Eisenstein Series, the Trace Formula, and the Modern Theory of Automorphic Forms*

1. Eisenstein Series and Automorphic L-functions

The modern theory of automorphic forms is a response to many different impulses and influences, above all the work of Hecke, but also class-field theory and the study of quadratic forms, the theory of representations of reductive groups, and of complex multiplication, but so far many of the most powerful techniques are the issue, direct or indirect, of the introduction by Maass and then Selberg of spectral theory into the subject.

The spectral theory has two aspects: (i) the spectral decomposition of the spaces L^{2}(Γ\G) by means of Eisenstein series; (ii) the trace formula, which can be viewed as a striking extension of the Frobenius reciprocity law to pairs (Γ, G), G a continuous group and Γ a discrete subgroup.

The attempt to discover a class of Euler products attached to automorphic forms that would include the Dirichlet series with Grössencharakter attached to real quadratic fields led Maass in 1946, under difficult circumstances in the chaos of immediate postwar Germany, to the study of eigenfunctions of the Laplacian

Δ = y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} )

on the upper half-plane, eigenfunctions that transform simply, or are even invariant, under discrete groups, Γ, especially subgroups of the modular group [27]. Apparently he was influenced to some extent by the work of Fueter on quaternionic function theory.

It is simplest to consider functions actually invariant under the discrete group. If it is Fuchsian and the fundamental domain is not compact, then Δ has a continuous spectrum, and the corresponding eigenfunctions are given by analytic continuation of functions defined by infinite series, the Eisenstein series. They are attached to cusps. If, for example, the cusp is at infinity so that the group

Γ_{∞} = \{γ = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) | γ ∈ Γ\}

* Appeared in Number Theory, Trace Formulas and Discrete Groups, Academic Press (1989)

is infinite, then the attached series is

∑_{γ∈Γ_{∞}\Γ} y^{(s+1)/2} = ∑ (\frac{y}{|cz + d|^{2}} )^{(s+1)/2} .  (1.1)

This series is easily seen to converge for Re s > 1 and to yield eigenfunctions of Δ, but not the ones needed for the spectral decomposition, for they correspond to parameters satisfying Re s = 0. Thus an analytic continuation is required. It is often carried out in two steps as in [22], the functions being first continued to the region Re s > 0, either with the help of the resolvent or the Green’s function, the method used by Roelcke, or by a truncation process as employed by Selberg. The continuation across Re s = 0 was first effected by Selberg by means of a further truncation and the reflection principle.

To recapitulate briefly, Maass’s work on Euler products and automorphic forms drew his attention to a problem in spectral theory that, in turn, led to a problem in analytic continuation. The purpose of the first part of this lecture is to recall how a much larger class of Euler products, the automorphic L-functions, one of the central notions of the modern theory of automorphic forms, arose, a little by accident, from the solution of the problem in analytic continuation.

Recall first that in the fifties there was a tremendous surge of interest in automorphic forms on groups of higher dimension – due largely, I suppose, to the papers of Siegel on orthogonal groups and the symplectic groups; so it is hardly surprising that Selberg and others attempted to extend to them his techniques and ideas, the analytic continuation of the Eisenstein series and the trace formula. Decisive progress, however, had to await the introduction by Gelfand in 1962 [15] of the general notion of cusp forms that lies at the center of the spectral theory in higher dimensions.

Although it is not necessary, it is extremely convenient, if only to avoid elaborate notational complications, to work with adelic groups. In addition, the spectral decomposition is then made with respect to the largest possible family of operators, including the Hecke operators and the differential operators. It entails working on the group rather than the symmetric space, but then the obvious symmetries are recognized and the same problem not solved repeatedly.

For example, if G = GL(n) then an automorphic form is a function ϕ on G(Q\G(A). It is a cusp form if, for every block decomposition of the n × n matrices and every g ∈ G(A), we have

∫ ϕ (( \begin{matrix}I & X \\ 0 & I \end{matrix} ) g) dx = 0.

Here X is an n_{1} × n_{2} matrix, n_{1} + n_{2} = n, with adelic entries.

The notion of a cusp form clearly isolated, the analytic continuation of the general Eisenstein series is effected in three steps.

A. Series in One Variable Attached to Cusp Forms. If, for example, G is SL(n) and n = n_{1} + n_{2}, then such a series is associated to a cusp form ϕ on

M(A) = \{m = ( \begin{matrix}m_{1} & 0 \\ 0 & m_{2} \end{matrix} ) ∈ G(A)\}

and to a parameter s. Here m_{i} is an n_{i} × n_{i} matrix. For simplicity, I am confining myself to functions invariant on the right under a maximal compact subgroup K.

