## Overview of the interplay of Harmonic Analysis and Number Theory

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I'm kind of disappointed that the question here was never sharpened.

The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ is the fundamental domain of, say, a congruence subgroup $\Gamma$ of $Sl_{2}(\mathbb{Z})$. Eigenfunctions of the discrete spectrum of $\Delta$ are real analytic solutions to $\Delta (\Psi)=\lambda \Psi$ that are $\Gamma$-equivariant functions in $L^{2}(D, dz)$, where $dz$ is the Poincare measure on the upper half-plane. These eigenfunctions evidently carry quite a bit of number theoretic information. Frankly, this point of view on number theory sounds incredibly interesting...

Question: Would someone please suggest a readable introductory account that tells this story?

(I imagine that answers will include the words Harish-Chandra, Langlands, etc...)

Also, if experts are inclined to write a short overview as an answer, that would also be much appreciated.

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## 4 Answers

I think it is also fair to say that things like hyperbolic $n$-spaces (and other symmetric spaces), and arithmetic quotients of them, are primordial, one-of-a-kind (well, not quite) objects. The opposite of "generic" mathematical objects. Partly because of the abruptly greater technical complexity in discussing the harmonic analysis on them, they are much less familiar than Euclidean spaces or their familiar quotients, circles and products thereof.

Apart from Langlands' program and direct, intentional discussion of $L$-functions, I find it provocative that the basic harmonic analysis of $SL_2(\mathbb Z)\backslash H$ is not merely far subtler than that of $\mathbb R^2$ or $\mathbb R^2/\mathbb Z^2$, but that those subtleties are directly related to profound unsolved problems. As a well-known example, while we easily understand the sup norm of exponentials in the harmonic analysis on the real line, sharp estimates on pointwise behavior of eigenfunctions for the Laplacian on the upper half-plane give Lindelof: e.g., the value of the Eisenstein series $E_s$ at $z=i$ is the zeta of the Gaussian integers (divided by $\zeta(2s)$).

Continuing, unlike the fact that the product of two exponentials is an exponential (that is, the tensor product of two one-dimensional irreducibles is still irreducible), decomposition of tensor products of waveforms, or of the repns they generate, is very tricky, and, again, is connected to serious outstanding problems. (Iwaniec' book mentions such examples.)

That is, apart from "big conjectures" about automorphic forms and L-functions by themselves, even-more-primitive number-theoretic things just arise unbidden when we try to do innocent, ordinary things that would be trivial in Euclidean space.

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That's how I see it. Please correct me if I am wrong:

1. Induce the trivial representation from a lattice $\Gamma$, e.g. $\mathrm{SL}_2( \mathbb{Z})$, to the group $\mathrm{SL}_2(\mathbb{R})$.

2. Then consider the $\mathrm{SO}_2(\mathbb{R})$-invariant subspace.(weights are actually associated to one dimensional representations of SO_2)

This representation is isomorphic to the representation $L^2(D,dz)$, you construct above.

How does the Laplace Beltrami operator enter the picture... It is the descent of the Casimir operator to this space, which is the generator of all invariant differential operators (= center of the universal envelopping algebra of \mathfrak{sl}_2(\mathbb{R})). So intuitevely the Casimir operator captures the $G$ structure on the Hilbert space.

Why did we became interested in such constructions? Maass discovered that the Mellin transforms of Dedekind zeta function associated to a quadratic real fields are Maass forms.

Now, since all interesting $L$ functions (ass. to Galois repr., elliptic curves,...) are conjectured to be associated to some representation to some group, it seems worthwhile: 1. To study if and how they are associated ... (e.g., Taniyama-Shimura conjecture) 2. To study their properties on either side and conclude about the other...

Mappings between groups and comparing their L-functions give also nice information about them (functoriality), e.g. they generalized Ramanujan conjecture would be implied by certain "functoriality" conjectures betweens general linear groups .

Perhaps I should conclude that with the adeles $\mathbb{A}$ the better picture is $$\mathcal{L}^{2} ( \mathrm{SL}_2(\mathbb{Q}) \backslash \mathrm{SL}_2(\mathbb{A}))^{\mathrm{K}(m)\mathrm{SO}(2)} \cong \mathcal{L}^2(\Gamma(m) \backslash \mathbb{H}),$$

where $\mathrm{K}(m)$ is the product over $\mathrm{K}_p(m)$, the group of elements in $\mathrm{SL}_2( \mathbb{Z}_p)$ which are the identy modulo $m$. This picture contains the Hecke operators more naturally...

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Thanks for this answer. – Jon Bannon Apr 1 2011 at 11:18

Ten Lectures on the interface between analytic number theory and harmonic analysis, by Hugh Montgomery is also worthwhile.

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This is good, thanks DJC! – Jon Bannon Apr 28 2011 at 21:22

I highly recommend Iwaniec's 1983 ICM lecture, which you can read here, and Peter Sarnak's article "Spectra of hyperbolic surfaces," which is here.

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Thank you, David! – Jon Bannon Mar 31 2011 at 23:28
These are exactly the sort of thing I was looking for. Thanks again! – Jon Bannon Mar 31 2011 at 23:29
Actually, Sarnak has been focussing on this interplay for a long time. His PhD thesis was about a generalization of Selberg's Trace Formula. – Denis Serre Apr 1 2011 at 5:26

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Wow, I did not know that all ICM talks are online (mathunion.org/ICM)! Thanks, David. – GH Apr 1 2011 at 8:52
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