Consequences of the Langlands program - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:26:30Z http://mathoverflow.net/feeds/question/78247 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78247/consequences-of-the-langlands-program Consequences of the Langlands program john 2011-10-16T04:57:27Z 2011-10-17T02:23:55Z <p>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.</p> <p>I would like to know are there any other important consequence of the Langlands program?</p> http://mathoverflow.net/questions/78247/consequences-of-the-langlands-program/78295#78295 Answer by Agol for Consequences of the Langlands program Agol 2011-10-16T22:59:20Z 2011-10-16T22:59:20Z <p>Langlands functoriality (base change for $GL(2)$) implies the <a href="http://en.wikipedia.org/wiki/Virtual_Haken_conjecture" rel="nofollow">virtual Haken conjecture</a> for closed <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ajm/1118669703" rel="nofollow">arithmetic hyperbolic 3-manifolds</a>. </p> http://mathoverflow.net/questions/78247/consequences-of-the-langlands-program/78304#78304 Answer by Emerton for Consequences of the Langlands program Emerton 2011-10-17T02:23:55Z 2011-10-17T02:23:55Z <p>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:</p> <ul> <li><p>The Selberg $1/4$ conjecture.</p></li> <li><p>The Ramanujan conjecture for cuspforms on $GL_n$ over arbitrary number fields.</p></li> <li><p>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.)</p></li> <li><p>Analogues of Sato--Tate for Frobenius eigenvalues on the $\ell$-adic cohomology of arbitrary varieties over number fields.</p></li> </ul>