Do $L$-functions of varieties over function fields have analytic continuation / functional equations?

Question

Suppose that $K$ is a function field (i.e: a finite extension of $\mathbf{F}_q(t)$) and $X$ is a smooth projective variety over $K$. I have a few questions about the $L$-function of $X/K$:

1. How do you define the global $L$-function of $X/K$, denoted $L(X/K, s)$? I'm vaguely aware that $L(X/K, s)$ is an Euler product of local zeta functions over finite fields. I can't seem to find the definition written down anywhere; does anyone have a reference?
2. Is it known that this $L$-function $L(X/K, s)$ has an analytic continutation to all of $\mathbf{C}$ and a functional equation? I'm vaguely aware that this should follow from some kind of Langlands correspondence over function fields (which I believe has been proven?) but I am quite fuzzy on the details.

Answer 1

$ \newcommand{\F}{\mathbb{F}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Jac}{\mathrm{Jac}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\sep}{\mathrm{sep}} \newcommand{\P}{\mathbb{P}} $

In general, we define the L-function of a Galois representation, not of a variety. In the case of a curve $C$, we typically take the $\ell$-adic Galois representation coming from the Tate module $V_\ell \mathrm{Jac}(C)$ of its jacobian. In any case, given a finite-dimensional $\Q_\ell$-vector space $V$ for some $\ell$ coprime to $q$ and a continuous representation $\rho : \mathrm{Gal}(\F_q(t)^{\sep}) / \F_q(t)) \to \GL(V)$ unramified almost everywhere, we define

$$L(\rho, T) := \prod_{x \in |\P^1|} \det( \mathrm{id} - \rho(\mathrm{Fr}_x) t^{\deg(x)} | V^{I_x} )$$

where $|\P^1|$ denotes the set of closed points of $\P^1$ (i.e., the set of Galois-orbits of $\overline{\F_q}$-rational points in $\P^1(\overline \F_q)$), $I_x$ is the inertia subgroup at the place $x$, and $\mathrm{Fr}_x$ is an arithmetic Frobenius conjugacy class.

It is known by works of Grothendieck et al. that $L(\rho, T)$ is a rational function, i.e. belongs to $\mathbb{Q}(T)$. Hence $s \mapsto L(\rho, |k|^{-s})$ defines a entire function on $\mathbb{C}$. See example 13.6 in Milne's book Etale Cohomology (there is a typo in the definition of the L-function, he missed the "$\deg(x)$").

The proof of Langlands correspondence over function fields by Lafforgue is a different story; it relates those Galois representations $\rho$ to automorphic ones.