Spectral Theory and the Trace Formula

by Daniel Bump

We give an account of a portion of the spectral theory Γ\SL(2, ℝ), particularly the Selberg trace formula, emphasizing ideas from representation theory. For simplicity, we will treat the trace formula only in the case of a compact quotient. The last section is of a different nature, intended to show a simple application of the trace formula to a lifting problem.

I would like to thank Yonatan Gutman and the referee for extensive and very helpful comments on an earlier draft. Preparation of this report was supported in part by NSF grant DMS-9970841.

1. The spectral problem. The group G = SL(2, ℝ) acts on ℌ = {x+iy|y > 0}:

( \begin{matrix}a & b \\ c & d \end{matrix} ) : z ↦ \frac{az + b}{cz + d}.

The stabilizer of i is K = SO(2) so ℌ ≅ G/K. The noneuclidean Laplacian

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

is a G-invariant differential operator. Let Γ be a discrete cocompact subgroup of G = SL(2, ℝ). Then X = Γ\ℌ is a compact Riemann surface. The question at hand is to describe the spectrum of Δ on X.

Proposition 1. The group Γ has a fundamental domain F on ℌ whose boundary consists of pairs of geodesic arcs α_{i} and γ_{i}(α_{i}), with γ_{i} ∈ Γ. When the boundary is traversed counterclockwise the congruent arcs α_{i} and γ_{i}(α_{i}) are traversed in opposite directions.

Proof (sketch). Choose a point P ∈ ℌ which is not fixed by any element of Γ except ±I. Let F be the set of points which are nearer P than to γ(P) in the noneuclidean metric for any γ ∈ Γ. Let N = {γ_{1}, ⋯, γ_{n}} be the set elements of Γ such that γ_{i}(F) is adjacent to F.

At first we assume no γ_{i}^{2} = 1. Evidently each γ_{i}^{−1} ∈ N, so N has an even number 2h of elements. We arrange it so that the γ_{i} so that γ_{i+h} = γ_{i}^{−1} (1 ≤ i ≤ h). Let α_{i} be the intersection of the geodesic consisting of the set of points which are equidistant from P and γ_{i}^{−1}(P) with the closure of F. Then γ_{i}(α_{i}) = α_{i+h} and α_{1}, ⋯, α_{h} satisfy our requirements.

If some γ_{i}^{2} = 1 then the arc α_{i} is self-congruent by γ_{i} and its midpoint is fixed by γ_{i}. So we split such arcs in two at their midpoints and we are done.

The Laplacian Δ acts on C^{∞}(Γ\ℌ).

Proposition 2. Δ is symmetric and positive with respect to the invariant metric y^{−2} dx ∧ dy.

Proof. Symmetry means that

⟨Δf, g⟩ − ⟨f, Δg⟩ = 0, f, g ∈ C^{∞}(Γ\ℌ).

Taking a fundamental domain F as in Proposition 1, this equals

∫_{F} (f (\frac{∂^{2}\overline{g}}{∂x^{2}} + \frac{∂^{2}\overline{g}}{∂y^{2}} ) − \overline{g} (\frac{∂^{2}f}{∂x^{2}} + \frac{∂^{2}f}{∂y^{2}} )) dx ∧ dy = ∫_{F} dω

where

ω = f \frac{∂\overline{g}}{∂x} dy − f \frac{∂\overline{g}}{∂y} dx − \overline{g} \frac{∂f}{∂x} dy + \overline{g} \frac{∂f}{∂y} dx.

By Stoke’s theorem, this is equal to the integral of ω around the boundary of F. By Proposition 1, the contributions of the boundary arcs cancel in pairs.

Positivity means that ⟨Δf, f⟩ ≥ 0 with equality only if f is constant. We compute

[|\frac{∂f}{∂x} |^{2} + |\frac{∂f}{∂y} |^{2} + (\frac{∂f^{2}}{∂x^{2}} + \frac{∂f^{2}}{∂y^{2}} ) \overline{f}] dx ∧ dy = d (\overline{f} (\frac{∂f}{∂x} dy − \frac{∂f}{∂y} dx)) .

By Stoke’s theorem we thus have

(2)  ⟨Δf, f⟩ ≥ − ∫_{∂F} \overline{f} (\frac{∂f}{∂x} dy − \frac{∂f}{∂y} dx) ,

with equality only if ∂f/∂x = ∂f/∂y = 0 identically, that is, if f is constant. Using the Cauchy-Riemann equations and the chain rule if one may check that if x + iy and u + iv are related by a holomorphic mapping, such as a linear fractional transformation then (∂f/∂x)dy − (∂f/∂y)dx = (∂f/∂u)dv − (∂f/∂v)du. It follows that the contributions of congruent boundary arcs cancel and (2) is zero.

We will see later that Δ extends to a self-adjoint unbounded operator on 𝔏 = L^{2}(Γ\ℌ, y^{−2} dx ∧ dy). This means that there is a dense subspace D of 𝔏 containing C^{∞}(Γ\ℌ) such that Δ extends to D, and that if v ∈ 𝔏 is such that u → ⟨v, Δu⟩ is continuous on D, then v ∈ D and ⟨Δv, u⟩ = ⟨v, Δu⟩.

Since Δ is a self-adjoint operator it has a nice spectral theory, which we want to develop. We will accomplish this by introducing integral operators which commute with Δ. We will show that these operators are of trace class, and we will prove the Selberg trace formula for them.

Finally we will consider the more difficult case where Γ has a cusp. In this case there are both discrete and continuous spectra, and the theory of Eisenstein series is an essential feature.

2. Rings of integral operators. The group G acts on functions f : G → ℂ by the right regular action (ρ(g)f)(x) = f(xg). Since we will soon be discussing differential operators let us at first restrict ourselves to f ∈ C^{∞}(G).

Because ℌ is canonically identified with the homogeneous space G/K, any function f on ℌ may be regarded as a function on G which is right invariant by K. The relevance of the regular representation of G to functions on ℌ may not be immediately clear, because the property of right K invariance is not preserved under right translation. However, there is a relevance which we now explain.

The Lie algebra of G consists of the vector space 𝔤 of 2 × 2 real matrices of trace zero. It acts on smooth functions f on G by:

Xf(g) = \frac{d}{dt}f(ge^{tX})|_{t=0}, X ∈ 𝔤.

This is a Lie algebra representation. This means that [X, Y] = XY − Y X (XY = matrix multiplication) has the same effect as X ∘ Y − Y ∘ X (X ∘ Y = composition of operators). This representation of 𝔤 is the differential form of the regular representation.

The universal enveloping algebra U(𝔤) is the associative ℝ-algebra generated by 𝔤 modulo the relations

[X, Y] − (X · Y − Y · X) = 0

(X · Y = multiplication in U(𝔤)). Any Lie algebra representation extends uniquely to a representation of U(𝔤). In particular the regular representation of 𝔤 extends to a representation of U(𝔤). Thus U(𝔤) is realized as a ring of differential operators on C^{∞}(G).

If D is an element of the center of U(𝔤) then it commutes with the regular representation. This is intuitively reasonable and proved in Bump [B], Proposition 2.2.4. In particular, if f is fixed under ρ(K), then so is Df. Therefore D acts on C^{∞}(ℌ).

A particular element of the center of U(𝔤) is the Laplace-Beltrami or Casimir element

(3)  −4Δ = H · H + 2R · L + 2L · R,
R = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ),  L = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ),  H = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

In fact, ℂ[Δ] is the center of U(𝔤). We have used the same letter Δ that we previously used for the noneuclidean Laplacian, for when this element of U(𝔤) is interpreted as a differential operator on ℌ they are the same. See Bump [B], Proposition 2.2.5.

Let H be the convolution ring of smooth, compactly supported functions on G. Let H^{∘} be the subring of K-bi-invariant functions. These are rings without unit. We call H^{∘} the spherical Hecke algebra but caution the reader that there are other natural and closely related rings which have also been called this. The ring H is noncommutative, but:

Theorem 3. The ring H^{∘} is commutative.

Proof. Matrix transposition preserves K so it induces an involution ι on H^{∘} such that ι(ϕ ∗ ψ) = ι(ψ) ∘ ι(ϕ), where (ιf)(g) = f(g^{t}). Every double coset in K\G/K has a diagonal representative. So ι is the identity map.

This theorem of Gelfand has a representation theoretic meaning. If (π, V) is a representation of G on a Banach space we will denote by V^{K} the vector subspace of K-fixed vectors. The algebra H acts on V by

π(ϕ) v = ∫_{G} ϕ(g) π(g) v dg, ϕ ∈ H, v ∈ V.

Here ∫_{G} dg is the Haar integral. If ϕ ∈ H^{∘} then π(ϕ) v ∈ V^{K}. In particular V^{K} is a module for H^{∘}.

Since K is compact, any irreducible representation ρ of K is finite-dimensional. The ρ-isotypic part V (ρ) of V is the direct sum of all K-invariant subspaces isomorphic to ρ as K-modules. A representation (π, V) of G is called admissible if V (ρ) is finite-dimensional for every ρ. In particular, if ρ is the trivial representation V (ρ) = V^{K} so V^{K} must be finite-dimensional.

Theorem 4. If (π, V) is an irreducible admissible representation of G then V^{K} is at most one-dimensional.

Proof. Since H^{∘} is commutative, its finite-dimensional irreducible modules are one-dimensional. Thus it is sufficient to show that V^{K} (if nonzero) is an irreducible module for H^{∘}. Suppose L ⊂ V^{K} is a closed nonzero H^{∘}-invariant subspace. If v ∈ V^{K} we will show that v ∈ L.

Let ϵ > 0 be given. Since V is irreducible and L is nonzero, the closure of π(H) L is V and so there exists ϕ ∈ H, w ∈ L such that π(ϕ) w = v_{1}, where |v_{1} − v| < ϵ. Let v_{2} = ∫_{K} π(k)v_{1} dk. Then v_{2} is K-fixed and since v is K-fixed,

|v_{2} − v| = | ∫_{K} π(k)(v_{1} − v) dk| ≤ ∫_{K} |π(k)(v_{1} − v)| dk = ∫_{K} |v_{1} − v| dk < ϵ.

(We normalize the Haar measure so K has volume 1.) Since w is K-fixed, π(k)π(ϕ)π(k′)w = π(k)v_{1} for all k, k′ ∈ K. Integrating over k and k′, we thus get π(ϕ_{0}) w = v_{2} where

ϕ_{0}(g) = ∫_{K×K} ϕ(kgk′) dk dk′.

Now ϕ_{0} ∈ H^{∘} and since L is H^{∘}-invariant this implies that v_{2} ∈ L. We can therefore approximate v arbitrarily closely by elements of L, and since L ⊂ V^{K} is finite-dimensional, hence closed, this implies that v ∈ L.

More generally, if (π, V) is a representation of G and k ∈ ℤ we will denote by V (k) the subspace of v ∈ K satisfying

π(κ_{θ}) v = e^{ikθ} v, κ_{θ} = ( \begin{matrix}cos(θ) & sin(θ) \\ − sin(θ) & cos(θ) \end{matrix} ) ∈ K.

(This is the ρ-isotypic subspace where ρ is the character e^{ikθ} of K.) If (π, V) is irreducible, then since −I ∈ K is central, it acts by a scalar (−1)^{ϵ} where ϵ = 0 or 1. Evidently the parity of k must be the same as ϵ for V (k) to be nonzero. We will call π even if ϵ = 0, and odd if ϵ = 1.

Proposition 5. If (π, V) is an irreducible admissible representation of G then V (k) is at most one-dimensional.

Proof (sketch). This may be proved along the same lines as Theorem 4. Because K = SO(2) is commutative, it may be seen that the convolution ring of smooth, compactly supported functions which satisfy f(κ_{θ}gκ_{θ′}) = e^{ik(θ+θ′)} f(g) is commutative. See Bump [B], Proposition 2.2.8.

We note that while Theorem 4 generalizes directly to arbitrary reductive Lie groups, Proposition 5 does not. Thus if (π, V) is an irreducible admissible representation of a reductive Lie group, the multiplicity of the trivial representation of its maximal compact subgroup is at most one; the other irreducible representations of the maximal compact each occur with finite multiplicity (this is admissibility), but not necessarily multiplicity one. Actually irreducible representations are automatically admissible, though this fact is not needed in the theory of automorphic forms, where admissibility of automorphic representations can be proved directly.

If (π, V) is an irreducible representation of G, we will denote by V_{fin} the direct sum of the V (k). This is the space of K-finite vectors. It is not invariant under the action of G, but it is invariant under the actions of the Lie algebra 𝔤 of G and of K. It is therefore a (𝔤, K)-module. See Bump [B], p. 200 for a discussion of this concept.

If π = ρ is the right regular representation ρ of G we will use the notation T_{ϕ} instead of ρ(ϕ). Thus

T_{ϕ}f(x) = ∫_{G} ϕ(g) f(xg) dg.

This is convolution with g → ϕ(g^{−1}). The notation is intended to suggest Hecke operators. These integral operators are important for us because they commute with Δ, yet they are easier to study than Δ. We note that L^{2}(Γ\G) is invariant under T_{ϕ}, and if ϕ ∈ H^{∘} then T_{ϕ} preserves the property of right invariance by K, so it can be regarded as an integral operator on L^{2}(Γ\ℌ).

