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.
Best Answer
I highly recommend Iwaniec's 1986 ICM lecture, which you can read here (on page 444; page 546 of the PDF), and Peter Sarnak's article "Spectra of hyperbolic surfaces," which is here.