# «A (very brief ) History of the Trace Formula James Arthur This note is a short summary of a lecture in the series celebrating the tenth anniversary ...»

A (very brief ) History of the Trace Formula

James Arthur

This note is a short summary of a lecture in the series celebrating

the tenth anniversary of PIMS. The lecture itself was an attempt to

introduce the trace formula through its historical origins. I thank Bill

Casselman for suggesting the topic. I would also like to thank Peter

Sarnak for sharing his historical insights with me. I hope I have not

distorted them too grievously.

As it is presently understood, the trace formula is a general identity (GTF) {geometric terms} = {spectral terms}.

The spectral terms contain arithmetic information of a fundamental nature. However, they are highly inaccessible, “spectral” actually, in the nonmathematical meaning of the word. The geometric terms are quite explicit, but they have the drawback of being very complicated.

There are simple analogues of the trace formula, “toy models” one could say, which are familiar to all. For example, suppose that A = (aij ) is a complex (n × n)-matrix, with diagonal entries {ui } = {aii } and eigenvalues {λj }. By evaluating its trace in two diﬀerent ways, we obtain an identity n n ui = λj.

i=1 j=1 The diagonal coeﬃcients obviously carry geometric information about A as a transformation of Cn. The eigenvalues are spectral, in the precise mathematical sense of the word.

For another example, suppose that g ∈ Cc (Rn ). This function ∞ then satisﬁes the Poisson summation formula g(u) = g (λ), ˆ u∈Zn λ∈2πiZn where λ ∈ Cn, g(x)e−xλ dx, g (λ) = ˆ Rn is the Fourier transform of g. One obtains an interesting application by letting g = gT approximate the characteristic function of the closed ball BT of radius T about the origin. As T becomes large, the left hand side approximates the number of lattice points u ∈ Zn in BT.

The dominant term on the right hand side is the integral g (0) = ˆ g(x)dx, Rn which in turn approximates vol(BT ). In this way, the Poisson summation formula leads to a sharp asymptotic formula for the number of lattice points in BT.

Our real starting point is the upper half plane H = z ∈ C : Im(z) 0.

The multiplicative group SL(2, R) of (2 × 2) real matrices of determinant 1 acts transitively by linear fractional transformations on H. The discrete subgroup Γ = SL(2, Z) acts discontinuously. Its space of orbits Γ\H can be identiﬁed with a noncompact Riemann surface, whose fundamental domain is the familiar modular region.

-1 0 1

and the hyperbolic Laplacian ∂2 ∂2 ∆ = −y +.

∂x2 ∂y 2 Modular forms are holomorphic sections of line bundles on Γ\H.

For example, a modular form of weight 2 is a holomorphic function f (z) on H such that the product f (z)dz descends to a holomorphic 1-form on the Riemann surface Γ\H. The classical theory of modular forms was a preoccupation of a number of prominent nineteenth century mathematicians. It developed many strands, which intertwine complex analysis and number theory.

In the ﬁrst half of the twentieth century, the theory was taken to new heights by E. Hecke (∼ 1920–1940)1. Among many other things, he introduced the notion of a cusp form. As objects that are rapidly decreasing at inﬁnity, cusp forms represent holomorphic eigensections of ∆ (for the relevant line bundle) that are square integrable on Γ\H.

The notion of an eigenform of ∆ calls to mind the seemingly simpler problem of describing the spectral decomposition of ∆ on the space of functions L2 (Γ\H). I do not know why this problem, which seems so natural to our modern tastes, was not studied earlier. Perhaps it was because eigenfunctions of ∆ are typically not holomorphic. Whatever the case, major advances were made by A. Selberg. I will attach his name to the ﬁrst of three sections, which roughly represent three chronological periods in the development of the trace formula.

that converge if Re(λ) 1. Selberg2 introduced general techniques, which showed that E(λ, z) has analytic continuation to a meromorphic function of λ ∈ C, that its values at λ ∈ iR are analytic, and that these values exhaust the continuous spectrum of ∆ on L2 (Γ\H). One can say that the function E(λ, z), λ ∈ iR, z ∈ Γ\H, plays the same role for L2 (Γ\H) as the function eλx in the theory of Fourier transforms.

(b) Trace formula for Γ\H (∼ 1955).

Selberg’s analysis of the continuous spectrum left open the question of the discrete spectrum of ∆ on L2 (Γ\H). About this time, examples of square integrable eigenfunctions of ∆ were constructed separately (and by very diﬀerent means) by H. Maass and C.L. Siegel. Were these examples isolated anomalies, or did they represent only what was visible of a much richer discrete spectrum?

A decisive answer was provided by the trace formula Selberg created to this end. The Selberg trace formula is an identity (STF) ai g(ui ) = bj g (λj ) + e(g), ˆ i j where g is any symmetric test function in Cc (R), {ui } are essentially3 ∞ the real eigenvalues of conjugacy classes in Γ, and {λi } are essentially3 the discrete eigenvalues of ∆ on L2 (Γ\H). The coeﬃcients {ai } and {bj } are explicit nonzero constants, and e(g) is an explicit error term (which contains both geometric and spectral data). The proof of (STF) was a tour de force. The function g gives rise to an operator on L2 (Γ\H), but the presence of a continous spectrum means that the operator is not of trace class. Selberg had ﬁrst to subtract the contribution of this operator to the continuous spectrum, something he could in principle do by virtue of (a). However, the modiﬁed operator is quite complicated. It is remarkable that Selberg was able to express its trace by such a relatively simple formula.