Let ϕ ∈ H, and let

(4)  K_{ϕ}(x, y) = ∑_{γ∈Γ} ϕ(x^{−1}γy).

At first we regard (x, y) as an element of G × G. If x and y are restricted to a compact set C, then ϕ(x^{−1}gy) vanishes for g off a compact set C′. Therefore only finitely many γ contribute. It follows that (4) is convergent and defines a smooth function of x and y.

A change of variables shows that K_{ϕ}(x, y) is invariant if either x or y is changed on the left by an element of γ, so we may regard this kernel as defined on either G × G or on Γ\G × Γ\G, and it is a continuous function. If ϕ ∈ H^{∘} then K_{ϕ} may even be regarded as a function on X × X. (Recall that X = Γ\ℌ.)

Proposition 6. We have

(5)  (T_{ϕ}f)(x) = ∫_{Γ\G} K_{ϕ}(x, y) f(y) dy.

Proof. The left side equals

∫_{G} ϕ(y) f(xy) dy = ∫_{G} ϕ(x^{−1}y) f(y) dy = ∑_{γ∈Γ} ∫_{Γ\G} ϕ(x^{−1}γy) f(y) dy,

where we have used f(γy) = f(y). Interchanging sum and integral gives (5).

If H is a Hilbert space, an operator T : H → H is compact if T maps bounded sets into compact sets.

Theorem 7. T_{ϕ} is a compact operator on L^{2}(Γ\G).

See Bump [B], Section II.3, particularly Theorem 2.3.2 and Proposition 2.3.1 for fuller details.

Proof. The kernel K_{ϕ} is continuous on the compact space (Γ\G) × (Γ\G), so it is certainly in L^{2}((Γ\G) × (Γ\G)). The well-known theorem of Hilbert and Schmidt asserts that if Z is any locally compact Borel measure space such that L^{2}(Z) is a separable Hilbert space then integral operator

(T f)(x) = ∫_{Z} K(x, y) f(y) dy

with the kernel K ∈ L^{2}(Z × Z) is compact.

If

(6)  ϕ(g^{−1}) = \overline{ϕ(g)}

then K_{ϕ}(x, y) = \overline{K_{ϕ}(y, x)}, so T_{ϕ} is self-adjoint.

Theorem 8. Let T be a compact self-adjoint operator on a separable Hilbert space H. Then H has an orthonormal basis ϕ_{i} (i = 1, 2, 3, ⋯) of eigenvectors of T, so that T ϕ_{i} = μ_{i}ϕ_{i}. The eigenvalues μ_{i} → 0 as i → ∞.

This is the Spectral Theorem for compact operators. See Bump [B], Theorem 2.3.1.

Thus if (6) is true then T_{ϕ} is a self-adjoint compact operator whose nonzero eigenvalues μ_{i} → 0. The Hilbert-Schmidt property implies more: ∑ |μ_{i}|^{2} < ∞. Later we will see that more is true: ∑ |μ_{i}| < ∞. This means that T_{ϕ} is trace class. This fact is important because of the Selberg trace formula.

Theorem 9. L^{2}(X) has a basis consisting of eigenvectors of Δ.

Proof. The operators T_{ϕ} with ϕ satisfying (6) are a commuting family of selfadjoint compact operators so they can be simultaneously diagonalized. By the spectral theorem the nonzero eigenspaces are finite-dimensional; there is no nonzero vector on which the operators T_{ϕ} are all zero, since ϕ can be chosen to be positive, of mass one and concentrated near the identity in which case T_{ϕ}f approximates f. Therefore the simultaneous eigenspaces of H^{∘} are finite-dimensional.

Let V be such an eigenspace. Since Δ commutes with the T_{ϕ}, it preserves V, and since it is symmetric it induces a self-adjoint transformation on V. Choose an orthonormal basis for each such V consisting of eigenvectors of Δ and put these together for all V.

Closely related to Theorem 9 is a representation-theoretic statement about L^{2}(Γ\G). The regular representation ρ is a unitary representation on this space.

Lemma 10. Let H be a closed nonzero G-invariant subspace of L^{2}(Γ\G). Then H contains an irreducible subspace.

Proof (Langlands). Since H is G-invariant, each T_{ϕ} induces an endomorphism of H. We show first that the restriction of T_{ϕ} to H is nonzero for suitable ϕ satisfying (6). If 0 ≠ ξ ∈ H then for g near the identity ρ(g)ξ is near ξ. Thus if we take ϕ satisfying (6) such that ϕ > 0, ∫_{G} ϕ(g) dg = 1 and such that the support of ϕ is nonzero then T_{ϕ}ξ is near ξ so T_{ϕ} is nonzero on H, and T_{ϕ} is self-adjoint.

Let L ⊂ H be the eigenspace of a nonzero eigenvalue of T_{ϕ}. It is finitedimensional by Theorem 8. Let L_{0} be a nonzero subspace minimal with respect to property of being the intersection of L with a nonzero closed invariant subspace of H. Let V be the smallest closed invariant subspace of H such that L∩V = L_{0}. We show V is irreducible. If not, let V_{1} be a proper, nonzero closed invariant subspace and let V_{2} be its orthogonal complement, so V = V_{1} ⊕ V_{2}. Let 0 ≠ f ∈ L_{0}. Write f = f_{1} + f_{2} with f_{i} ∈ V_{i}. Since 0 = T_{ϕ}f − λf = (T_{ϕ}f_{1} − λf_{1}) + (T_{ϕ}f_{2} − λf_{2}) and T_{ϕ}f_{i} − λf_{i} ∈ V_{i} we have T_{ϕ}f_{i} − λf_{i} = 0. Thus f_{i} ∈ L ∩ V_{i}. By the minimality of L_{0}, L_{0} = L ∩ V_{i} for some i, say L_{0} = L ∩ V_{1}. Now the minimality of V is contradicted.

Theorem 11. L^{2}(Γ\G) decomposes as a direct sum of closed, irreducible subspaces. Each affords an irreducible admissible representation of G.

Proof. By Zorn’s Lemma, let S be a maximal set of orthogonal closed irreducible subspaces. Let H = ⨁_{V ∈S} V. If H is proper, applying Lemma 10 to its orthogonal complement contradicts the maximality of S. We leave admissibility to the reader.

Each of these closed irreducible subspaces has at most one K-fixed vector by Theorem 4.

Theorem 11 may be thought of as a more satisfactory extension of Theorem 9. Indeed, if ϕ is an eigenfunction of Δ occurring in L^{2}(Γ\ℌ) then its right translates span an irreducible subspace of L^{2}(Γ\G). Conversely, Δ acts by a scalar on each irreducible subspace. If that subspace happens to have a K-fixed vector in it, that vector will be one of the basis elements in Theorem 9.

There will, however, be some irreducible subspaces of L^{2}(Γ\G) which have no K-fixed vectors. These can be constructed from holomorphic modular forms as follows. Let f : Γ\ℌ → ℂ be a holomorphic function satisfying

f(\frac{az + b}{cz + d}) = (cz + d)^{k} f(z), ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

If k is sufficiently large, such f will always exist, as may be shown from the Riemann-Roch theorem. Regarded as a function on G, the function f is right invariant by K, but it is not left invariant by Γ, so it is not a function on Γ\G. However we may modify it, sacrificing the right invariance by K to obtain true left invariance by Γ. Define

F ( \begin{matrix}a & b \\ c & d \end{matrix} ) = (ci + d)^{−k} f(\frac{ai + b}{ci + d}) .

Then F (γg) = F (g) for γ ∈ Γ, while F (g κ_{θ}) = e^{ikθ}F (g). The irreducible representation spanned by F has not K-fixed vector. This is the weight k holomorphic discrete series representation.

The representation-theoretic approach has another generalization, its extension to automorphic forms on adele groups. Assume that Γ = SL(2, ℤ). Let 𝔸 be the adele ring of ℚ. The inclusion of SL(2, ℝ) → GL(2, 𝔸) at the infinite place induces a homeomorphism

Γ\G → GL(2, ℚ)Z_{𝔸}\GL(2, 𝔸)/_{∏p}K_{p},

where Z_{𝔸} is the center of GL(2, 𝔸) and K_{p} = GL(2, ℤ_{p}) is a maximal compact subgroup of GL(2, ℚ_{p}). Thus functions on Γ\G may be reinterpreted as functions on GL(2, ℚ)Z_{𝔸}\GL(2, 𝔸), and in particular we may embed L^{2}(Γ\G) → L^{2}(GL(2, ℚ)Z_{𝔸}\GL(2, 𝔸)).

Now the study of L^{2}(GL(2, ℚ)Z_{𝔸}\GL(2, 𝔸)) may be carried out along exactly the same lines as we’ve applied above. The class of integral operators is larger now, however. In addition to the ring H, we have its p-adic analogs, which are rings of Hecke operators. To see this, fix a prime p. Let H_{p} be the convolution ring of smooth (i.e. locally constant) compactly supported functions on GL(2, ℚ_{p}). This ring is not commutative, but the subring H_{p}^{∘} of functions which are K_{p} bi-invariant is commutative. (Compare Theorem 3.) For example, the characteristic function of K_{p} \binom{p 0}{0 1} K_{p} is an element of this ring, and this is the adelization of classical Hecke operator T_{p} which picks off the p-th Fourier coefficient of a Hecke eigenform. We will return to this point of view in the final section when we study such operators using the trace formula.

3. Green’s functions and the spectral resolvent. References for this section are Hejhal [H2], Section 6 and Bump [B], Chapter 2 Section 3. We are interested in functions f on ℌ with the following

Property S. The function f is left invariant by K = SO(2), possibly singular at K’s fixed point at i, and is an eigenfunction of the Laplacian.

We will see that for each eigenvalue λ of Δ there are two linearly independent such functions, one of which (the Green’s function) has nice behavior at the boundary, the other of which (the spherical function) is continuous at i. Each has its uses. Since H ≅ G/K, these may be regarded as functions on K\G/K (possibly undefined on K) which are eigenfunctions of the Laplace Beltrami operator.

We can map the upper half plane into the unit disk 𝔇 by the Cayley transform z ↦ w = (z − i)/(z + i). Let r = |w|. Then since f as in Property S is left invariant by K, f(z) depends only on r. Denote f(z) = W (r). Thus a function f with Property S is determined by the function W on (0, 1) such that

W (r) = f( \begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix} ) ,

where r = (y − 1)/(y + 1) ∈ (0, 1). The eigenvalue property amounts to the differential equation

(7)  W′′(r) + \frac{1}{r}W′(r) + \frac{4λ}{(1 − r^{2})^{2}} W (r) = 0.

This differential equation has regular singular points at (0, 1) and there are two solutions of interest. One is nicely behaved at 0, the other at 1. (We assume familiarity with regular singular points of second order linear differential equations, particularly the indicial equation, for which see Whittaker and Watson [WW], Section 10.3.) In this section we will be concerned with the solution which has nice behavior on the boundary, that is, at r = 1; the other one will occupy the next section.

Let λ = −s where s is a positive real number. At r = 1 the roots of the indicial equation of (7) for the singularity at r = 1 are \frac{1}{2}(1 ± \sqrt{1 + 4s}). Only one is positive, so there is a unique (up to multiple) solution g to (7) which vanishes near the boundary. We can use it to study the resolvent of the Laplacian.

Lemma 12. g_{s}(r) has a logarithmic singularity at r = 0.

Proof. The roots of the indicial equation at r = 0 are 0 with multiplicity 2, so one solution has a logarithmic singularity, another is analytic. If g_{s} does not have the logarithmic singularity, then g_{s}(r) is real and analytic on [−1, 1] hence has a maximum or minimum. At such a point g′_{s}(r) = 0 and since λ = −s < 0, equation (7) implies that g_{s} and g′′_{s} have the same sign, impossible at the maximum or minimum because g_{s}(−1) = g_{s}(1) = 0.

Let

g_{s}(z, ζ) = g_{s}(|\frac{z − ζ}{z − \overline{ζ}} |) .

z, ζ ∈ ℌ. This is a Green’s function.

Theorem 13.

[−y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + s] g(z, ζ) = 0;

g_{s}(z, ζ) is singular on the diagonal z = ζ;

g_{s}(z, ζ) → 0 as y → 0;
g_{s}(z, ζ) = g_{s}(ζ, z);

(8)  g_{s}(h(z), h(ζ)) = g_{s}(z, ζ),  h ∈ SL(2, ℝ).

See Bump [B], Proposition 2.3.4 on p. 181 for fuller details. Nothing here is very surprising, for example the first property boils down to (7), the second property comes from Lemma 12, the third follows from the boundary behavior of g_{s}, and the last two properties follow since |(z − ζ)/(z − \overline{ζ})| is unchanged if z and ζ are interchanged, or if an element of SL(2, ℝ) is applied to both.

Since g_{s} has a logarithmic singularity at 0 it can be normalized so g_{s}(r) − \frac{1}{2π} log(r) is bounded as r → 0. It follows that g′_{s}(r) − \frac{1}{2πr} is analytic near r = 0.

