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 which is again an element of $\Lambda ^\natural $.
\[ \sum \nolimits _{x\in X(k)} \text{Tr}(\pi _ X |_{K_{\overline{x}}}) \]
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., in $\Lambda ^\natural $.
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}
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 $\ell ^ n$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, $H^2(C, \mu _{\ell ^ n}) = \mathop{\mathrm{Pic}}\nolimits (C)/\ell ^ n\mathop{\mathrm{Pic}}\nolimits (C) \cong \mathbf{Z}/\ell ^ n \mathbf{Z}$ 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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