Lemma 63.23.1. Let $X$ be a smooth, projective, geometrically irreducible curve over a finite field $k$. Then

1. the $L$-function $L(X, \mathbf{Q}_\ell )$ is a rational function,

2. the eigenvalues $\alpha _1, \ldots , \alpha _{2g}$ of $\pi _ X^*$ on $H^1(X_{\bar k}, \mathbf{Q}_\ell )$ are algebraic integers independent of $\ell$,

3. the number of rational points of $X$ on $k_ n$, where $[k_ n : k] = n$, is

$\# X(k_ n) = 1 - \sum \nolimits _{i = 1}^{2g}\alpha _ i^ n + q^ n,$
4. for each $i$, $|\alpha _ i| < q$.

Proof. Part (3) is Theorem 63.20.5 applied to $\mathcal{F} = \underline{\mathbf{Q}_\ell }$ on $X \otimes k_ n$. For part (4), use the following result. $\square$

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).