Theorem 14. If f ∈ C_{c}^{∞}(ℌ) then (writing ζ = ξ + iη)

∫_{ℌ} g_{s}(z, ζ) [−η^{2} (\frac{∂^{2}}{∂ξ^{2}} + \frac{∂^{2}}{∂η^{2}} ) + s] f(ζ) \frac{dξ ∧ dη}{η^{2}} = f(z) (ζ = ξ + iη).

Proof. See Bump [B], Proposition 2.3.4 on p. 181 for fuller details. We review the proof quickly. Let w = (z − ζ)/(z − \overline{ζ}) = u + iv ∈ 𝔇. Let F : 𝔇 → ℂ be defined by F (w) = f(z). In the w coordinates we must prove

∫_{𝔇} g_{s}(|w|) [− (\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} ) + \frac{4s}{(1 − |w|^{2})^{2}} ] F (w) du ∧ dv = F (0).

Let B_{r} be the disk of radius r, and let R < 1 be large enough that the support of F is contained in B_{R}. The left side equals

lim_{ϵ→0} ∫_{B_{R}−B_{ϵ}} g_{s}(|w|) [− (\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} ) + \frac{4s}{(1 − |w|^{2})^{2}} ] F (w) du ∧ dv.

Using Stoke’s theorem as in Proposition 2, this is

lim_{ϵ→0} ∫_{B_{R}−B_{ϵ}}F (w)[−(\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂v^{2}} ) + \frac{4s}{(1 − |w|^{2})^{2}} ] g_{s}(|w|) du ∧ dv+

lim_{ϵ→0} ∫_{C_{ϵ}} F (w) (\frac{∂g_{s}(|w|)}{∂u} dv − \frac{∂g_{s}(|w|)}{∂v} du) −

lim_{ϵ→0} ∫_{C_{ϵ}} g_{s}(|w|) (\frac{∂F (w)}{∂u} dv − \frac{∂F (w)}{∂v} du) ,

where C_{ϵ} is the path circling the origin counterclockwise around the circle with radius ϵ. (There would also be terms integrating around C_{R}, but these are zero because they lie outside the support of F.)

The first term vanishes by Theorem 13. The last term vanishes because the length of the arc shrinks faster than g_{s} blows up (logarithmically).

To evaluate the middle term let w = re^{iθ}. By the chain rule,

\frac{∂g_{s}(|w|)}{∂u} dv − \frac{∂g_{s}(|w|)}{∂v} du = r g′_{s}(r) dθ.

We obtain

lim_{ϵ→0} ∫_{0}^{2π} F (ϵe^{iθ}) dθ ϵ g′_{s}(ϵ) = F (0),

since g′_{s}(ϵ) ∼ 1/(2πϵ) as ϵ → 0.

Proposition 15. The series

G_{s}(z, ζ) = ∑_{γ∈{±1}\Γ}g_{s}(z, γ(ζ)) = ∑_{γ∈{±1}\Γ} g_{s}(γ(z), ζ).

is absolutely convergent provided z and ζ are not Γ equivalent.

See Bump [B], Proposition 2.3.5 on p. 184.

G_{s}(z, ζ) is the automorphic Green’s function. We will see that it is an integral kernel for the resolvent of the Laplacian.

Theorem 16. G_{s} is defined and real analytic for all values of (z, ζ) except where ζ = γ(z) for some γ ∈ Γ. We have

[−y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + s] G_{s}(z, ζ) = 0;

G_{s}(h(z), h(ζ)) = G_{s}(z, ζ), h ∈ G,

G_{s}(z, ζ) = G_{s}(γ(z), ζ) = G_{s}(z, γ(ζ)), γ ∈ Γ,
G_{s}(z, ζ) = \overline{G_{s}(ζ, z)},

and G_{s}(z, ζ) ∼ \frac{e}{2π} log |z−ζ| near z = ζ, where e is the order of the isotropy subgroup of ζ in Γ. For f ∈ C^{∞}(Γ\ℌ)

(9) ∫_{Γ\ℌ} G_{s}(z, ζ) [−η^{2} (\frac{∂^{2}}{∂ξ^{2}} + \frac{∂^{2}}{∂η^{2}} ) + s] f(ζ) \frac{dξ ∧ dη}{η^{2}} = f(z)  (ζ = ξ + iη).

See Bump [B] Proposition 2.3.5 and Hejhal [H2], Proposition 6.5 on p. 33.

Proof. Most of these properties follow from the corresponding properties of g_{s}. We prove (9). We will need a function F ∈ C_{c}^{∞}(ℌ) such that f(z) = ∑_{γ∈Γ} F (γz). To construct F, let u be a function on ℌ which is smooth, nonnegative, and has compact support containing a fundamental domain of Γ. Then for all z, the function ∑_{γ∈Γ} u(γz) is positive, and for z restricted to a compact set this sum is finite. It is thus a smooth, positive valued function and we can divide by it. Now

F (z) = \frac{u(z) f(z)}{∑_{γ∈Γ} u(γz)}

has the required property.

Substituting this and the definition of G_{s} and using (8) gives

∑_{γ,δ∈Γ} ∫_{Γ\ℌ} g_{s}(δ(z), γ(ζ)) [−η^{2} (\frac{∂^{2}}{∂ξ^{2}} + \frac{∂^{2}}{∂η^{2}} ) + s] F (γ(ζ)) \frac{dξ ∧ dη}{η^{2}} .

One of the summations may be collapsed with the integration to give

∑_{γ∈Γ} ∫_{ℌ} g_{s}(δ(z), ζ) [−η^{2} (\frac{∂^{2}}{∂ξ^{2}} + \frac{∂^{2}}{∂η^{2}} ) + s] F (ζ) \frac{dξ ∧ dη}{η^{2}} = ∑_{δ} F (δ(z)) = f(z).

This completes the proof.

Let s > 0 be a positive constant. As is easily checked, the logarithmic singularity along the diagonal is not sufficient to cause divergence of the integral

∫_{Γ\ℌ} ∫_{Γ\ℌ} |G_{s}(z, ζ)|^{2} \frac{dx ∧ dy}{y^{2}} \frac{dξ ∧ dη}{η^{2}} < ∞.

Thus the corresponding integral operator, which we shall denote R(s, Δ), is Hilbert-Schmidt. It is the resolvent of the Laplacian. If ϕ is an eigenfunction of Δ with eigenvalue λ, then it follows from (9) that it is also an eigenfunction of R(s, Δ) with eigenvalue (λ + s)^{−1}.

Theorem 17. (i) The eigenvalues λ_{i} of Δ on L^{2}(Γ\ℌ) tend to ∞, and satisfy ∑ λ_{i}^{−2} < ∞. (We exclude the eigenvalue λ_{0} = 0 corresponding to the constant function from this summation.) (ii) The Laplacian Δ has an extension to a self-adjoint operator on the Hilbert space L^{2}(Γ\ℌ).

Proof. Assume re(s) > 0. Since R(s, Δ) is Hilbert-Schmidt, ∑_{i}(λ_{i} + s)^{−2} < ∞, whence ∑_{i} λ_{i}^{−2} < ∞. By Theorem 9, let ϕ_{i} be a basis of H = L^{2}(Γ\ℌ) consisting of eigenvectors of Δ, with corresponding eigenvalues λ_{i}.

We prove (ii). Let 𝔇_{Δ} be the linear subspace of ℌ consisting of elements of the form ∑ a_{i}ϕ_{i} such that ∑ λ_{i}^{2}|a_{i}|^{2} < ∞; on this space, define

Δ(∑ a_{i}ϕ_{i}) = ∑ λ_{i} a_{i} ϕ_{i}.

Since the λ_{i} tend to infinity, and in particular are bounded away from zero, it is not hard to check that this operator is closed and in fact self-adjoint. This completes the proof.

We arrange the eigenvalues of Δ in ascending order:

λ_{0} = 0 < λ_{1} ≤ λ_{2} ≤ λ_{3} ≤ · · · .

The eigenvalue λ_{0} corresponds to the constant function and has multiplicity exactly one. Eigenvalues in the range (0, \frac{1}{4}) are called exceptional eigenvalues. They are qualitatively different. For example, the spherical functions corresponding to exceptional eigenvalues are nontempered—they grow faster than spherical functions corresponding to λ ≥ \frac{1}{4}. (See Section 4). Exceptional eigenvalues correspond to zeros of the Selberg zeta function on the real line between 0 and 1. By contrast zeros corresponding to λ ≥ \frac{1}{4} satisfy the Riemann hypothesis. (See Section 7.)

Randol [R] proved that for some X, exceptional eigenvalues do occur. On the other hand, Selberg [S4] conjectured that exceptional eigenvalues do not occur in the cuspidal spectrum of congruence subgroups of SL(2, ℤ), and it would follow from this that they do not occur in the spectrum of compact quotients Γ\ℌ associated to quaternion division algebras.

4. Spherical functions. As we pointed out, Property S reduces to a second order differential equation which has two independent solutions. One solution, having nice behavior at the boundary, is the Green’s function. Another, having nice behavior at i, is the so-called spherical function. The substance of Lemma 12 is that these two solutions are not the same. In this section we will study the spherical function.

Definition. Let λ be a complex number. We call a function σ on SL(2, ℝ) a λ-spherical function if it is smooth, K-bi-invariant, and Δσ = λσ.

Here Δ is the Laplace-Beltrami operator. Before we show that such a function exists and is unique up to constant multiple, let us explain briefly how such a function fits into the representation theory.

Suppose that (π, V) is an irreducible representation of G. Let (π̂, V̂) be its contragredient. Thus there exists an invariant bilinear pairing ⟨ , ⟩ : V × V̂ → ℂ.

By Theorem 4, if V^{K} and V̂^{K} are nonzero, they are one-dimensional, and we will assume this. (Actually if one is nonzero the other is too.) Let v^{∘} ∈ V^{K} and v̂^{∘} ∈ V̂^{K} be nonzero K-fixed vectors. The function

σ(g) = ⟨π(g) v^{∘}, v̂^{∘}⟩

is evidently K-bi-invariant. Moreover, Regarding Δ as an element of the center of the universal enveloping algebra U(𝔤), it acts by a scalar λ on V. and σ inherits this property. So it is a spherical function.

Without reference to this construction, we now show that spherical functions exist and are unique.

Theorem 18. (i) Let λ ∈ ℂ. Then there is a unique smooth K-bi-invariant function ω_{λ} on SL(2, ℝ) such that Δω_{λ} = λω_{λ} and ω_{λ}(1) = 1. (ii) If f : G → ℂ is any smooth function such that Δf = λf, then

(10)  ∫_{K×K} f(kgk′) dk dk′ = f(1) ω_{λ}(g).

(iii) If f is right K-invariant and Δf = λf, then

(11)  ∫_{K} f(hkg) dk = f(h) ω_{λ}(g).

Proof. To satisfy Δω = λω we need

W (r) = ω( \begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix} ) , r = \frac{y − 1}{y + 1}

to satisfy (7). As we have seen, this differential equation has a regular singular point at the origin, and one solution is bounded there, whereas the other has a logarithmic singularity. Hence ω_{λ}, if it exists, is unique.

To show that such a function exists, let f be any continuous function on G which is an eigenfunction of Δ. The left hand side of (10) is a K-bi-invariant function which is an eigenfunction of Δ. If f(1) = 1, this will satisfy (i), proving existence. For example we can take f = f_{s} where λ = s(1 − s) and

(12)  f_{s} (\binom{y^{1/2}  ∗}{y^{−1/2}} k) = y^{s},  y > 0, k ∈ K.

Now that the existence and uniqueness of ω are established, we note that the left sides of both (10) and (11) are smooth K-bi-invariant eigenfunctions of Δ, hence constant multiples of ω_{λ}. In both cases, the constant may be evaluated by taking g = 1. This proves both (ii) and (iii).

Theorem 19. Suppose that f is a smooth function on G which is right invariant by K and such that Δf = λf. Then for ϕ ∈ H^{∘} we have T_{ϕ}f = χ_{λ}(ϕ)f where

(13)  χ_{λ}(ϕ) = ∫_{G} ϕ(g) ω_{λ}(g) dg.

Proof. By Theorem 18, we have

(14)  ∫_{K} f(hkg) dk = f(h) ω_{λ}(g).

We note that T_{ϕ}f is an average of right translates of f, and right translation commutes with left translation. Hence we may apply T_{ϕ} to both sides of (14) to obtain

∫_{K}(T_{ϕ}f)(hkg) dk = f(h) (T_{ϕ}ω_{λ})(g).

We take g = 1 in this identity. Since T_{ϕ}f is right K-invariant, the integrand on the left side becomes constant when g = 1 and so the left side becomes just (T_{ϕ}f)(h). On the other hand (T_{ϕ}ω_{λ})(1) equals the integral (13), so T_{ϕ}f(h) = χ_{λ}(ϕ) f(h).

Theorem 20. If ϕ_{1} and ϕ_{2} ∈ H^{∘}, then

χ_{λ}(ϕ_{1} ∗ ϕ_{2}) = χ_{λ}(ϕ_{1}) χ_{λ}(ϕ_{2}).

Proof. This follows from Theorem 19 on applying T_{ϕ} to any eigenfunction f, for example f_{s} as in (12).

Thus the function χ_{λ} : H^{∘} → ℂ is a character of H^{∘}. We return to the point of view introduced at the beginning of the section to explain the meaning of χ(ϕ) in terms of representations. First, we recall the construction and parametrization of a class of irreducible representations of SL(2, ℝ), the principal series.

