I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any variety $X/\mathbb Z$, we should be able to find an automorphic form $f$ so that $\zeta(X) = \zeta(f)$ where $\zeta(f)$ is the Hasse-Weil zeta function and $\zeta(f)$ is the L-function associated to an automorphic form.
Question 1: What automorphic objects do the zeta functions of varieties over finite fields correspond to?
Tentative answer: It seems to me that these should correspond to euler factors at a prime of an automorphic form. (Lift the variety to characteristic 0, find the zeta function of that and take the corresponding automorphic form).
Is this on the right track?
Question 2: If so, what property of the automorphic forms does the Riemann hypothesis for the Weil conjectures over finite fields (proved by Deligne) correspond to? More generally, what do the Weil Conjectures correspond to on the automorphic side.
I can see two possibilities:
1) They correspond to some conjectural property of automorphic forms and the proof of the Weil conjectures actually tells us something about (a subclass of) automorphic forms.
2) They correspond to some known (easy to prove?) property of automorphic forms and the "reason" Deligne/Grothendieck had to work so hard is because we don't know how to associate automorphic forms to motivic L functions.
Question 3: Which one is right?