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 $.

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 $.

**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

**Proof.**
The number $V(\varphi )$ is the number of fixed points of $\varphi $, it is equal to

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

**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

where

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

The aforementioned correspondence is the following. We denote by $\sigma $ the automorphism of $C \times C$ that switches the factors.

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

Now, on the one hand

and on the other hand, since $R$ is the orthogonal of the ample divisor $\{ c_0\} \times C + C \times \{ c_0\} $,

Recapitulating, we have

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

**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

compatibly with the action of the geometric Frobenius. On the other hand, $H^1(C, \mu _{\ell ^ n}) = J[\ell ^ n]$. Therefore,

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

Thus we have

and the corollary follows from Theorem 64.14.4. $\square$

An alternative way to prove this corollary is to show that

defines a Weil cohomology theory on smooth projective varieties over $k$. Then the trace formula

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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