Let s be a complex number. Let P_{s}^{+} be the set of functions f : G → ℂ such that

(15)  f(\binom{y^{1/2} xy^{−1/2}}{y^{−1/2}} g) = (−1)^{ϵ sgn(y)} y^{s}f(g),

where ϵ = 0, and such that the restriction of f to K is square integrable. Similarly let P_{s}^{−} be the space of functions satisfying (15) with ϵ = 1. G acts on these spaces by right translation. Let (π_{s}^{±}, P_{s}^{±}) be these representations. The representation π_{s}^{±} is irreducible except in the case where 2s is an integer and 2s is even for π_{s}^{+}, odd for π_{s}^{−}. We call π_{s}^{+} and π_{s}^{−} the odd and even principal series of representations, respectively.

Suppose that re(s) > \frac{1}{2}. An intertwining integral M(s) : P_{s}^{±} → P_{1−s}^{±} is given by

(16)  (M(s)f)(g) = ∫_{−∞}^{∞} f(( \begin{matrix}& −1 \\ 1 \end{matrix} ) \binom{1 x}{1} g) dx .

It may be checked that the integral is absolutely convergent. Now M(s) extends by analytic continuation to an intertwining map for all s such that 2s is not an integer congruent to ϵ mod 2. Actually we only claim that it extends to an infinitesimal equivalence, that, is, an isomorphism of the underlying (𝔤, K) module, but it is a true isomorphism in the important special case re(s) = \frac{1}{2}. Thus π_{1/2+it}^{±} ≅ π_{1/2−it}^{±}.

Let k ≡ ϵ modulo 2. Let

(17)  f_{s,k}(\binom{y^{1/2} x}{y^{−1/2}} κ_{θ}) = sgn(y)^{ϵ} y^{s} e^{ikθ}.

This is a K-finite vector in P_{s}^{±} where ± is (−1)^{ϵ}. Let f̃_{1−s,k} = M(s) f_{s,k}. One may compute

(18)  f̃_{k,1−s} = (−i)^{k} \sqrt{π} \frac{Γ(s) Γ(s − \frac{1}{2})}{Γ(s + \frac{k}{2} ) Γ(s − \frac{k}{2} )} f_{k,1−s}.

See Bump [B], Proposition 2.6.3.

The representation π_{1/2+it}^{±} is unitary if t is real. These representations are called the unitary principal series representations. Also π_{s}^{+} is unitary if s is a real number between 0 and 1. These are the complementary series representations. They correspond to exceptional eigenvalues of the Laplacian in the automorphic spectrum. There are a few other irreducible representations, namely the discrete series (related to holomorphic modular forms) and the trivial representations. But only the even principal series π_{s}^{+} have K-fixed vectors. All of these facts are proved in Bump [B], Chapter 2.

The K-fixed vector in P_{s}^{+} is precisely the function f_{s} in (12). Now if ϕ ∈ H^{∘} then since ϕ is K-bi-invariant, the operator π^{+}(ϕ) maps all of P_{s}^{+} onto the unique K-fixed vector f_{s}. Thus the trace of the rank one operator π_{s}^{+}(ϕ) is just its eigenvalue on this vector f_{s}, so

(19)  tr π_{s}^{+}(ϕ) = χ_{λ}(ϕ).

5. The Plancherel formula. Let ϕ ∈ H^{∘}. Define

(20)  g(u) = e^{u/2} ∫_{−∞}^{∞} ϕ(( \begin{matrix}e^{u/2} \\ & e^{−u/2} \end{matrix} ) \binom{1 x}{1}) dx,

and let

(21)  h(t) = ∫ g(u) e^{iut} dt

be its Fourier transform.

Theorem 21. The functions g and h are even, and g is compactly supported. If λ = \frac{1}{4} + t^{2}, then

(22)  χ_{λ}(ϕ) = h(t).

Proof. Let f_{s} be as in (12) with s = \frac{1}{2} + it so that λ = \frac{1}{4} + t^{2} = s(1 − s). By Theorem 18,

∫_{K} f_{s}(kg) dk = ω_{λ}(g),

and

χ_{λ}(ϕ) = ∫_{G} ∫_{K} ϕ(g) f_{s}(gk) dk dg.

Interchanging the order of integration and making the variable change g → gk^{−1}, since ϕ is right k invariant, we obtain

χ_{λ}(ϕ) = ∫_{G} ϕ(g) f_{s}(g) dg.

Now we use the coordinates

(23)  g = ( \begin{matrix}e^{u/2} \\ & e^{−u/2} \end{matrix} ) \binom{1 x}{1} κ_{θ},  dg = \frac{1}{2π} du dx dθ.

Noting that in these coordinates f_{s}(g) = e^{u/2} e^{iut}, we obtain

χ_{λ}(ϕ) = ∫_{∞}^{∞} g(u) e^{itu} du,

proving (22). We note that ω_{λ}, and therefore the character χ_{λ}(ϕ) is unchanged if s → 1 − s, that is, if t → −t. Hence (22) implies that h is an even function. By Fourier inversion, so is g.

Theorem 22. (i) We have

ϕ(1) = \frac{1}{2π} ∫_{0}^{∞} t h(t) tanh(πt) dt.

(ii) If ϕ_{1}, ϕ_{2} are K-invariant and compactly supported then

∫_{G} ϕ_{1}(g) \overline{ϕ_{2}(g)} dg = \frac{1}{2π} ∫_{0}^{∞} t h_{ϕ_{1}}(t) \overline{h_{ϕ_{2}}(t)} tanh(πt) dt.

Proofs may be found in Knapp [K], Chapter 11, Gelfand, Graev and Piatetski-Shapiro [GGP], Chapter 2 Section 6 and Varadarajan [V] Theorem 39 on p. 205. We will give a proof (after some Lemmas) which uses no Lie theory. A portion of the argument parallels Proposition 4.1 on p. 15 of Hejhal [H1], and we have made our notation consistent with his.

Theorem 22 (i) is the Fourier inversion formula on the noncommutative group SL(2, ℝ). It is sometimes called the Plancherel formula because it implies (ii), which is the true Plancherel formula. The measure \frac{1}{2π} t tanh(πt) dt is called the Plancherel measure on the even unitary principal series. (Since we have only considered ϕ ∈ H^{∘} we do not need the other irreducible unitary representations.) The Plancherel measure is closely related to the intertwining integrals (16). Indeed, M(\frac{1}{2} − it) ∘ M(\frac{1}{2} + it) is an G-equivariant endomorphism of the irreducible space P_{1/2+it}^{+}, so by Schur’s Lemma it is a scalar. One may check using (18) that the reciprocal of this scalar is \frac{1}{π} t tanh(πt). See Knapp and Stein [KS].

Lemma 23. Let η ≥ y > 1. Then

ϕ(\binom{1 x}{1} ( \begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix} )) = ϕ( \begin{matrix}η^{1/2} \\ & η^{−1/2} \end{matrix} ) ,

where

x = ±\sqrt{y(η + η^{−1} − y − y^{−1})}.

Proof. Identifying K\G/K with K\H, we want to find x such that the images ηi and z = x + iy of the two matrices in H are in the same SO(2) orbit. Mapping H to the unit disk by the Cayley transform z → (z − i)/(z + i), their images must therefore be equidistant from the origin. That is,

\frac{η − 1}{η + 1} = |\frac{z − i}{z + i}| = \sqrt{\frac{a − 2y}{a + 2y}} , a = x^{2} + y^{2} + 1.

Applying the map t → (1+t^{2})/(1−t^{2}) to both sides of this equation, η+η^{−1} = a/y. This equation can now be solved for x.

Define a function Φ on ℝ^{+} by

(24)  Φ(e^{v} + e^{−v} − 2) = ϕ( \begin{matrix}e^{v/2} \\ & e^{−v/2} \end{matrix} ) .

Lemma 24. If U = e^{u} + e^{−u} − 2, then g(u) = Q(U) where

(25)  Q(U) = ∫_{U}^{∞} \frac{Φ(V) dV}{\sqrt{V − U}} .

Proof. Let V = e^{v} + e^{−v} − 2. With y = e^{u}, η = e^{v} and x as in Lemma 23, think of x as function of V ≥ U. Then dx = \frac{1}{2} e^{u} (V − U)^{−1/2} dV. We integrate V from U to infinity and double the result to account for both positive and negative x.

Lemma 25. We have

(26)  Φ(U) = −\frac{1}{π} ∫_{U}^{∞} \frac{Q′(V) dV}{\sqrt{V − U}} .

Proof. We’d like to integrate under the integral sign in (25) but since the left endpoint depends on U we must be careful. Integrate (25) by parts to obtain

Q(U) = −2 ∫_{U}^{∞} Φ′(V) \sqrt{V − U} dV.

The integrand now vanishes at the left endpoint, so we may differentiate under the integral sign, then integrate by parts again to obtain

Q′(U) = ∫_{U}^{∞} \frac{Φ′(V)}{\sqrt{V − U}} dV = −2 ∫_{U}^{∞} Φ′′(V) \sqrt{V − U} dV.

We substitute this into the right side of (26), then switch the order of integration:

(27) − \frac{1}{π} ∫_{U}^{∞} \frac{Q′(V) dV}{\sqrt{V − U}} = \frac{2}{π} ∫_{U}^{∞} ∫_{V}^{∞} \sqrt{\frac{W − V}{V − U}} Φ′′(W) dW dV =

\frac{2}{π} ∫_{U}^{∞} ∫_{U}^{W} \sqrt{\frac{W − V}{V − U}} dV Φ′′(W) dW.

To evaluate the inner integral, let V = U + (W − U) v. We have

∫_{U}^{W} \sqrt{\frac{W − V}{V − U}} dV = (W − U) ∫_{0}^{1} \sqrt{\frac{1 − v}{v}} dv = \frac{π}{2} (W − U)

since ∫_{0}^{1} v^{−1/2}(1 − v)^{1/2} dv = B(\frac{1}{2}, \frac{3}{2}) = \frac{π}{2} . Thus (27) equals

∫_{U}^{∞}(U − W) Φ′′(W) dW = − ∫_{U}^{∞} Φ′(W) dW = Φ(U),

where we have integrated twice by parts.

Proof of Theorem 22. Take U = 0 and write Lemma 25 in the form

(28)  ϕ(1) = Φ(0) = −\frac{1}{π} ∫_{0}^{∞} \frac{g′(u) du}{e^{u/2} − e^{−u/2}} .

Since g is even and g′ is odd, we have the Fourier inversion formula

g(u) = \frac{1}{π} ∫_{0}^{∞} h(t) e^{−itu} dt, g′(u) = \frac{1}{iπ} ∫_{0}^{∞} t h(t) e^{−itu} dt,

and we may change the limits in (28) to (−∞, ∞), dividing by 2, then interchange the order of integration to obtain

ϕ(1) = \frac{1}{2π} ∫_{0}^{∞} t h(t) \frac{i}{π} ∫_{−∞}^{∞} \frac{e^{−iut} du}{e^{u/2} − e^{−u/2}} dt.

The inner integral passes through the pole at u = 0 of the integrand and is interpreted as the principal value. The Plancherel formula now follows from

(29)  ∫_{−∞}^{∞} \frac{e^{−iut} du}{e^{u/2} − e^{−u/2}} = −iπ tanh(πt),

which we may prove as follows. Since t > 0, the numerator e^{−iut} is small for u in the lower half plane, and we may move the path of integration downwards. The left side of (29) −2πi times the sum of the residues at u = −2πin of the integrand in the lower half plane. The residue at u = 0 is only counted half since the path of integration passes through this point. We get

−2πi (\frac{1}{2} + ∑_{k=1}^{∞}(−1)^{k}e^{−2πkt}) = −πi tanh(πt)

proving (i).

To prove (ii) define ϕ′_{2}(g) = \overline{ϕ_{2}(g^{−1})}. Since ϕ_{2} is K-bi-invariant, and since the K-double cosets are stable under g → g^{−1} this equals \overline{ϕ_{2}(g)}. Now apply (i) to ϕ = ϕ_{1} ∗ ϕ′_{2}, then ϕ(0) = ⟨ϕ_{1}, ϕ_{2}⟩_{2}. On the other hand h_{ϕ}(t) = h_{ϕ_{1}}(t) \overline{h_{ϕ_{2}}(t)} by Theorem 20 so (ii) follows.

Proposition 26. Let g be an even, compactly supported function on ℝ. There exists ϕ ∈ H^{∘} such that (20) is true.

Proof. Let Q : ℝ^{+} → ℂ be defined by Q(e^{u} + e^{−u} − 2) = g(u). We define Φ : ℝ^{+} → ℂ by (26) or, integrating by parts and substituting W = V − U,

Φ(U) = \frac{2}{π} ∫_{0}^{∞} Q′′(W + U) \sqrt{W} dW.

We may differentiate under the integral sign arbitrarily many times, so Φ is smooth, even at U = 0. Now u → Φ(e^{u} + e^{−u} − 2) is a smooth, even, compactly supported function, and it follows that there is a unique smooth, compactly supported K-bi-invariant ϕ satisfying (24). Now the statement follows from Lemma 24.

