I have been reading the book Fearless Symmetry by Ash and Gross.It talks about Langlands program, which it says is the conjecture that there is a correspondence between any Galois representation coming from the etale cohomology of a Z-variety and an appropriate generalization of a modular form, called an “automorphic representation".

Even though it appears to be interesting, I would like to know that are there any important immediate consequences of the Langlands program in number theory or any other field. Why exactly are the mathematicians so excited about this?

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    $\begingroup$ You can formulate the Langslands programme in any dimension. In dimension 1 it is equivalent to the main results of class field theory. That's one reason it is interesting. In dimension 2 it implies the infamous Taniyama-Shimura conjecture proved by Wiles. I don't know much else but a programme which already implies such big results in low dimension must be interesting, right? $\endgroup$ – shaye Oct 9 '11 at 15:00
  • $\begingroup$ Well,I know that 2-dimensional case implies TS conjecture. But are other any other important consequences? $\endgroup$ – Mohan Oct 9 '11 at 15:02
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    $\begingroup$ @shaye, why "infamous"? $\endgroup$ – lhf Jun 26 '12 at 2:01
  • $\begingroup$ Please have a look at mathoverflow.net/questions/78247. $\endgroup$ – Watson Apr 23 '18 at 16:57

There are many applications of the Langlands program to number theory; this is why so many top-level researchers in number theory are focusing their attention on it.

One such application (proved six or so years ago by Clozel, Harris, and Taylor) is the Sato--Tate conjecture, which describes rather precisely the deviation of the number of mod $p$ points on a fixed elliptic curve $E$, as the prime $p$ varies, from the "expected value" of $1 + p$.

Further progress in the Langlands program would give rise to analogous distribution results for other Diophantine equations. (The key input is the analytic properties of the $L$-functions mentioned in Jeremy's answer.)

At a slightly more abstract level, one can think of the Langlands program as providing a classification of Diophantine equations in terms of automorphic forms.

At a more concrete level, it is anticipated that such a classification will be a crucial input to the problem of developing general results on solving Diophantine equations. (E.g. all results in the direction of the Birch--Swinnerton-Dyer conjecture take as input the modularity of the elliptic curve under investigation.)


In very general terms the Langlands correspondence implies that the L-functions of algebraic varieties are automorphic, and therefore they have analytic continuations and functional equations generalizing the properties of Riemann's Zeta Function.

The analytic consequence of the fact that "all elliptic curves over the rationals are modular" is that the Hasse-Weil $L$-function of such curves has an analytic continuation and functional equation.

I suppose this counts as an "application" in number theory.


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