# In what way do the Weil Conjectures pertain to Langlands?

For a relative variety $X$ over a ring of integers $O_K$, we can define a zeta function. This zeta function is defined as the product of the zeta functions of the variety specialized to $O_K/\mathfrak{p}$ as $\mathfrak{p}$ runs over $Spec(O_K)$. In turn, zeta functions of varieties over finite fields are easy to define using the counting of rational points. As Grothendieck proved, these zeta functions can be expressed as a product of $L$-functions indexed by $i$, where the $i^{th}$ $L$-function is related to the $i^{th}$ (Weil) cohomology of $X_{O_K/\mathfrak{p}}$. The $i^{th}$ $L$-function of $X$ over $O_K$ is defined to be the product over $\mathfrak{p}$ of the $i^{th}$ $L$-function of $X_{O_K/\mathfrak{p}}$.

The Weil conjectures give us a lot of information about the zeta functions of varieties over finite fields, and in fact about their $L$-functions.

The Langlands program is about properties of $L$-functions of $X$ over $O_K$.

Is it possible to interpret the Weil conjectures as telling us something meaningful about the Langlands program?

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James, do you mean something other than proving the Ramanujan conjecture for cohomological automorphic forms? (For the modular form case, see the last part of Freitag and Kiehl's "Etale cohomology and the Weil conjecture") Also, if zeta functions of algebraic varieties over finite fields were not rational functions with a certain functional equation, they couldn't possibly be related to automorphic L-functions... –  B R Oct 25 '11 at 20:01

If $\pi$ is a regular algebraic cuspidal automorphic representation of $GL_n/K$ with K totally real or CM, and $\pi$ satisfies a certain self-duality condition, then $\pi_v$ is tempered for all finite $v$. This monumental theorem is a vast generalization of Deligne's proof of the Ramanujan conjecture, and the proof ultimately appeals to the Weil conjectures, by proving an identity $L(s,\pi)=L(s,M(\pi))$ where $M(\pi)$ is (essentially) a submotive of the cohomology of a Shimura variety.