6. The Selberg trace formula. We have already shown that the integral operators T_{ϕ} are Hilbert-Schmidt, hence compact. More is true: they are trace class. A compact operator is trace class if it can be factored as the composite of two Hilbert-Schmidt operators. If it is self-adjoint, and has eigenvalues λ_{i}, it is easy to see that this is equivalent to ∑ |λ_{i}| < ∞. Lang’s [L1] contains much useful material about trace class operators.

Let f_{i} be a basis of L^{2}(Γ\ℌ) consisting of eigenfunctions of T_{ϕ} which are also eigenfunctions of Δ. Assuming that ϕ ∈ H^{∘} satisfies (6), let μ_{i} be the eigenvalues of T_{ϕ} with respect to this basis. Making a Fourier expansion we have

(30)  K_{ϕ}(z, w) = ∑ μ_{i}f_{i}(z) \overline{f_{i}(w)}.

Theorem 27. If ϕ ∈ H^{∘} then T_{ϕ} is trace class.

Proof. A linear combination of trace class operators is trace class. Hence it is sufficient to prove this with ϕ replaced by \frac{1}{2}(ϕ(g)+\overline{ϕ(g^{−1})}) and by \frac{1}{2i}(ϕ(g)−\overline{ϕ(g^{−1})}). We may thus assume that ϕ satisfies (6) and so T_{ϕ} is self-adjoint. Let μ_{i} be its (nonzero) eigenvalues. Let λ_{i} be the corresponding eigenvalues of Δ. Thus ∑ λ_{i}^{−2} < ∞.

Applying Δ to K_{ϕ}(z, w) in the first variable gives a new kernel Δ_{z}K_{ϕ}. We will show that

(31)  (Δ_{z}K_{ϕ})(z, w) = ∑ μ_{i}λ_{i}f_{i}(x) \overline{f_{i}(y)}.

Formally this follows from (30) by termwise differentiation. However this must be justified since the convergence in (30) is in the L^{2} sense, and Δ is an unbounded operator. Note that Δ_{x}K_{ϕ}(x, y) is continuous hence has an expansion

(Δ_{z}K_{ϕ})(z, w) = ∑ ν_{i}f_{i}(x) \overline{f_{i}(y)}.

Let s > 0. Consider

∫_{Γ\ℌ} G_{s}(z, ζ) [(Δ_{ζ}K_{ϕ})(ζ, w) + sK_{ϕ}(ζ, w)] \frac{dξ ∧ dη}{η^{2}} .

On the one hand by (9) this is just (30). On the other hand, the term in square brackets is ∑_{i}(ν_{i} + sμ_{i})f_{i}(ζ) \overline{f_{i}(w)}, and since the resolvent is compact, may apply it term by term. Using R(s, Δ)f_{i} = (s + λ_{i})^{−1}f_{i} we get (ν_{i} + sμ_{i})(λ_{i} + s)^{−1} = μ_{i}, so ν_{i} = λ_{i}μ_{i} proving (31).

Since this function Δ_{z}K(z, w) is continuous, it is Hilbert-Schmidt, and so we obtain the bound

(32)  ∑ |μ_{i}λ_{i}|^{2} < ∞.

Now ∑ |μ_{i}| < ∞ follows from ∑ |λ_{i}|^{−2} < ∞ and (32) by Cauchy-Schwarz.

The trace tr T of a self-adjoint trace class operator T is by definition the sum of its eigenvalues.

Theorem 28. If ϕ ∈ H^{∘} satisfies (6), and if μ_{i} are the eigenvalues of T_{ϕ}, the trace

(33)  tr T_{ϕ} = ∫_{Γ\ℌ} K_{ϕ}(z, z) \frac{dx ∧ dy}{y^{2}} .

Proof. This follows from orthonormality on integrating (30).

The Selberg trace formula is a more explicit formula for its trace. Let {γ} denote a set of representatives for the conjugacy classes of Γ. Let Z_{Γ}(γ) denote the centralizer in Γ of γ.

Theorem 29. We have

(34)  tr T_{ϕ} = ∑_{{γ}} ∫_{Z_{Γ}(γ)\G} ϕ(g^{−1}γg) dg.

Proof. We rewrite the right side of (33) as

∑_{γ∈Γ} ∫_{G} ϕ(g^{−1}γg) dg = ∑_{{γ}} ∑_{δ∈Z_{Γ}\Γ} ∫_{G} ϕ(g^{−1} δ^{−1} γ δ g) dg.

Combining the integral and the summation gives (34).

This is a primitive form of the trace formula. To make this more explicit, we must study more explicitly the orbital integrals on the right side. An element 1 ≠ γ ∈ Γ is hyperbolic if its eigenvalues are real, elliptic if complex of absolute value 1. Γ is hyperbolic if each 1 ≠ γ ∈ Γ is hyperbolic. For example, let X be a compact Riemann surface of genus ≥ 2. Its universal cover is ℌ and Γ = π_{1}(X) acts with quotient X. These examples are precisely the hyperbolic groups.

We assume now that Γ is a hyperbolic group. If 1 ≠ γ ∈ Γ define N = N(γ) by asking that γ be conjugate to

(35)  ( \begin{matrix}N^{1/2} \\ & N^{−1/2} \end{matrix} ) .

for some N. Let N_{0} = N_{0}(γ) be such that

( \begin{matrix}N_{0}^{1/2} \\ & N_{0}^{−1/2} \end{matrix} )

is conjugate to a generator of Z_{Γ}(γ). We may obviously assume that N and N_{0} are > 1. We note that Z_{G}(γ) is conjugate to the diagonal subgroup. Its image in X is a closed geodesic. So the numbers N(γ) are thus the lengths of closed geodesics, and the numbers N_{0}(γ) are the lengths of prime geodesics.

Theorem 30. With g the function in (20),

(36)  ∫_{Z_{Γ}(γ)\G} ϕ(g^{−1}γg) dg = \frac{log N_{0}}{N^{1/2} − N^{−1/2}} g(log N).

Proof. We may assume that γ equals (35). Using Iwasawa coordinates (23) the integral is

∫_{0}^{log N_{0}} du ∫_{−∞}^{∞} ϕ (\binom{1 −x}{1} ( \begin{matrix}N^{1/2} \\ & N^{−1/2} \end{matrix} ) \binom{1 x}{1}) dx =

| log(N_{0})| ∫_{−∞}^{∞} ϕ (( \begin{matrix}N^{1/2} \\ & N^{−1/2} \end{matrix} ) \binom{1 (1 − N^{−1})x}{1}) dx,

and a change of variables proves (36).

Theorem 31. Let g be a smooth, even, compactly supported function and let h be its Fourier transform, defined by (21). If \frac{1}{4} + t_{i}^{2} are the eigenvalues of Δ on ℌ, and if log(N) runs through the lengths of closed geodesics of Γ, where for each N we let N_{0} be the length of the corresponding prime geodesic, we have

(37) ∑ h(t_{i}) = \frac{vol(Γ\ℌ)}{4π} ∫_{−∞}^{∞} t h(t) tanh(πt) dt + ∑_{N} \frac{log N_{0}}{N^{1/2} − N^{−1/2}} g(log N).

This is the Selberg trace formula.

Proof. We choose ϕ as in Proposition 26. By Theorem 19 and Theorem 21, the h(t_{i}) are the eigenvalues of T_{ϕ} on the eigenfunctions of the Laplacian, so the left side of (37) is the trace of T_{ϕ}. By Theorem 22 and Theorem 30, the right side of (37) is the sum of the orbital integrals. Thus the identity follows from Theorem 29.

7. The Selberg zeta function. In order to get useful applications the class of functions g and h in the trace formula must be expanded.

Theorem 32. The trace formula Theorem 31 remains true provided h is an even function analytic in the strip im(z) ≤ \frac{1}{2} + δ, such that h(r) = O(1 + |r|)^{−2−δ} in this strip. This assumption implies that the Fourier transform g(u) = O(e^{−( \frac{1}{2} +δ)|u|}).

See Hejhal [H1], Chapter 1 Section 7 for the proof by an approximation argument that Theorem 31 implies this stronger statement.

Theorem 33. (Weyl’s law). The number of j with t_{j} ≤ x is asymptotically \frac{1}{4π} vol(Γ\ℌ) x^{2}.

This must be mentioned as the first significant application of the trace formula. Noting that λ_{j} = \frac{1}{4} + t_{j}^{2}, this means that the number of λ_{j} ≤ x is asymptotically \frac{1}{4π} vol(Γ\ℌ) x.

We will not prove Weyl’s law. Briefly, taking h(t) = e^{−t^{2}T}, the hyperbolic contributions in Theorem 31 are of smaller magnitude and the first term in (37) predominates. One obtains ∑ e^{−λ_{j}T} ≅ \frac{1}{4πT} vol(Γ\ℌ), and Weyl’s law follows from a Tauberian theorem. See Hejhal [H1], Chapter 2 Section 2.

The trace formula may be regarded as a duality between the length spectrum of Γ\ℌ (that is, the set of lengths N of closed geodesics) and the numbers t_{i} such that \frac{1}{4} + t_{i}^{2} are the eigenvalues of the Laplacian. A similar duality, which we next discuss, pertains between the set of prime numbers and the zeros of the Riemann zeta function.

Theorem 34. Let g be compactly supported, smooth and even, and let h(t) = ∫_{−∞}^{∞} e^{itu} g(u) du. Let \frac{1}{2} + it_{i} denote the zeros of ζ in the cricital strip. Then

(38) h(−\frac{i}{2}) − ∑ h(t_{i}) + h(\frac{i}{2}) = −\frac{1}{2π} ∫_{−∞}^{∞} h(t)\frac{Γ′(\frac{1+2it}{4})}{Γ(\frac{1+2it}{4})} dt

+ g(0) log(π) + 2 ∑ \frac{log(p)}{\sqrt{p^{n}}} g( log(p^{n})).

Since h is assumed even h(−i/2) = h(i/2). However there are good reasons for writing the formula this way. (See Remark 4 below.) Special cases (“explicit formulae”) were found by Riemann, von Mangoldt, Hadamard, de la Vallee Poussin, and Ingham. These are discussed in Ingham [Ing]. Weil [W1] formalized the duality.

Proof. To prove this, let ξ(s) = Γ_{ℝ}(s) ζ(s), where Γ_{ℝ}(s) = π^{−s/2}Γ(\frac{s}{2}). The function h(t) is entire and we write h(t) = H(\frac{1}{2} + it), H(s) = h( − i(s − 1/2)). Consider, for δ > 0

\frac{1}{2πi} ∫_{1+δ−i∞}^{1+δ+i∞} \frac{ξ′(s)}{ξ(s)} H(s) ds.

Moving the path of integration to re(s) = −δ and using the functional equation ξ(s) = ξ(1 − s), we obtain the negative of this integral plus the sum of the residues; so

\frac{1}{πi} ∫_{1+δ−i∞}^{1+δ+i∞} \frac{ξ′(s)}{ξ(s)} H(s) ds = ∑ h(t_{i}) − h(\frac{i}{2}) − h(−\frac{i}{2}) .

We have ξ′/ξ = Γ′_{ℝ}/Γ_{ℝ} + ζ′/ζ, and the integral over Γ_{ℝ} can be moved left to re(s) = \frac{1}{2}. We have

\frac{1}{πi} ∫_{\frac{1}{2} −i∞}^{\frac{1}{2} +i∞} \frac{Γ′_{ℝ}(s)}{Γ_{ℝ}(s)} h( − i(s − \frac{1}{2})) ds = −g(0) log(π) + \frac{1}{2π} ∫_{−∞}^{∞} h(t) \frac{Γ′(\frac{1+2it}{4})}{Γ(\frac{1+2it}{4})} dt.

On the other hand, we have −ζ′(s)/ζ(s) = ∑ Λ(n) n^{−s} where Λ(p^{k}) = log(p), p prime, while Λ(n) = 0 if n is not a prime power. We have

\frac{1}{2πi} ∫_{2−i∞}^{2+i∞} H(s) n^{−s} ds = \frac{g( log(n))}{\sqrt{n}},

and assembling the pieces we get (38).

Remark 1. Moving the line of integration requires knowing that a path from 2+iT to −1+iT can be found for arbitrarily large T where ζ′/ζ is not too large. This can be accomplished by choosing the path to lie about half way between a pair of zeros that are not too close together. See Ingham [Ing], Theorem 26 on p. 71, where he proves that one can always find t near T such that ζ/ζ′ is O(log^{2}(T)) on the line from 2 + it to −1 + it.

Remark 2. In practice the condition that g be compactly supported is too strong. It is sufficient that g and h be as in Theorem 32.

Remark 3. If H is not too big in the left half plane we have

\frac{1}{2π} ∫_{−∞}^{∞} h(t)\frac{Γ′(\frac{1+2it}{4})}{Γ(\frac{1+2it}{4})} dt = \frac{1}{2πi} ∫_{\frac{1}{2} −i∞}^{\frac{1}{2} +i∞} H(s)\frac{Γ′(\frac{s}{2})}{Γ(\frac{s}{2})} ds = 2 ∑_{n=0}^{∞} H(−2n),

where the sum is over the residues at the poles of Γ(s/2).

