The Stacks project

63.19 L-functions

Definition 63.19.1. Let $X$ be a scheme of finite type over a finite field $k$. Let $\Lambda $ be a finite ring of order prime to the characteristic of $k$ and $\mathcal{F}$ a constructible flat $\Lambda $-module on $X_{\acute{e}tale}$. Then we set

\[ L(X, \mathcal{F}) := \prod \nolimits _{x\in |X|} \det (1 - \pi _ x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \Lambda [[ T ]] \]

where $|X|$ is the set of closed points of $X$, $\deg x = [\kappa (x): k]$ and $\bar x$ is a geometric point lying over $x$. This definition clearly generalizes to the case where $\mathcal{F}$ is replaced by a $K \in D_{ctf}(X, \Lambda )$. We call this the $L$-function of $\mathcal{F}$.

Remark 63.19.2. Intuitively, $T$ should be thought of as $T = t^ f$ where $p^ f = \# k$. The definitions are then independent of the size of the ground field.

Definition 63.19.3. Now assume that $\mathcal{F}$ is a $\mathbf{Q}_\ell $-sheaf on $X$. In this case we define

\[ L(X, \mathcal{F}) := \prod \nolimits _{x \in |X|} \det (1 - \pi _ x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \mathbf{Q}_\ell [[T]]. \]

Note that this product converges since there are finitely many points of a given degree. We call this the $L$-function of $\mathcal{F}$.

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Your email address will not be published.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 03UU. Beware of the difference between the letter 'O' and the digit '0'.