# Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

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I'm trying to learn the Selberg trace formula, but have very little background in harmonic analysis. I was referred to Dennis Hejhal's The Selberg Trace Formula for $PSL(2, \mathbb{R})$ but just got the book and was dismayed to learn that that the author assumes familiarity with Selberg's original paper (which I don't have access to - would welcome a pointer to an online copy).

There's much in the first few pages that I don't know. For example, the author states without proof that the spectrum of the Laplacian on a compact hyperbolic surface is discrete. He gives a reference to a 1912 book by Hilbert, but aside from the fact that I don't read German, it's not clear to me that this is the best place to learn from (in light of the fact that Hejhal's book is from 1976 and many books have been written since).

Does anyone have a suggestion as to what to read before Hejhal's book?

Deitmar and Echterhoff "Principles in Harmonic Analysis" Chapter 9 and 11 for the cocompact case (Hejhal I). It requires some familiarity with representation theory, but you seem to be more interested in a conceptual approach. It is the only source which really made me understand the underlying concepts of the trace formula.

In chapter 9, the main result is as follows:

Given a cocompact subgroup $\Gamma$ in a unimodular group $G$, convolution operators $T_\phi$ for functions $\phi \in C_c^\infty(G)$ on $L^2( \Gamma\backslash G)$ are trace class operators. The formula for the trace is given as $$tr T_\phi = \sum\limits_{conj.classes in \Gamma} vol(\Gamma_\gamma \backslash G_\gamma) \int\limits_{G_\gamma \backslash G} \phi(g \gamma g^{-1}) \,d\,g.$$ The space $L^2( \Gamma\backslash G)$ decomposes of course into irreducible reps. It is a nice exercise to deduce the Poisson summation formula from this.

In chapter 11, they specialize this to the situation $G=SL(2,\mathbb{R})$ and $\Gamma$ has only hyperbolic elements.

The non-compact situation is more difficult (Hejhal II). The main idea (working with $G$) is mostly hidden in Selberg's and Hejhal's presentation, and the underlying computations with special functions can be avoided until a certain point (Chapter 11 is the analogue). This is where these guys start. Iwaniec "Spectral methods" is pretty close to Selberg's Göttingen lecture notes. Hejhal and Selberg are in my opinion a terrible point to enter the subject for an algebraic person. Hejhal remains the most important reference for researchers. Similar route is done for quadratic imaginary fields in Elstrodt, Grunewald, Mennicke "Hyperbolic Groups acting...".

It will be probably more useful to understand the Arthur trace formula as presented in Jacquet-Gelbart "Analytic aspects...." given your background. Knightly and Li "Trace of Hecke operators" are useful here for the Hecke eigenvalues, and the lecture notes by Gelbart for the general theory.

My thesis might be interesting to you, because I generalize the computations of Hejhal II/Knightly-Li to the number field case and "arbitrary" congruence subgroups by using Arthur's trace formula and the adelic framework: http://webdoc.sub.gwdg.de/diss/2012/palm/. As an example, I derive the Selberg trace formula from the Arthur trace formula there.

Well, the modern viewpoint relays on the interpretation of "fourier transform" (in any generalized fashion you like to define "fourier transform") in representation-theoretic language. As a consequence, there are several approaches today to get the trace formula (either more analytic by Green's functions or the more general representation theoretic manner).

A nice introductory account can be found here by Marklof - http://arxiv.org/pdf/math/0407288v2.pdf Another representation-free approach is done in Iwaniec's "spectral methods of automorphic forms".

The theorem that Hejhal mentioned is very well known for general (compact, closed) manifolds (follows from Poincare inequality), but in the (cocompact) homogeneous case, one can overcome many analytical complications by just mimicking the proof of the Peter-Weyl theorem in representation theory of compact groups (Hilbert-Schmidt operators and so on). In particular, no Sobolev computations whatsoever, that shows one simple example of the advantages of using representation theory.

A more advanced approach (which uses some representation theory) is found in Knapp's article - http://sporadic.stanford.edu/bump/match/trace.pdf Probably a good introduction to this article is Bump's book about automorphic representations (chapters 1-2 I guess, you only need the real part for this article).

For books I recommend Audrey Terras' "Harmonic Analysis on Symmetric Spaces and Applications, I." It begins with Fourier analysis on $\mathbb R^m$, includes automorphic forms both Maass and classical, and finishes with the Selberg Trace formula. The orientation is towards number theory, but there are lots of applications and extensive references to the literature. Also lots of exercises.

For an expository paper, I recommend H.P. McKean "Selberg's trace formula applied to a compact Riemann surface" in Comm. Pure & Appl. Math., v. 25 (1975), 225-246; with errata in v. 27 (1974) p. 134.

Both of these are older than the more modern references in the other answers, but closer perhaps in style to Hejhal's approach.

To address the narrow question of the discreteness of the spectrum of the laplacian on a compact riemannian manifold: this is part of the elliptic package, which is most of the proof of the Hodge theorem and is proved along the way in many books on differential geometry or topology. I find these treatments too ad hoc. The introduction I greatly prefer is Zimmer's Essential Results of Functional Analysis. This has just enough analytic theory for the geometric applications, which, in turn, helps motivate the theory. The only drawback is that since it really is an analysis book, it leaves the case of compact manifolds as an exercise.

When I tried to read Hejhal's books around 1983, I enjoyed the book $SL_2(\mathbb{R})$ (where $SL$ stands for Serge Lang) for background.