If

\begin{aligned}N & = \{( \begin{matrix}I & X \\ 0 & I \end{matrix} )\} \\ P & = \{( \begin{matrix}A & X \\ 0 & B \end{matrix} )\} \end{aligned}

then

P = MN = NM.

Set

F_{s}(nmk) = ϕ(m) (\frac{|det m_{1}|^{n_{2}}}{|det m_{2}|^{n_{1}}} )^{s+1/2} .  (1.2)

Then the Eisenstein series is

E_{s}(g) = ∑_{γ∈P ()\G()} F_{s}(γg).  (1.3)

Provided that ϕ is a cusp form, the methods used for the Eisenstein series on the upper halfplane will deal with those that like (1.3) are associated to maximal parabolic subgroups P and thus involve only a single parameter, although the extension should perhaps not be thought of as entirely routine.

B. Series in Several Variables Associated to Cusp Forms. They are attached to non maximal parabolic subgroups, thus for SL(n), to partitions n = n_{1} + n_{2} + ⋯ + n_{r} with more than two elements. The group M is given by

M = \{( \begin{matrix}m_{1} \\ & m_{2} \\ & & . & . & . \\ & & & & & m_{r} \end{matrix} ) | det m_{i} = 1\},

and P and N are defined accordingly. The functions F_{s} and the series E_{s} now depend on several parameters s = (s_{1}, …, s_{r}), ∑ n_{i}s_{i} = 0.

F_{s}(nmk) = ϕ(m) ∏_{i=1}^{r} |det m_{i}|^{s_{i}+ρ_{i}}  (1.4)

where the ρ_{i} are real numbers chosen to simplify the formulas for the functional equations. To deal with these Eisenstein series, one combines the results from A with forms of Hartog’s lemma. I observe that Hartog’s lemma was introduced into the subject quite early, and by several mathematicians independently (cf. Appendix I of [22]). C. The General Eisenstein Series. Apart from a more complicated dependence on k ∈ K, these are defined by functions like (1.4), with ϕ being any square-integrable automorphic form on M(A). That ϕ is no longer necessarily a cusp form entails altogether new difficulties. Even for classical groups of low dimension, the analytic continuation of these series and the spectral decomposition involve quite different ideas than those that suffice for A or B [22].

One need not look far for fatuous and misleading comments on the techniques involved in the three steps. They are regrettable, but fortunately need not concern us here, for it is the series of step A, or rather the calculation of their constant term, that led to the introduction of the general automorphic L-functions and the principle of functoriality. Thus these ideas could have appeared before the general theory of Eisenstein series and independently of it, but the psychological inhibitions may have been too great.

A perhaps too simple formulation of the principle of functoriality is that certain natural operations on the L-series attached to automorphic forms reflect possible operations on the forms themselves. The usual Hecke theory, for example, associates to an automorphic form an Euler product

∏_{ρ∉S} \frac{1}{(1 − α_{p}p^{−s})(1 − β_{p}p^{−s})}.  (1.5)

Since it is only the unordered pair {α_{p}, β_{p}} that matters, we may think of it as a conjugacy class {t_{p}} of complex 2 × 2 matrices with these eigenvalues. A natural operation on elements of GL(2) is to let them act on symmetric tensors of a given degree n. This transforms conjugacy classes in GL(2) into conjugacy classes in GL(n + 1). So we pass from

\{t_{p}\} = \{( \begin{matrix}α_{p} & 0 \\ 0 & β_{p} \end{matrix} )\}

to

\{( \begin{matrix}α_{p}^{n} \\ & α_{p}^{n−1}β_{p} \\ & & . & . & . \\ & & & & & α_{p}β_{p}^{n−1} \\ & & & & & & β_{p}^{n} \end{matrix} )\}

This leads us from the Euler product (1.5) to

∏_{p∉S} \frac{1}{(1 − α_{p}^{n}p^{−s})(1 − α_{p}^{n−1}β_{p}p^{−s})(1 − β_{p}^{n}p^{−s})},

and the principle of functoriality predicts that there is an automorphic form on GL(n+1) to which this series is attached.

Before indicating how the calculation of the constant term of Eisenstein series suggested the introduction of automorphic L-functions, I recall, for it is easy to forget, that 20 years ago it was by no means clear how, or even whether, the Hecke theory could be extended to groups other than GL(2). Ideas of varying quality were proposed, and it is surprising that this calculation, carried out more to pass the time than with any precise aim, should yield not only specific series whose analytic continuation and functional equation could be proved, but also a class of series with a natural completeness. Repeated efforts to rework the Hecke theory had led nowhere, except perhaps to a clearer understanding of how it functioned, which could only later be turned to profit.