Remark 4. Let X be a nonsingular complete curve of genus g over the finite field k = 𝔽_{q}. If P is a prime divisor of degree d(P) we denote N(P) = q^{d(P)}. The zeta function of X is

∏(1 − N(P)^{−s})^{−1} = \frac{∏_{j=1}^{2g}(1 − α_{j}q^{−s})^{−1}}{(1 − q^{−s})(1 − q^{1−s})} ,

where α_{i} are the eigenvalues of the Frobenius map in H^{1}(X, ℚ_{l}), for any prime l ≠ p. The Riemann hypothesis is that |α_{j}| = \sqrt{q}. We write α_{j} = q^{\frac{1}{2} +iθ_{j}}. Let g(n) : ℤ → ℂ be a sequence which is even and nonzero for only finitely many n, and let

h(t) = ∑_{n} g(n) e^{itn log(q)}.

Thus h is entire, even, and periodic with period 2π/ log(q). Then

h (−\frac{i}{2}) − ∑ h(θ_{j}) + h (\frac{i}{2}) = −(2g − 2) g(0) + ∑_{P} ∑_{m=1}^{∞} \frac{d(P) g(m d(P))}{\sqrt{q^{md(P)}}}.

This function field analog of Theorem 34 may be proved by considering

\frac{1}{2πi} ∫_{2}^{2+2πit/ log(q)} (\frac{Z′(s)}{Z(s)} + (g − 1) log(q)) H(s) ds,

where h(t) = H(\frac{1}{2} + it). It may be regarded as an application of the Lefschetz-Grothendieck fixed point formula applied to a correspondence on X, and the three contributions h(−i/2), ∑ h(θ_{j}) and h(i/2) are the traces of the correspondence applied to H^{0}(X), H^{1}(X) and H^{2}(X). See Patterson [P], Chapter 5 and the first part of Connes [C].

The analogy between the Selberg trace formula and the explicit formulae has been the source of much speculation. Whether this analogy is misleading or not remains to be seen. One major difference between the explicit formulae and the Selberg trace formula is that in the trace formula, ∑ h(t_{i}) appears with a positive sign, while in (38), it appears with a negative sign. As Connes [C] points out, this difference may be very significant.

One tangible fruit of the analogy is Selberg’s discovery of a zeta function which bears a relationship to the trace formula similar to that of the Riemann zeta function to the explicit formulae. With notation as in the previous section, Selberg considered

Z(s) = ∏_{{N_{0}}} ∏_{k=0}^{∞}(1 − N_{0}^{−s−k}),

where log(N_{0}) runs through the lengths of prime geodesics.

Theorem 35. Z(s) has analytic continuation to all s, with zeros at the negative integers and at \frac{1}{2} ± it_{k}, where \frac{1}{4} + t_{k}^{2} are the eigenvalues of the Laplacian on Γ\ℌ.

Proof and discussion. To motivate the introduction of the Selberg zeta function, we would like to take

g(u) = e^{−|u|(s−1/2)}

in the trace formula. Unfortunately we cannot use this function, but if we could, the geometric side of the trace formula would be the logarithmic derivative of Z(s):

Lemma 36.

∑_{\{N\}} \frac{log(N_{0}) N^{−(s−1/2)}}{N^{1/2} − N^{−1/2}} = \frac{Z′(s)}{Z(s)}.

Proof. Writing N = N_{0}^{m}

∑_{\{N_{0}\}} ∑_{m=1}^{∞} \frac{log(N_{0}) N_{0}^{−m(s−1/2)}}{N_{0}^{m/2} − N_{0}^{−m/2}} = ∑_{\{N_{0}\}} ∑_{m=1}^{∞} \frac{log(N_{0}) N_{0}^{−ms}}{1 − N_{0}^{−m}} .

Substituting (1−N_{0}^{−m})^{−1} = ∑_{k=0}^{∞} N_{0}^{−ks} and interchanging the order of summation, this equals

∑_{\{N_{0}\}} \frac{log(N_{0}) N_{0}^{−(s+k)}}{1 − N_{0}^{−(s+k)}} = \frac{Z′(s)}{Z(s)}.

This completes the proof.

Although we cannot use g(u) = e^{−|u|(s−1/2)}, we may use

g(u) = \frac{e^{−|u|(s−1/2)}}{2s − 1} − \frac{e^{−|u|(σ−1/2)}}{2σ − 1} ,

where s and σ are distinct. This device of subtraction eliminates the discontinuity in g′ at u = 0. We will hold σ fixed and vary s. Initially, they are both assumed to have large real part. The Fourier transform h(t) = ∫_{−∞}^{∞} g(u) e^{itu} du is

h(t) = \frac{1}{(s − \frac{1}{2})^{2} + t^{2}} − \frac{1}{(σ − \frac{1}{2})^{2} + t^{2}} .

Next we prove, assuming that s and σ have real part ≥ \frac{1}{2}, that

(39)  ∫_{−∞}^{∞} h(t) t tanh(πt) dt = 2 ∑_{k=0}^{∞} [\frac{1}{s + k} − \frac{1}{σ + k} ] .

To prove this, use the partial fraction decomposition

\frac{t}{(s − \frac{1}{2})^{2} + t^{2}} = \frac{1}{2i} [\frac{1}{(s − \frac{1}{2}) − it} − \frac{1}{(s − \frac{1}{2}) + it}] .

The left side of (39) equals

\frac{1}{2i} ∫_{−∞}^{∞} [\frac{1}{(s − \frac{1}{2}) − it} − \frac{1}{(σ − \frac{1}{2}) − it}] tanh(πt) dt

− \frac{1}{2i} ∫_{−∞}^{∞} [\frac{1}{(s − \frac{1}{2}) + it} − \frac{1}{(σ − \frac{1}{2}) + it}] tanh(πt) dt.

Since tanh is odd, the two contributions are equal, and the first integral may be evaluated by moving the path of integration down into the lower half plane, where the only poles are at the poles −i(k + \frac{1}{2}) of tanh(πt), k = 0, 1, 2, · · ·. The residue of tanh(πt) at these points is π^{−1}, and we obtain (39).

We now use the Selberg Trace Formula. We note that the volume of the fundamental domain is 4π(g − 1). So for these functions g and h we get:

\frac{1}{2s − 1} \frac{Z′(s)}{Z(s)} − \frac{1}{2σ − 1} \frac{Z′(σ)}{Z(σ)} =

− (2g − 2) ∑_{k=0}^{∞} [\frac{1}{s + k} − \frac{1}{σ + k} ] + ∑ [\frac{1}{(s − \frac{1}{2})^{2} + t_{i}^{2}} − \frac{1}{(σ − \frac{1}{2})^{2} + t_{i}^{2}} ] .

Fixing σ and varying s, the right side has meromorphic continuation to all s. Multiplying by 2s−1 see that the poles of Z′/Z are simple and have integer residues. Hence

Z(s) = exp ∫_{∞}^{s} \frac{Z′(s)}{Z(s)} ds

has analytic continuation to all s.

8. Groups with one cusp. If the group Γ is of cofinite volume but has a cusp, the spectral theory is complicated by a continuous spectrum, coming from the Eisenstein series. However the cuspidal spectrum is discrete. For simplicity assume that ∞ is the only cusp of Γ, and the stabilizer of ∞ in Γ is

Γ_{∞} = \{\binom{1 n}{1} |n ∈ ℤ\} .

We will also denote G_{∞} = \{\binom{1 x}{1} | x ∈ ℝ\} . Let L_{0}^{2}(Γ\G) be the subspace of “cusp forms” f ∈ L^{2}(Γ\G) satisfying

(40)  ∫_{0}^{1} f(\binom{1 x}{1} g) dx = 0.

Also let L_{0}^{2}(Γ\ℌ) be the right K-invariant elements of L_{0}^{2}(Γ\G), which may be regarded as functions on ℌ.

In contrast with Theorem 7, the operators T_{ϕ} are no longer compact. However we still have:

Theorem 37. (Gelfand, Graev and Piatetski-Shapiro). If ϕ ∈ H, the restriction of T_{ϕ} to L_{0}^{2}(Γ\G) is a compact operator.

Proof. We refer to Bump [B] Section 3.2 for an exposition of Godement’s proof.

Theorem 38. (i) L_{0}^{2}(Γ\ℌ) has a basis consisting of eigenfunctions of Δ.

(ii) L_{0}^{2}(Γ\G) decomposes as a direct sum of irreducible invariant subspaces. Any K-finite element of one of these spaces is of rapid decay.

Proof (sketch). The proofs Theorem 9 and Theorem 11 are easily adapted. For the second assertion in (ii), it is sufficient to show that if V ⊂ L_{0}^{2}(Γ\G) is an irreducible subspace and f ∈ V (k) then f is of rapid decay. We may find ϕ ∈ H satisfying ϕ(κ_{θ}gκ_{σ}) = e^{ik(θ+σ)}ϕ(g), and such that

y → ϕ( \begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix} )

is a positive function of mass 1 concentrated near y = 1. Then T_{ϕ}f is near f, therefore nonzero, and it is in V (k), which is one-dimensional by Proposition 5, so it is proportional to f. It follows from the rapid decay of the kernel K_{0}(g, h), defined in Proposition 3.2.3 of Bump [B] that T_{ϕ}f is of rapid decay, thus so is f.

As in the case of compact quotient, the Laplacian acts by scalars on each irreducible one-dimensional subspace. The cuspidal spectrum behaves much as the entire spectrum in the compact case. On the other hand the orthogonal complement of L_{0}^{2}(Γ\G) contains a continuous spectrum. The eigenfunctions of the Laplacian relevant to the spectral theorem are themselves not square integrable.

To understand how this can be, and to get some intuition as to the nature of the continuous spectrum, consider the following example. The group ℝ acts on itself by translation, and the Laplacian −d^{2}/dx^{2} is an invariant differential operator. It has eigenfunctions f_{a}(x) = e^{2πiax} with eigenvalues a^{2}. Any L^{2} function has a Fourier expansion

ϕ(x) = ∫_{−∞}^{∞} ϕ̂(a) f_{a}(x) da,

but f_{a} is itself not L^{2}. If T ⊂ ℝ is measurable, the Fourier transforms of L^{2} functions supported on T form an invariant subspace. There are no minimal invariant subspaces, so L^{2}(ℝ) doesn’t decompose as a direct sum of irreducible representations.

In order to formulate the basic theorem about the Eisenstein series which belong to both the odd and the even principal series representations, let us slightly generalize our setup. We assume that the discrete Γ contains −I, which acts trivially on ℌ. Fix a unitary character χ of Γ such that χ(−I) ≡ (−1)^{ϵ} mod 2 and such that χ|Γ_{∞} is trivial, where ϵ = 0 or 1. Let k ≡ ϵ modulo 2 and let ± = (−1)^{ϵ}. Recall that f_{s,k} was defined by (15).

(41)  E(g, f_{s,k}, χ) = ∑_{Γ_{∞}\Γ} \overline{χ(γ)} f_{s,k}(γg).

Proposition 39. The series (41) converges absolutely if re(s) > 1.

Proof (sketch). Let σ be the real part of s. What we need to show is that if σ > 1 then

(42)  ∑_{Γ_{∞}\Γ} f_{σ,0}(γg) < ∞.

We define a measure μ_{σ} on the upper half plane by μ_{σ} = y^{σ−2} dx ∧ dy. Let B be a small neighborhood around z ∈ ℌ. The Jacobian of the map z → γ(z), where γ = \binom{a b}{c d} ∈ SL(2, ℝ) is |cz + d|^{−4}. Hence the μ_{σ}-volume of γ(B) is (approximately) |cz + d|^{−2σ} vol(B).

We may choose the representatives γ ∈ Γ_{∞}\Γ so that the images γ(B) all lie within the rectangle 0 ≤ x ≤ 1, 0 < y ≤ C for some constant C. (Actually this is not quite true. If one γ(B) happens to lie on the left or right edge of this region, cut it into two pieces along this edge and move one piece by ±1 back into the region.) This rectangle has finite volume, so ∑_{Γ_{∞}\Γ} |cz + d|^{−2σ} < ∞, which implies (42).

The analytic continuation of the Eisenstein series is closely connected with the analytic continuation of its constant term

(43)  E_{0}(g, f_{s,k}, χ) = ∫_{0}^{1} E(\binom{1 t}{1} g, f_{s,k}, χ) dt.

We recall that an intertwining operator M(s) : P_{s}^{±} → P_{1−s}^{±} defined by (16).

Proposition 40. Assume re(s) > 1. There exists an analytic function c(s) independent of k which is bounded on vertical strips (to the right of 1) such that

(44)  E_{0}(g, f_{s,k}, χ) = f_{s,k}(g) + c(s) M(s) f_{s,k}.

Proof. Substitute the definition of E(g, f_{s,k}, χ) into (43). The coset Γ_{∞} in Γ_{∞}\Γ contributes f_{s,k}. The remaining terms contribute

∫_{0}^{1} ∑_{\begin{matrix}γ∈Γ_{∞}\Γ/Γ_{∞} \\ γ∉Γ_{∞} \end{matrix}} ∑_{δ∈Γ_{∞}} \overline{χ(γδ)} f_{s,k}(γδ \binom{1 x}{1} g) dx

= ∑_{\begin{matrix}γ∈Γ_{∞}\Γ/Γ_{∞} \\ γ∉Γ_{∞} \end{matrix}} \overline{χ(γ)} ∫_{−∞}^{∞} f_{s,k}(γ \binom{1 x}{1} g) dx.

