Definition 64.14.1. 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 .
64.14 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 64.14.2. With \Lambda , X, k, K as in Definition 64.14.1. 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 64.3, 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 64.14.3 (Lefschetz Trace Formula). 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.,
64.14.3.1
\begin{equation} \label{trace-equation-trace-formula} \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. \square
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 64.21).
We only stated the formula for curves, and in some weak sense it is a consequence of the following result.
Theorem 64.14.4 (Weil). 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{\mathrm{Pic}}^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]. \square
Example 64.14.5. 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 \mathop{\mathrm{Pic}}\nolimits (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} \text{End}(J)
& R
\\ \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} }
\\ \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 64.14.6. Consider the situation of Theorem 64.14.4 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.
Sketch of 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 ^ nth 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}) = \mathop{\mathrm{Pic}}\nolimits (C)/\ell ^ n\mathop{\mathrm{Pic}}\nolimits (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 64.14.4. \square
An alternative way to prove this corollary is to show that
X \mapsto H^* (X, \mathbf{Q}_\ell ) = \mathbf{Q}_\ell \otimes \mathop{\mathrm{lim}}\nolimits _ 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).
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