# 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 consequences of the Langlands program?

• The Artin conjecture on L-functions. See the Wikipedia page on Artin L-functions. Oct 16, 2011 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 Oct 16, 2011 at 5:36

• The Selberg $1/4$ conjecture.
• The Ramanujan conjecture for cuspforms on $GL_n$ over arbitrary number fields.
• Analogues of Sato--Tate for Frobenius eigenvalues on the $\ell$-adic cohomology of arbitrary varieties over number fields.
Langlands functoriality (base change for $GL(2)$) implies the virtual Haken conjecture for closed arithmetic hyperbolic 3-manifolds.