If γ = \binom{a b}{c d} ∉ Γ_{∞} then c ≠ 0 and

γ = \binom{c^{−1} a}{c} ( \begin{matrix}& −1 \\ 1 \end{matrix} ) \binom{1 d/c}{1} ,

so the variable change x → x − d/c shows that

∫_{−∞}^{∞} f_{s,k}(γ \binom{1 x}{1} g) dx = |c|^{−2s} M(s) f_{s,k}.

Thus (44) is satisfied where

(45)  c(s) = ∑_{\begin{matrix}γ∈Γ_{∞}\Γ/Γ_{∞} \\ γ∉Γ_{∞} \end{matrix}} χ(γ) |c|^{−2s}.

Since we are within the region of absolute convergence of the Eisenstein series this Dirichlet series is convergent if re(s) > 1, and (45) shows that it is an analytic function bounded in vertical strips.

We may rewrite this in the form

(46)  E_{0}(g, f_{s,k}, χ) = f_{s,k}(g) + A_{k}(s) f_{1−s,k},

where A_{k} is c times the constant on the right side in (18). It follows from Stirling’s formula and the boundedness of c(σ + it) that

A(σ + it) = O(|t|^{−1/2})

when σ > 1.

The analytic continuation of E(g, f_{s,k}, χ) and c(s) are closely connected with each other. Selberg proved:

Theorem 41. A_{k}(s) and E(g, f_{s,k}, χ) have meromorphic continuations and the same poles. They are analytic for re(s) > \frac{1}{2} except possibly for a finite number of poles on the real axis in (\frac{1}{2}, 1]. On the line re(s) = \frac{1}{2} they are holomorphic. The Eisenstein series satisfies the functional equation

E(g, f_{s,k}, χ) = A_{k}(s) E(g, f_{1−s,k}, χ).

We will only prove this in the special case where Γ = SL(2, ℤ) (Theorem 44). For the general case, there are various accounts in the literature. See Borel [Bo2], Cohen and Sarnak [CS], Colin de Verdier [CdV], Efrat [Ef], Elstrodt [El], Fadeev [F], Harish-Chandra [Ha], Hejhal [H2] (Chapter 6 and Appendix F, with discussion of the literature on p. 225), Kubota [K], Jacquet [J], Langlands [LES], Lax and Phillips [LP], Moeglin and Waldspurger [MW], Osborne and Warner [OW], Venkov [V1] and Wong [Wo]. The most general treatments are Langlands’ historically important work [LES] and the careful modern treatise of Moeglin and Waldspurger [MW].

This body of literature all owes something to Selberg, who (according to Hejhal [H2]) found three proofs. Generally speaking, one shows the analytic continuation of A_{k}(s) and E(g, f_{s,k}, χ) in parallel. The basic principle is that the resolvent of an operator has analytic continuation to the complement of its spectrum. Applied as for example in Kubota [K], Venkov [V1] or Appendix IV of Langlands [LES] this gives the analytic continuation of A_{k}(s) and E(g, f_{s,k}, χ) to the region re(s) > \frac{1}{2}, s ∉ (\frac{1}{2}, 1]. In this approach, obtaining the meromorphic continuation to the entire plane then presents some difficulties.

Selberg and Bernstein realized independently that these difficulties could be avoided by combining the resolvent principle with another one, the insight that if a system of inhomogeneous linear equations having analytic coefficients has a unique solution, then the solution has meromorphic continuation to wherever the coefficients in the linear equations do. Selberg’s published comments on this idea are in his introduction (written in 1988) to his Göttingen lectures [S3]. In addition to the analytic continuation of Eisenstein series Bernstein gave other applications, such as to the analytic continuation of the intertwining integrals M(s) and their p-adic analogs ([Be1], [Be2]), and his method has become an standard technique in the representation theory of p-adic groups.

Assuming the analytic continuation, we note the following two propositions:

Proposition 42. The functions E(g, s, f_{s,k}) for all k ≡ ϵ modulo 2 comprise a (𝔤, K)-module isomorphic to P_{s}^{±}.

Proof. First suppose that s > 1. Then E(g, s, f_{s,k}) is constructed by averaging left translates of f_{s,k}. The representation of the group and its Lie algebra on functions is via right translation. The two operations commute with each other, so the statement is clear in this case. The identities this imply are preserved under analytic continuation.

Proposition 43. If s is not a pole of E(g, f_{s,k}, χ) then E − E_{0} is of rapid decay.

Proof (sketch). Let ϕ ∈ H. The function

Ẽ(g) = E(g, f_{s,k}) − E_{0}(g, f_{s,k})

is not automorphic with respect to Γ but modifying the proof of Proposition 3.2.3 of Bump [B], where the kernel K_{0} is defined, we have

T_{ϕ}Ẽ(g) = ∫_{Γ_{∞}\G} K_{0}(g, h) Ẽ(h, f_{s,k}, χ) dh.

By the same reasoning as in Theorem 38 (ii) we may choose ϕ so that T_{ϕ}f̃_{s,k} is a nonzero multiple of f_{s,k}. Now Ẽ lies in V (k), where (π, V) is a space of functions isomorphic to π_{s}^{±}, so T_{ϕ}Ẽ is the same constant multiple of Ẽ. The rapid decay now follows from the corresponding property of the kernel K_{0}(g, h).

The asymptotic behavior of E(g, s, f_{s}) near the cusp is therefore determined by its constant term. The constant term, we see, consists of two parts, one of the order y^{s} and the other of order y^{1−s}. The smallest growth evidently occurs on the line re(s) = \frac{1}{2}, where it is of order \sqrt{y}. On this line, the Eisenstein series is almost but not quite L^{2}.

There is a substantial difference between the arithmetic case (such as Γ = SL(2, ℤ), χ = 1 and the nonarithmetic case). In the arithmetic case it is possible to normalize the Eisenstein series. This means there is a multiple of the Eisenstein series which has a simple functional equation, and only a finite number of poles.

For example, suppose that if Γ = SL(2, ℤ), χ = 1 and k = 0. Then the Eisenstein series

E(z, s) = ∑_{γ∈Γ_{∞}\SL(2,ℤ)} f_{s,0}(γz)

and the normalized Eisenstein series is

(47)  E^{∗}(z, s) = π^{s} Γ(s) ζ(2s) E(z, s)

Theorem 44. E^{∗}(z, s) has analytic continuation to all s except s = 0, 1, where it has simple poles; the residue at s = 1 is the constant function \frac{1}{2}. It satisfies the functional equation E^{∗}(z, s) = E^{∗}(z, 1 − s).

Proof (sketch). If z = x + iy ∈ H and t > 0, let

Θ(t) = ∑_{(m,n)∈ℤ^{2}} e^{−π|mz+n|^{2}t/y}.

It follows from Euler’s integral for Γ that

E^{∗}(z, s) = \frac{1}{2} ∫_{0}^{∞}(Θ(t) − 1) t^{s} \frac{dt}{t}.

The Poisson summation formula implies that Θ(t) = t^{−1} Θ(t^{−1}). From this one gets

E^{∗}(z, s) = \frac{1}{2} ∫_{0}^{∞}Θ(t) (t^{s} + t^{1−s}) \frac{dt}{t} − \frac{1}{2s} − \frac{1}{2 − 2s}.

This expression gives the analytic continuation and functional equation.

Theorem 44 is a special case of Theorem 41. It is typical of the arithmetic case, but not of the general case.

9. Spectral expansion. The spectral expansion was obtained by Roelcke [Ro], modulo the analytic continuation of the Eisenstein series. Numerous accounts are in the literature, of which Godement [Go1], [Go2] is a good and influential one. In this section we discuss the spectral expansion for L^{2}(Γ\ℌ) where as in Section 6 the group has a single cusp. We will assume the analytic continuation of the Eisenstein series (Theorem 41).

Let ϕ be a K-finite element of C_{c}^{∞}(G_{∞}\G). The incomplete theta series

(48)  θ_{ϕ}(g) = ∑_{Γ_{∞}\Γ} ϕ(γg)

is something like an Eisenstein series but it is not a Δ-eigenfunction. (Godement’s term “incomplete theta series” seems something of a misnomer.) If f ∈ C^{∞}(Γ\G) let

f_{0}(g) = ∫_{Γ_{∞}\G_{∞}} f(ug) du

be its constant term, which is in C^{∞}(G_{∞}\G).

Proposition 45. The incomplete theta series and constant term maps are adjoints; that is

∫_{Γ\G} θ_{ϕ}(g) \overline{f(g)} dg = ∫_{G_{∞}\G} ϕ(g) \overline{f_{0}(g)} dg.

Proof. The left side is

∫_{Γ\G} ∑_{γ∈Γ_{∞}\Γ} ϕ(γg) \overline{f(g)} dg = ∫_{Γ_{∞}\G} ϕ(g) \overline{f(g)} dg =

∫_{G_{∞}\G} ∫_{Γ_{∞}\G_{∞}} ϕ(ug) \overline{f(ug)} du dg = ∫_{G_{∞}\G} ϕ(g) ∫_{Γ_{∞}\G_{∞}} \overline{f(ug)} du dg

which equals the right side.

The incomplete theta series are compactly supported modulo Γ, hence are square-integrable. For this reason they are easier to work with than the Eisenstein series themselves.

Proposition 46. L_{0}^{2}(Γ\G) is the orthogonal complement in L^{2}(Γ\G) of the closed subspace spanned by the incomplete theta series.

Proof. Immediate from Proposition 45, since the cuspidal spectrum is characterized by vanishing of its constant terms.

For the remainder of this section, we make the simplifying assumption that the character χ to be trivial and we only consider functions on G which are right invariant by K, that is, which may be regarded as functions on ℌ. We will therefore denote E(g, f_{s,0}, 1) as just E(g, s), or as E(z, s) where z = g(i). We will also denote f_{s} = f_{s,0}. We will write the constant term in the form

(49)  E_{0}(g, s) = f_{s}(g) + A(s) f_{1−s}(g).

(See Proposition 40.)

Using Theorem 38, let ξ_{i} be a basis of L_{0}^{2}(Γ\G). Also, note that by Theorem 41 there may be a finite number of poles of the Eisenstein series E(z, s) at locations s_{j} ∈ (\frac{1}{2}, 1] (j = 1, ⋯, N). These are also poles of A(s). Let α_{j} be the residue of A(s) at s = s_{j}, and let

(50)  η_{j} = \frac{1}{\sqrt{|α_{j}|}} res_{s=s_{j}}E(g, s).

Proposition 47. If ϕ is a cusp form then

(51)  ∫_{Γ\G} ϕ(g) E(g, s) dg = 0

Proof. Assume first re(s) > 1. Then (51) is

∫_{Γ\G} ∑_{γ∈Γ_{∞}\Γ} ϕ(g) f_{s}(γg) dg = ∫_{Γ_{∞}\G} ϕ(g) f_{s}(g) dg =

∫_{G_{∞}\G} ∫_{Γ_{∞}\G_{∞}} ϕ(ug) f_{s}(ug) du dg = ∫_{G_{∞}\G} f_{s}(g) ∫_{Γ_{∞}\G_{∞}} ϕ(ug) du dg = 0

since ϕ is a cusp form. The general case follows by analytic continuation.

Proposition 48. (i) The constant term of η_{j} is \sqrt{|α_{j}|} f_{1−s_{j}}. (ii) The functions η_{j} are real valued, square integrable and orthogonal to the cusp forms.

Proof. Part (i) is immediate from (49) since the first term on the right has no pole but the second one does. The η_{j} are real valued since the s_{j} are real. It follows from Proposition 43 that η_{i} is asymptotic to its constant term near the cusp so using (i) and the fact that s_{j} > \frac{1}{2}, we find that η_{i} is square integrable. Taking the residue in Proposition 47 shows that the η_{j} are orthogonal to the cusp forms.

Proposition 49. Let ϕ ∈ C_{c}^{∞}(G_{∞}\G/K). Define

(52)  ϕ̂(s) = ∫_{G_{∞}\G} ϕ(g) f_{1−s}(g) dg.

This is an entire function of s and t → ϕ̂(σ + it) is of Schwartz class for all real σ. For real σ we have

(53)  ϕ(g) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞} ϕ̂(s) f_{s}(g) ds.

(The contour integral is over the vertical line with real part σ.)

Proof. We have

ϕ̂(s) = ∫_{0}^{∞} ϕ( \begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix} ) y^{−s} \frac{dy}{y} .

Since ϕ is compactly supported and smooth, t → ϕ̂(σ + it) is the Fourier transform of a Schwartz function, hence is Schwartz itself. By the Mellin inversion formula

(54)  ϕ( \begin{matrix}y^{1/2} \\ & y^{−1/2} \end{matrix} ) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞} ϕ̂(s) y^{s} ds.

This verifies (53) on the diagonal, and since both sides are left invariant by G_{∞} and right invariant by K, the general case follows.

Proposition 50. We have

⟨θ_{ϕ}, η_{j}⟩ = \sqrt{|α_{j}|} ϕ̂(s_{j}).

Proof. This follows from Proposition 45, Proposition 48 and from the definition of ϕ̂.

