TITLE: Overview of the interplay of Harmonic Analysis and Number Theory
QUESTION [25 upvotes]: 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.

REPLY [5 votes]: 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.