# Relation between Weil Conjecture and Langlands Program

Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the associated L-function will coincide with the L-function associated to a certain automorphic cusp form (in Silverman's "Advanced topics in the arithmetic of elliptic curves" there is an alternative statement using Grössencharakter).

Since both Weil conjecture and Langlands program consist rationality and functional equation and by the preceding result (BTW, some approaches are similar), are there some inner relations between the two conjectures?

• There is a misunderstanding here. The $L$-function of an elliptic curve over a finite field is very simple analytically, namely of the form $(1-\alpha p^{-s})(1-\beta p^{-s})$. (Depending on the definition you use, this is either trivial or a simple fact.) Hasse-Weil $L$-functions are similarly simple in this regard, namely they are rational functions of $p^{-s}$. (This fact is due to Dwork and lies much deeper of course.) The $L$-function of a cusp form, on the other hand, is an infinite product over all the primes. (Continued in next comment.) – GH from MO Nov 5 '15 at 19:25
• In a sense a Hasse-Weil $L$-function of a variety over the $p$-element field is much like the reciprocal of a single Euler factor (namely the one at $p$) of an automorphic $L$-function. In short, your second sentence is in error: we do not associate cusp forms to elliptic curves over $\mathbb{F}_p$. Instead, we associate cusp forms to elliptic curves over $\mathbb{Q}$. – GH from MO Nov 5 '15 at 19:25

The Weil conjectures asks for rationality, functional equation, Riemann hypothesis, and degree relating to the Hasse-Weil zeta function of a nonsingular projective variety $X$ over a finite field $k$, which is a product of L-functions. Since you are reading Gelbart, as Artin you can think of these as L-functions associated to the Galois representation of Gal$(\bar k/k)$ acting on each degree of the (etale) cohomology of $X$ respectively.
The Langlands program does not ask for rationality, but to each automorphic representation it does ask for a functional equation and meromorphic continuation for the associated automorphic $L$-function. (This also depends on the representation of the $L$-group.) It is generally believed that these $L$-functions also satisfy the Riemann hypothesis, but this I do not think is explicitly a part of the program.
What you do get is that the functoriality conjecture for $H=\{1\}$ and $G=GL_n$ (p.208 of Gelbart) is what some call Langlands reciprocity: to every Galois representation is associated an automorphic representation, so if you like, you may think of the Hasse-Weil zeta function as (conjecturally) a product of automorphic $L$-functions instead, though I am not sure what one gains from this.
• "I am not sure what one gains from this": The $L$-function of an irreducible cuspidal representation of $GL_n$ is automatically analytic on the entire complex plane and satisfies the usual functional equation, hence what you call Langlands reciprocity would imply that Artin $L$-functions also have these properties. – GH from MO Nov 5 '15 at 19:13
• Do you mean automorphic $L$-functions over a function field? Because over a number field a product of automorphic $L$-functions is never a rational function. – GH from MO Nov 6 '15 at 1:44