64.23 Constant sheaves
Let $k$ be a finite field, $X$ a smooth, geometrically irreducible curve over $k$ and $\mathcal{F} = \underline{\mathbf{Q}_\ell }$ the constant sheaf. If $\bar x$ is a geometric point of $X$, the Galois module $\mathcal{F}_{\bar x} = \mathbf{Q}_\ell $ is trivial, so
\[ \det (1-\pi _ x^*\ T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} = \frac{1}{1-T^{\deg x}}. \]
Applying Theorem 64.20.2, we get
\begin{align*} L(X, \mathcal{F}) & = \prod _{i = 0}^2 \det (1 - \pi _ X^*T |_{H_ c^ i(X_{\bar k}, \mathbf{Q}_\ell )})^{(-1)^{i+1}} \\ & = \frac{\det (1 - \pi _ X^*T |_{H_ c^1(X_{\bar k}, \mathbf{Q}_\ell )})}{ \det (1 - \pi _ X^*T |_{H_ c^0(X_{\bar k}, \mathbf{Q}_\ell )}) \cdot \det (1 - \pi _ X^*T |_{H_ c^2(X_{\bar k}, \mathbf{Q}_\ell )})}. \end{align*}
To compute the latter, we distinguish two cases.
Projective case. Assume that $X$ is projective, so $H_ c^ i(X_{\bar k}, \mathbf{Q}_\ell ) = H^ i(X_{\bar k}, \mathbf{Q}_\ell )$, and we have
\[ H^ i(X_{\bar k}, \mathbf{Q}_\ell ) = \left\{ \begin{matrix} \mathbf{Q}_\ell
& \pi _ X^* = 1
& \text{if }i = 0,
\\ \mathbf{Q}_\ell ^{2g}
& \pi _ X^* = ?
& \text{if }i = 1,
\\ \mathbf{Q}_\ell
& \pi _ X^* = q
& \text{if }i = 2.
\end{matrix} \right. \]
The identification of the action of $\pi _ X^*$ on $H^2$ comes from Étale Cohomology, Lemma 59.69.2 and the fact that the degree of $\pi _ X$ is $q = \# (k)$. We do not know much about the action of $\pi _ X^*$ on the degree 1 cohomology. Let us call $\alpha _1, \ldots , \alpha _{2g}$ its eigenvalues in $\bar{\mathbf{Q}}_\ell $. Putting everything together, Theorem 64.20.2 yields the equality
\[ \prod \nolimits _{x \in |X|} \frac{1}{1 - T^{\deg x}} = \frac{\det (1- \pi _ X^* T|_{H^1(X_{\bar k}, \mathbf{Q}_\ell )})}{(1-T)(1-qT)} = \frac{(1 - \alpha _1 T) \ldots (1 - \alpha _{2g}T)}{(1-T)(1-qT)} \]
from which we deduce the following result.
Lemma 64.23.1. Let $X$ be a smooth, projective, geometrically irreducible curve over a finite field $k$. Then
the $L$-function $L(X, \mathbf{Q}_\ell )$ is a rational function,
the eigenvalues $\alpha _1, \ldots , \alpha _{2g}$ of $\pi _ X^*$ on $H^1(X_{\bar k}, \mathbf{Q}_\ell )$ are algebraic integers independent of $\ell $,
the number of rational points of $X$ on $k_ n$, where $[k_ n : k] = n$, is
\[ \# X(k_ n) = 1 - \sum \nolimits _{i = 1}^{2g}\alpha _ i^ n + q^ n, \]
for each $i$, $|\alpha _ i| < q$.
Proof.
Part (3) is Theorem 64.20.5 applied to $\mathcal{F} = \underline{\mathbf{Q}_\ell }$ on $X \otimes k_ n$. For part (4), use the following result.
$\square$
Exercise 64.23.2. Let $\alpha _1, \ldots , \alpha _ n \in \mathbf{C}$. Then for any conic sector containing the positive real axis of the form $C_\varepsilon = \{ z \in \mathbf{C} \ | \ |\arg z| < \varepsilon \} $ with $\varepsilon > 0$, there exists an integer $k \geq 1$ such that $\alpha _1^ k, \ldots , \alpha _ n^ k \in C_\varepsilon $.
Then prove that $|\alpha _ i| \leq q$ for all $i$. Then, use elementary considerations on complex numbers to prove (as in the proof of the prime number theorem) that $|\alpha _ i| < q$. In fact, the Riemann hypothesis says that for all $|\alpha _ i| = \sqrt{q}$ for all $i$. We will come back to this later.
Affine case. Assume now that $X$ is affine, say $X= \bar X-\left\{ x_1, \ldots , x_ n\right\} $ where $j : X \hookrightarrow \bar X$ is a projective nonsingular completion. Then $H_ c^0(X_{\bar k}, \mathbf{Q}_\ell ) = 0$ and $H_ c^2(X_{\bar k}, \mathbf{Q}_\ell ) = H^2(\bar X_{\bar k}, \mathbf{Q}_\ell )$ so Theorem 64.20.2 reads
\[ L(X, \mathbf{Q}_\ell ) = \prod _{x \in |X|}\frac{1}{1 - T^{\deg x}} = \frac{\det (1-\pi _ X^*T |_{H_ c^1(X_{\bar k}, \mathbf{Q}_\ell )})}{1 - qT}. \]
On the other hand, the previous case gives
\begin{eqnarray*} L(X, \mathbf{Q}_\ell ) & = & L(\bar X, \mathbf{Q}_\ell )\prod _{i = 1}^ n\left(1-T^{\deg x_ i}\right) \\ & = & \frac{\prod _{i = 1}^ n(1-T^{\deg x_ i})\prod _{j = 1}^{2g}(1-\alpha _ jT)}{(1-T)(1-qT)}. \end{eqnarray*}
Therefore, we see that $\dim H_ c^1(X_{\bar k}, \mathbf{Q}_\ell ) = 2g+\sum _{i = 1}^ n \deg (x_ i)-1$, and the eigenvalues $\alpha _1, \ldots , \alpha _{2g}$ of $\pi _{\bar X}^*$ acting on the degree 1 cohomology are roots of unity. More precisely, each $x_ i$ gives a complete set of $\deg (x_ i)$th roots of unity, and one occurrence of 1 is omitted. To see this directly using coherent sheaves, consider the short exact sequence on $\bar X$
\[ 0\to j_!\mathbf{Q}_\ell \to \mathbf{Q}_\ell \to \bigoplus _{i = 1}^ n \mathbf{Q}_{\ell , x_ i}\to 0. \]
The long exact cohomology sequence reads
\[ 0\to \mathbf{Q}_\ell \to \bigoplus _{i = 1}^ n \mathbf{Q}_\ell ^{\oplus \deg x_ i} \to H_ c^1(X_{\bar k}, \mathbf{Q}_\ell ) \to H_ c^1(\bar X_{\bar k}, \mathbf{Q}_\ell )\to 0 \]
where the action of Frobenius on $\bigoplus _{i = 1}^ n \mathbf{Q}_\ell ^{\oplus \deg x_ i}$ is by cyclic permutation of each term; and $H_ c^2(X_{\bar k}, \mathbf{Q}_\ell ) = H_ c^2(\bar X_{\bar k}, \mathbf{Q}_\ell )$.
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