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.,

63.14.3.1
\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 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

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

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 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).


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03TW. Beware of the difference between the letter 'O' and the digit '0'.