# Consequences of the Langlands program

In the one-dimensional case the Langlands program is equivalent to the class field theory and the two-dimensional case implies the Taniyama-Shimura conjecture.

I would like to know are there any other important consequence of the Langlands program?

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The Artin conjecture on L-functions. See the Wikipedia page on Artin L-functions. –  KConrad Oct 16 '11 at 5:14
Benedict Gross has been giving a lecture series on more or less this topic. Videos of the lectures are available online at math.columbia.edu/~staff/EilenbergVideos/index.html –  Alison Miller Oct 16 '11 at 5:36

There are many, many consequences of the general Langlands program (which I'll interpret to mean both functoriality for automorphic forms and reciprocity between Galois representations and automorphic forms). Some of these are:

• The Selberg $1/4$ conjecture.

• The Ramanujan conjecture for cuspforms on $GL_n$ over arbitrary number fields.

• Modularity of elliptic curves over arbitrary number fields. (Indeed, Langlands reciprocity is essentially the statement that all Galois representations coming from geometry are attached to automorphic forms.)

• Analogues of Sato--Tate for Frobenius eigenvalues on the $\ell$-adic cohomology of arbitrary varieties over number fields.

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Langlands functoriality (base change for $GL(2)$) implies the virtual Haken conjecture for closed arithmetic hyperbolic 3-manifolds.

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