_X-2)\), this means \[q^N + 1 + (\# G)(2g_X - 1) + 1)q^{N/2}\geq \# G (q^N + 1 - 2g_Xq^{N/2})\] Thus we see that \(G\) has to be the trivial group if \(N\) is large enough.
Weird Question. Set \(W_X = \deg^{-1}(\mathbf{Z})\subset \pi_1(X)\). Is it true that for some finite set of closed points \(x_1, \ldots, x_n\) of \(X\) the set of all frobenii corresponding to these points algebraically generate \(W_X\)?
By a Baire category argument this translates into the same question for all Frobenii.
How many points are there really?
If the genus of the curve is large relative to \(q\), then the main term in the formula \(\# X(k) = q - \sum \omega_i + 1\) is not \(q\) but the second term \(\sum \omega_i\) which can (a priori) have size about \(2g_X\sqrt{q}\). In the paper [Drinfeld-number] the authors Drinfeld and Vladut show that this maximum is (as predicted by Ihara earlier) actually at most about \(g\sqrt{q}\).
Fix \(q\) and let \(k\) be a field with \(k\) elements. Set \[A(q) = \limsup_{g_X \to \infty} \frac{\# X(k)}{g_X}\] where \(X\) runs over geometrically irreducible smooth projective curves over \(k\). With this definition we have the following results:
RH \(\Rightarrow A(q)\leq 2\sqrt{q}\)
Ihara \(\Rightarrow A(q)\leq \sqrt{2q}\)
DV \(\Rightarrow A(q)\leq \sqrt{q}-1\) (actually this is sharp if \(q\) is a square)
Proof
Given \(X\) let \(w_1, \ldots, w_{2g}\) and \(g = g_X\) be as before. Set \(\alpha_i = \frac{w_i}{\sqrt{q}}\), so \(|\alpha_i| = 1\). If \(\alpha_i\) occurs then \(\overline{\alpha}_i = \alpha_i^{-1}\) also occurs. Then \[N = \# X(k) \leq X(k_r) = q^r + 1 - (\sum_i \alpha_i^r) q^{r/2}\] Rewriting we see that for every \(r \geq 1\) \[-\sum_i \alpha_i^r \geq Nq^{-r/2} - q^{r/2} - q^{-r/2}\] Observe that \[0 \leq |\alpha_i^n +\alpha_i^{n-1} +\ldots +\alpha_i +1|^2 = (n + 1) + \sum_{j = 1}^n (n + 1 - j) (\alpha_i^j + \alpha_i^{-j})\] So \[\begin{align*} 2g(n+1) & \geq - \sum_i \left(\sum_{j = 1}^n (n+1-j)(\alpha_i^j +\alpha_i^{-j})\right)\\ & =-\sum_{j = 1}^n (n+1-j)\left(\sum_i\alpha_i^j +\sum_i\alpha_i^{-j}\right) \end{align*}\] Take half of this to get \[\begin{align*} g(n+1)& \geq - \sum_{j = 1}^n (n+1-j)(\sum_i\alpha_i^j)\\ & \geq N\sum_{j = 1}^n (n+1-j)q^{-j/2}-\sum_{j = 1}^n (n+1-j)(q^{j/2}+q^{-j/2}) \end{align*}\] This gives \[\frac{N}{g}\leq \left(\sum_{j = 1}^n \frac{n+1-j}{n+1}q^{-j/2} \right)^{-1} \cdot \left( 1 + \frac{1}{g} \sum_{j = 1}^n \frac{n + 1 - j}{n + 1}(q^{j/2} + q^{-j/2}) \right)\] Fix \(n\) let \(g\to \infty\) \[A(q)\leq \left(\sum_{j = 1}^n \frac{n+1-j}{n+1}q^{-j/2}\right)^{-1}\] So \[A(q)\leq \lim_{n\to\infty}(\ldots) = \left(\sum_{j = 1}^\infty q^{-j/2}\right)^{-1}=\sqrt{q}-1\]