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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.

This is a brief answer; possibly others have different opinions about this.

Question 1: The Langlands conjectures gives a correspondence between Galois representations and automorphic forms. So a very naive way to view the zeta function of a variety over a finite fields from this philosophy is to look for a Galois representation which it comes from. However, this is exactly the point of the Weil conjectures; the zeta function has a description in terms of the action of the absolute Galois group of the finite field on the etale cohomology of the variety. These indeed give the Euler factors.

Question 2: The Riemann hypothesis for the Weil conjectures over finite fields corresponds to the Ramanjuan conjecture for automorphic forms. In fact, this was one of Deligne's original applications of the Weil conjectures, to proving the Ramanjuan conjecture for the Ramanjuan tau function.