The Stacks project

63.14 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 63.14.1. 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 $.

Definition 63.14.2. With $\Lambda , X, k, K$ as in Definition 63.14.1. 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 63.3, 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 63.14.3 (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.,
\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 63.21).

We only stated the formula for curves, and in some weak sense it is a consequence of the following result.

Theorem 63.14.4 (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 63.14.5. 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

\[ A^1(C \times C) = \text{pr}_1^* (A^1(C)) \oplus \text{pr}_2^* (A^1(C)) \oplus R \]


\[ R = \{ Z \in A^1(C \times C) \ | \ Z|_{C \times \{ c_0\} } \sim _\text {rat} 0 \text{ and } Z|_{\{ c_0\} \times C} \sim _\text {rat} 0 \} . \]

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 63.14.6. Consider the situation of Theorem 63.14.4 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 63.14.4. $\square$

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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