Selberg’s original application of (STF) came by choosing g so that g ˆ approximated the characteristic function of a large symmetric interval in R. The result was a sharp asymptotic formula π Λj = 4 − λ 2 ≤ T ∼ vol(Γ\H)T j for the number of eigenvalues Λj in the discrete spectrum. This is an analogue of Weyl’s law (which applies to compact Riemannian manifolds) for the noncompact manifold Γ\H. In particular, it shows that the congruence arithmetic quotient Γ\H has a rich discrete spectrum, something subsequent experience has shown is quite unusual for noncompact Riemannian manifolds.

Ramiﬁcations (∼ 1955–1960).

(i) Selberg seems to have observed after his discovery of (STF) that a similar but simpler formula could be proved for any compact Riemann surface Γ \H. (and indeed, for any compact, locally symmetric space).

For example, one could take the fundamental group Γ to be a congruence group inside a quaternion algebra Q over Q with Q(R) ∼ M2 (R).

= The trace formula in this case is similar to (STF), except that the explicit error term e(g) is considerably simpler.

(ii) Selberg also observed that (STF) could be extended to the Hecke operators {Tp : p prime} on L2 (Γ\H). These operators have turned out to be the most significant of Hecke’s many contributions. They are a commuting family of operators, parametrized by prime numbers p, which also commute with ∆. The corresponding family of simultaneous eigenvalues {tp,j } carries arithmetic information. They can be regarded as the analytic embodiment of data that govern fundamental arithmetic phenomena.

Selberg’s generalization of (STF) includes terms on the right hand side that quantify the numbers {tp,j }. It also holds more generally if L2 (Γ\H) is replaced by the space of square integrable sections of a line bundle on Γ\H. In this form, it can be applied to the space of classical cusp forms of weight 2k on Γ\H. It yields a ﬁnite closed formula for the trace of any Hecke operator on this space.

(iii) Selberg also studied generalizations of Eisenstein series and (STF) to some spaces of higher dimension.

II. Langlands (a) General Eisenstein series (∼ 1960–1965).

Motivated by Selberg’s results, R. Langlands set about constructing continuous spectra for any locally symmetric space Γ\X of ﬁnite volume. Like the special case Γ\H, the problem is to show that absolutely convergent Eisenstein series have analytic continuation to meromorphic functions, whose values at imaginary arguments exhaust the continuous spectrum. The analytic diﬃculties were enormous. Langlands was able to overcome them with a remarkable argument based on an interplay between spectral theory and higher residue calculus. The result was a complete description of the continuous spectrum of L2 (Γ\X) in terms of discrete spectra for spaces of smaller dimension.

(b) Comparison of trace formulas (∼ 1970–1975).

Langlands changed the focus of applications of the trace formula.

Instead of taking one formula in isolation, he showed how to establish deep results by comparing two trace formulas with each other. He treated three diﬀerent kinds of comparison, following special cases that had been studied earlier by M. Eichler and H. Shimizu, Y. Ihara, and H. Saito and T. Shintani. I shall illustrate each of these in shorthand, with a symbolic correpondence between associated data for which the comparison yields a reciprocity law. In each case, the left hand side represents some form of the trace formula (STF), while the right hand side represents another trace formula.

(i) (Γ\H) ↔ (Γ \H) {λj, tp,j } ↔ {λj, tp,j }.

Here Γ \H represents a compact Riemann surface attached to a congruence quaternion group Γ. The reciprocity law, established by Langlands in collaboration with H. Jacquet, is a remarkable correspondence between spectra of Laplacians on two Riemann surfaces, one noncompact and the other compact, and also a correspondence between eigenvalues of associated Hecke operators.

(ii) (Γ\H) ↔ (Γ\H)p {tp,j } ↔ {Φp,j }.

Here (Γ\H)p represents an algebraic curve over Fp, obtained by reduction mod p of a Z-scheme associated to Γ\H. The relevant trace formula is the Grothendieck-Lefschetz ﬁxed point formula, and {Φp,j } represent eigenvalues of the Frobenius endomorphism on the -adic cohomology of (Γ\H)p. The reciprocity law illustrated in this case gives an idea of the arithmetic signiﬁcance of eigenvalues {tp,j } of Hecke operators (iii) (Γ\H) ↔ (ΓE \HE ) {λj, tp,j } ↔ {λE,j, tp,j }.

Here, (ΓE \HE ) is a higher dimensional locally symmetric space attached to a cyclic Galois extension E/Q, and p denotes a prime ideal in OE over p. The relevant formula is a twisted trace formula, attached to the diﬀeomorphism of ΓE \HE deﬁned by a generator of the Galois group of E/F. The reciprocity law it yields (and its generalization with Q replaced by an arbitrary number ﬁeld F ) is known as cyclic base change. It has had spectacular consequences. It led to the proof of a famous conjecture of E. Artin on representations of Galois groups, in the special case of a two dimensional representation of a solvable Galois group. This result, known as the Langlands-Tunnell theorem, was in turn a starting point for the work of A. Wiles on the ShimuraTaniyama-Weil conjecture and his proof of Fermat’s last theorem.

My impressionistic review of the three kinds of comparison is not to be taken too literally. For example, it is best not to ﬁx the congruence subgroup Γ of SL(2, R). The correspondences are really between a (topological) projective limit

and its three associated analogues. Moreover, the group SL(2) should actually be replaced by GL(2). Nevertheless, the basic idea is as stated, to compare a formula like (STF) with something else. One deduces relations between data on the spectral sides from a priori relations between data on the geometric sides. We recall that the geometric terms in (STF) are indexed by conjugacy classes in the discrete group Γ.

Before going to the next stage, I need to recall some other foundational ideas of Langlands. To maintain a sense of historical ﬂow, I shall divide these remarks artiﬁcially into two time periods.

**Between II(a) and II(b) (∼ 1965–1970).**