## 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 $\mathcal{F}$.

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