The Riemann Hypothesis and the Langlands program

Question

On page 263 of this book review appears the following:

Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic number theory from the standpoint of L-functions and their analytic properties), but in fact the properties of L-functions traditionally of interest to analytic number theorists - for example, the location of zeroes in the critical strip (the Generalized Riemann Hypothesis) - have historically had little to do with the preoccupations of the Langlands program. Thanks largely to the efforts of a few charismatic and determined individuals, this is beginning to change and Langlands himself has in recent years turned to methods in analytic number theory in an attempt to get beyond the visible limits of the techniques developed over the last few decades.

I'd like to ask for a big picture exposition of how such questions about the location of zeroes of L-functions appear and interact with the Langlands program. My interest is mainly cultural and the answer should be tailored for the outsider to number theory (I'm viewing Langlands program algebraically as the pursuit of a nonabelian class field theory.)

A more crude question is:

Does the Langlands program say anything about the Grand Riemann Hypothesis or vice versa?

This is almost certainly too crude a question for MO, but Langlands seems to have such an amazing unifying appeal, that I feel a temptation to see how much it subsumes. I fully expect an answer like "It is impossible to coherently discuss this without years of training". Thank you for any attempt to explain things to someone who is not a number theorist, in advance!

Answer 1

One can use Langlands functoriality to eliminate the so-called Siegel zeros of an automorphic $L$-function. For example, Hoffstein-Ramakrishnan (IMRN 1995) proved that the $L$-function of a $GL(n)$ cusp form for $n>1$ has no Siegel zero if all $GL(m)\times GL(n)$ $L$-functions are $GL(mn)$ $L$-functions. There are several unconditional results along this line, e.g. in the same paper it is shown that the $L$-function of a $GL(2)$ cusp form has no Siegel zero.

Answer 2

Number theory can seem to the beginner like a very random collection of results, and it is only fairly recently in its 5000 year history that the larger picture has begin to emerge. A report from the NAS in the early 1990s opened my eyes to the fact that number theory now is centered around three questions, each having to do with $L$-functions.

The first area is the Riemann hypothesis, and its generalizations to more general $L$-functions. Questions about the vertical as well as horizontal distribution of the zeros initiated by Montgomery and influenced by random matrix theory fall in this area.

The second area is the Langlands program. Unlike the Riemann hypothesis, the Langlands program is named for the most advanced part of the theory, not the original question. But the roots of the Langlands program go all the way back to Gauss' Law of Quadratic Reciprocity: Given an odd prime $q$, let $\epsilon=\pm 1$ so that $\epsilon q\equiv 1\bmod 4$. Then for an odd prime $p$, the Legendre symbols $(\frac{p}{q})$ and $(\frac{\epsilon q}{p})$ are equal. More abstractly, the Galois representation arising from the Kronecker symbol $(\frac{\epsilon q}{*})$ has the same $L$ function as the Dirichlet character $(\frac{*}{q})$. Langlands interprets the latter as an automorphic from on $GL(1)$.

The third area is the Bloch-Beilinson conjectures, which include the Birch and Swinnerton-Dyer conjecture as a special case. The simplest manifestation of these is the Dirichlet Class Number Formula.

As GH's answer, different areas relate to each other. The possibility of a Landau-Siegel zero prevents us from getting the lower bound on class numbers we expect. The fact CM elliptic curves were known to be automorphic allowed Birch and Swinnerton-Dyer to be able to compute sufficient examples to make a conjecture.

Answer 3

This doesn't really address the question, but it seems worth mentioning that in order for the Riemann hypothesis to be well formulated for a motivic L-function, one has to know that the L-function is also an automorphic L-function. For example, a priori all one knows about the L-function of an elliptic curve is that it converges for $Re(s) > 3/2$ (by Hasse's bound). One needs automorphicity to extend the L-function to $Re(s) = 1$, and the Riemann hypothesis for the L-function of an elliptic curve states that all (nontrivial?) zeros of the L-function lie on the line $Re(s) = 1$.

Answer 4

The analogue of the Riemann hypothesis for the Selberg zeta function for $\Gamma(N) \backslash \mathbb{H}$ is known as the Selberg eigenvalue conjecture. It would follow from the Langlands functoriality. However, all but finitely many of these zeros lie on $\Re s =1/2$ beforehand, so this is very different from the Riemann zeta function.