Theorem 51. Let ψ ∈ L^{2}(Γ\ℌ). Assume also that ψ(g) f_{1/2}(g) is integrable over the Siegel set S. Then

(55) ψ(g) =

∑_{j=0}^{N} ⟨ψ, η_{j}⟩ η_{j}(g) + ∑_{i=0}^{∞} ⟨ψ, ξ_{i}⟩ ξ_{i}(g) + \frac{1}{4π} ∫_{−∞}^{∞} ⟨ψ, E( - , \frac{1}{2} + it)⟩ E(g, \frac{1}{2} + it) dt.

The integrability assumption on that ψ(g) f_{1/2}(g) guarantees that the inner products occurring in the expansion are convergent. Correctly interpreted, the expansion is valid for all f ∈ L^{2}(Γ\ℌ). This is like the Fourier inversion formula for ℝ, which is valid for all L^{2}(ℝ) though the usual definition of the Fourier transform as an integral is only strictly correct on L^{2}(ℝ) ∩ L^{1}(ℝ).

Proof (sketch). By Proposition 48 and Proposition 47 this is true for cusp forms. By Proposition 46 it is therefore sufficient to prove it for incomplete theta series; that is,

(56)  θ_{ϕ}(g) = ∑_{j=1}^{N} ⟨θ_{ϕ}, η_{j}⟩ η_{j}(g) + \frac{1}{4π} ∫_{0}^{∞} ⟨θ_{ϕ}, E( - , \frac{1}{2} + it)⟩ E(g, \frac{1}{2} + it) dt

when ϕ ∈ C_{c}^{∞}(G_{∞}\ℌ).

Indeed, both sides are left invariant by G_{∞}, right invariant by K, and agree on the diagonal by (54), so this follows by the Iwasawa decomposition. We take σ > 1 and sum over γg with γ ∈ Γ_{∞}\Γ to obtain

(57)  θ_{ϕ}(g) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞} ϕ̂(s) E(g, s) ds.

Now we move the path of integration to the left. The pole at s = s_{j} contributes a residue which by (50) and Proposition 50 equals ⟨θ_{ϕ}, η_{j}⟩ η_{j}. Thus

(58)  θ_{ϕ}(g) = ∑_{j=1}^{N} ⟨θ_{ϕ}, η_{j}⟩ η_{j}(g) + \frac{1}{2π} ∫_{−∞}^{∞} ϕ̂(\frac{1}{2} + it) E(g, \frac{1}{2} + it) dt

Now let us look at the integral in (56). Using Proposition 45, (49) and (52) the integrand is

\frac{1}{4π} ∫_{Γ\G} θ_{ϕ}(h) E(h, \frac{1}{2} − it) dh · E(g, \frac{1}{2} + it) =

\frac{1}{4π} (ϕ̂(\frac{1}{2} + it) E(g, \frac{1}{2} + it) + ϕ̂(\frac{1}{2} − it) A(\frac{1}{2} − it) E(g, \frac{1}{2} + it)) =

\frac{1}{4π} (ϕ̂(\frac{1}{2} + it) E(g, \frac{1}{2} + it) + ϕ̂(\frac{1}{2} − it) E(g, \frac{1}{2} − it))

Integrating with respect to t, combining the duplicate contributions and applying (58) now proves (56) .

10. Liftings and the trace formula. One of the most interesting applications of the trace formula is to liftings of automorphic forms. The method involves comparison of two different trace formulae, on different groups, leading to the conclusion that automorphic forms on one group can be lifted to automorphic forms on the other.

Jacquet and Langlands [JL] gave an early application in a lifting from automorphic forms on a division algebra to GL(2). A variation of this theme noted by Gelbart and Jacquet [GJ] is probably the simplest example of this type, since in this case neither trace formula involves a continuous spectrum. It will be convenient to switch to an adelic point of view, but we think the reader who has read Section 5 will not have trouble making the transition.

If D_{1} and D_{2} are central division algebras over a field F, then D_{1} ⊗ D_{2} ≅ Mat_{k}(D_{3}) for some D_{3} and k, and D_{1}, D_{2} → D_{3} is an associative multiplication on the set B(F) of isomorphism classes of central division algebras. Thus B(F) becomes a group, called the Brauer group.

If D is a central division algebra over F then the dimension of D is a square d^{2}, and if E/F is any field extension of degree d which can be embedded in D then E ⊗ D ≅ Mat_{d}(E). Thus a division ring is a Galois twisted form of a matrix ring. The composite map

D → E ⊗ D ≅ Mat_{d}(E) → E,

the last map being either the trace or determinant, takes values in F, and gives us the reduced trace or reduced norm.

The Brauer group of a local or global field F admits a simple and beautiful description related to the reciprocity laws of class field theory. See Section 1 of Chapter 6 (“Local Class Field Theory” by J.-P. Serre) and Section 9 of Chapter 7 (“Global Class Field Theory” by J. Tate) in Cassels and Fröhlich [CF].

Let F be a global field, 𝔸 its adele ring, and D a central division algebra of degree p^{2} over F, where p is a prime. Let Z ≅ F^{×} be the center of D^{×}. Let S be the finite set of places where D_{v} is a division ring. If v ∉ S we identify D_{v} = Mat_{p}(F_{v}).

Let H be the set of functions on D_{𝔸}^{×} = ∏_{v} D_{v}^{×} which are finite linear combinations of functions of the form ∏_{v} ϕ_{v} where for each v, ϕ_{v} : D_{v}^{×} → ℂ is smooth and compactly supported modulo Z_{v}, satisfies ϕ_{v}(z_{v}g_{v}) = ϕ_{v}(g_{v}) when z_{v} ∈ Z_{v}, and agrees with the characteristic function of Z_{v}Mat_{p}(𝔬_{v}) for almost all places v of F. The ring H contains the classical Hecke operators as well as the integral operators introduced in Section 7 above.

Z_{𝔸}D_{F}^{×}\D_{𝔸}^{×} is compact. As with SL(2, ℝ), L^{2}(Z_{𝔸}D_{F}^{×}\D_{𝔸}^{×}) admits integral operators T_{ϕ} for ϕ ∈ H:

(T_{ϕ}f)(g) = ∫_{Z_{𝔸}\D_{𝔸}^{×}} ϕ(h) f(gh) dh.

Let {γ} be a set of representatives for the conjugacy classes of D_{F}^{×}. We denote by C_{γ} the centralizer of γ in D_{F}^{×}. It is an algebraic group, so C_{γ}(𝔸) ⊂ D_{𝔸}^{×} will denote its points in 𝔸.

Theorem 52. (Selberg trace formula).

(59)  tr T_{ϕ} = ∑_{{γ}} vol(C_{γ}\C_{γ}(𝔸)) ∫_{C_{γ}(𝔸)Z_{𝔸}\D_{𝔸}^{×}} ϕ(g^{−1}γg) dg.

Proof. The proof of Proposition 6 goes through without change, so

(T_{ϕ}f)(g) = ∫_{Z_{𝔸}D_{F}^{×}\D_{𝔸}^{×}} K_{ϕ}(g, h) f(h) dh,

K_{ϕ}(g, h) = ∑_{γ∈D_{F}^{×}/Z_{F}^{×}} ϕ(g^{−1}γh).

As with SL(2, ℝ), the operator T_{ϕ} is thus Hilbert-Schmidt, and with more work may be shown to be trace class. As in Theorem 11,

tr T_{ϕ} = ∫_{Z_{𝔸}D_{F}^{×}\D_{𝔸}^{×}} K_{ϕ}(g, g) dg

Now (59) follows as in Theorem 28.

The conjugacy classes of D_{F}^{×} are easily described.

Proposition 53. If α ∈ D_{F}^{×} − Z_{F}, then F (α) is a field extension of F of degree p. Elements α and β are conjugate in D_{F}^{×} if and only if there is a field isomorphism F (α) → F (β) such that α ↦ β. If F (α) is a field extension of degree p, then F (α) may be embedded in D_{F} if and only if [F_{v}(α) : F_{v}] = p for all v ∈ S.

Proof. The conjugacy of α and β follows from the Skolem-Noether Theorem (Herstein [He], p. 99). The last statement follows from (i) ↔ (ii) in Weil [W2], Proposition VIII.5 on p. 253.

The trace formula can be used to prove functorial liftings in many cases. Let E be another division algebra of degree p^{2}, and assume that the set of

places where E_{v} is a division ring agrees with the set S of places where D_{v} is. If p = 2 this implies that D and E are isomorphic, but not in general. (This follows from the computation of the Brauer group in the global class field theory. See Chapters 6 and 7 in [CF].) Thus we want p > 2.

We will show that two spaces of automorphic forms on D and on E are isomorphic.

Suppose that π = ⊗_{v}π_{v} is an irreducible constituent of L^{2}(Z_{𝔸}D_{F}^{×}\D_{𝔸}^{×}). Since Z_{v}\D_{v}^{×} is compact for v ∈ S, π_{v} is finite-dimensional at these places. We assume that π_{v} is trivial when v ∈ S.

If v ∉ S, then D_{v} ≅ E_{v} ≅ Mat_{p}(F_{v}). We may therefore identify π_{v} with an irreducible representation π′_{v} of E_{v} when v ∉ S, and if v ∈ S we take π′_{v} = 1. Let π′ = ⊗π′_{v}. It is an irreducible representation of E_{𝔸}^{×}.

Theorem 54. π′ occurs in L^{2}(Z_{𝔸}E_{F}^{×}\E_{𝔸}^{×}).

The correspondence π → π′ is a functorial lift of automorphic forms in the sense of Langlands (Langlands [LPAF], Borel [Bo1]).

Proof (sketch). If v ∈ S, then Z_{v}\D_{v}^{×} is compact, so the constant function ϕ_{v}^{∘}(g_{v}) = 1 is in C_{c}^{∞}(F_{v}). Let H_{S} be the subalgebra of H spanned by functions ∏ ϕ_{v} such that ϕ_{v} = ϕ_{v}^{∘} for v ∈ S. It is isomorphic to the corresponding Hecke ring on E. Let ϕ → ϕ′ denote this isomorphism.

By Proposition 53, noncentral conjugacy classes in D_{F}^{×} and E_{F}^{×} are both in bijection with the set of Galois equivalence classes of elements α of field extensions [F (α) : F] = p such that [F_{v}(α) : F_{v}] = p for all p ∈ S. This intrinsic characterization shows that we may identify the conjugacy classes of D_{F} and E_{F}, and compare trace formulae to get

(60)  tr T_{ϕ} = tr T′_{ϕ}.

This is almost but not quite as easy as we’ve made it sound, because one must show that the volumes on the right side of (59) are the same for the two trace formulae.

It follows from (60) that the representations of H_{S} on the spaces L^{2}(Z_{𝔸}D_{F}^{×} ∏_{v∈S} D_{v}^{×}\D_{𝔸}^{×}) and L^{2}(Z_{𝔸}E_{F}^{×} ∏_{v∈S} E_{v}^{×}\E_{𝔸}^{×}) are isomorphic, and the theorem follows.

Underlying the final step is the fact that two representations of rings are characterized by their traces. For example if R is an algebra over a field of characteristic zero and if M_{1}, M_{2} are finite-dimensional semisimple R-modules, and if for every α ∈ R the induced endomorphisms of M_{1} and M_{2} have the same trace, then the modules are isomorphic (Lang [L2], Corollary 3.8 on p. 650). This statement is not directly applicable here but it gives the flavor.

For the remainder we take p = 2, and review the Jacquet-Langlands correspondence. Let D be as before. The Jacquet-Langlands correspondence is a lifting of automorphic representations from D^{×} to GL(2, F).

There is a local correspondence for v ∈ S. D_{v}^{×} is compact modulo its center, so its irreducible representations are finite-dimensional. These lift to irreducible representations of GL(2, F_{v}) having the same central character. The lifting was constructed by Jacquet and Langlands by use of the theta correspondence. Indeed, Z_{v}\D_{v}^{×} is a quotient of the orthogonal group GO(4) while GL(2) is the same as GSp(2), and theta correspondence GO(4) ↔ GSp(2) gives the Jacquet-Langlands correspondence. Its image is the square integrable representations (that is, the supercuspidals and the Steinberg representation).

Jacquet and Langlands constructed a global correspondence from automorphic forms on D^{×} to automorphic forms on GL(2) first using the converse theorem in Section 14 of their book. To prove functional equations of L-functions on D^{×}, they use the Godement-Jacquet construction, because the Hecke integral is not available in this context.

Finally, they reconsidered the lifting from the point of view of the trace formula. This allowed them to characterize the image of the lift. They sketched a proof (and later Gelbart and Jacquet completed) of:

Theorem 55. An automorphic representation π of GL(2, 𝔸) is the lift of an automorphic representation of D_{𝔸}^{×} if and only if π_{v} is square integrable for every v ∈ S.

The remarkable fact is that their proof of this fact uses so many different techniques which have proved important in the subsequent 30 years: the Hecke and Godement-Jacquet integral constructions of L-functions, the Weil representation and the trace formula.

The trace formula on GL(2) is harder than on the division ring because of the presence of the continuous spectrum. We’ve avoided this problem by proving Theorem 54 instead of Theorem 55.

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Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA e-mail: bump@math.stanford.edu.