Introduction to Spectral Theory on Hyperbolic …fuchs-braun.com/media/7c48db97e18c2326ffff802affffffef.pdfIntroduction to Spectral Theory on Hyperbolic Surfaces David Borthwick Contents

Introduction to Spectral Theory on Hyperbolic Surfaces

David Borthwick

Contents

1. Hyperbolic geometry 12. Fuchsian groups and hyperbolic surfaces 43. Spectrum and resolvent 114. Spectral theory: finite-area case 165. Spectral theory: infinite-area case 206. Selberg trace formula 237. Arithmetic surfaces 33References 44

1. Hyperbolic geometry

In complex analysis, we learn that the upper half-plane H = Im z > 0 has alarge group of conformal automorphisms, consisting of Mobius transformations ofthe form

(1.1) γ : z 7→ az + b

cz + d,

where a, b, c, d ∈ R and ad − bc > 0. The Schwarz Lemma implies that all au-tomorphisms of H are of this type. Since the transformation is unchanged by asimultaneous rescaling of a, b, c, d, the conformal automorphism group of H is iden-tified with the matrix group

PSL(2,R) := SL(2,R)/±I.Under the PSL(2,R) action, H has an invariant metric,

ds2H =

dx2 + dy2

y2,

often called the Poincare metric. To see the invariance, it is convenient to switchto the complex notation, where

ds2H =

|dz|2

(Im z)2.

2000 Mathematics Subject Classification. Primary 58J50, 35P25; Secondary 47A40.Supported in part by NSF grant DMS-0901937.

1

2 D. BORTHWICK

The pullback of the metric tensor under the map (1.1) is

γ∗(ds2H) =

|γ′(z) dz|2

(Im γ(z))2,

so the invariance of ds2H follows from the easily checked identities,

γ′(z) =1

(cz + d)2, Im γ(z) =

Im z

|cz + d|2.

1.1. Geometry of the hyperbolic plane. Because PSL(2,R) acts transi-tively on H, by isometries, it is immediately clear that the Gaussian curvatureK must be constant for ds2

H. Using the formula provided by Gauss’s TheoremaEgregium, we can easily check that this value is K = −1. This is the definingfeature of a 2-dimensional hyperbolic metric, and for this reason the Poincare upperhalf space H is also called the hyberbolic plane.

In this section we will introduce just a few of the most basic geometric conceptsin H. First, the Riemanian measure associated to ds2

H is

dA(z) =dx dy

y2.

Obviously, this metric inherits PSL(2,R)-invariance from the metric. The metricalso determines angles, and conveniently the angles measured with respect to ds2

Hare the same as Euclidean angles, since the metrics are conformally related (ds2

H =y−2dsR2).

A geodesic is a curve which is locally length minimizing within the class ofpiecewise smooth curves. It is not hard to see, by direct computation, that verticallines in H have this property. One simply notes that for a smooth curve η(t) =(x(t), y(t)), with η(0) = ia and η(1) = ib, with b > a, we have

`(η|[0,1]) =∫ 1

0

√x2 + y2

ydt.

Clearly the minimum is achieved by setting x ≡ 0 and restricting to y > 0, whichshows that the y-axis is a geodesic and gives the hyperbolic distance

(1.2) dH(ia, ib) = logb

a.

(Distance is defined as the infimum of the lengths of connecting paths.)We can then find the other geodesics just by moving this one around using the

group action. It is useful to think of H as a hemisphere within the Riemann sphereC ∪ ∞. Vertical lines then correspond to circles on the Riemann sphere thatintersect the boundary ∂H := R ∪ ∞ at right angles and pass through ∞. SinceMobius transformations preserve circles on the Riemann sphere and also angles,one can then easily deduce that the geodesics of H are precisely the arcs of circlesintersecting ∂H = R ∪ ∞ orthogonally, as illustrated in Figure 1.

By applying PSL(2,R) transformations to (1.2), it is not hard to work out thegeneral formula for the distance function in H explicitly,

dH(z, w) = log|z − w|+ |z − w||z − w| − |z − w|

.

Now consider a geodesic triangle ABC with vertex angles α, β, and γ. asin Figure 2. Vertices are allowed to lie on ∂H, in which case the angle is zero.

INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 3

Figure 1. Geodesics in H

αβ

γ αβ

γ

Figure 2. Geodesic triangles

The Gauss-Bonnet theorem (for any two-dimensional metric) says that the totalcurvature of a triangle T is equal to the angle deficit,∫

T

K dA = α+ β + γ − π.

Thus the fact that the metric is hyperbolic is equivalent to the area formula:

Area(T ) = π − α− β − γ,

for any geodesic triangle.

1.2. Classification of isometries. Elements of PSL(2,R) are classified ac-cording to their fixed points. For γ ∈ PSL(2,R), the fixed point equation z = γz isquadratic:

cz2 + (d− a)z − b = 0,

so there are exactly 2 solutions in C ∪∞. We break down the cases as follows:(1) elliptic: one fixed point in H, the other is the complex conjugate. An

elliptic transformation acts as a rotation centered at the fixed point, asshown in Figure 3. The conjugacy class is determined by the rotationangle.

(2) parabolic: a single degenerate fixed point, which must lie in ∂H. Theaction is illustrated in Figure 4. Any parabolic transformation is conjugateto the map z 7→ z + 1 (for which ∞ is the degenerate fixed point).

(3) hyperbolic: two distinct fixed points in ∂H. The transformation mapspoints away from one fixed point and toward the other, as shown in Fig-ure 5. A hyperbolic transformation is conjugate to the dilation z 7→ e`zfor some ` ∈ R.

4 D. BORTHWICK

Figure 3. An elliptic transformation rotates hyperbolic circlesaround a fixed center.

Figure 4. A parabolic transformation fixes a point on ∂H.

Figure 5. A hyperbolic transformation translates between twofixed points on ∂H. The axis (unique fixed geodesic) is shown ingray.

Notes. For background material on two-dimensional Riemannian geometry,see do Carmo [10] or Pressley [34]. Ratcliffe [35] gives a thorough intoduction ofhyperbolic geometry.

2. Fuchsian groups and hyperbolic surfaces

A Fuchsian group is a subgroup Γ ⊂ PSL(2,R) that is discrete with respectto the matrix topology (which is equivalent to Euclidean R4). It follows from thediscreteness that Γ must act properly discontinuously on H, meaning that eachorbit Γz is locally finite. The converse is also fairly easy to argue, as follows. If Γacts properly discontinuously on H, then the triangle inequality implies that onlyfinitely many points in any compact neighborhood could be fixed by Γ−I. But ifthere is a sequence Tk → I in Γ, and w ∈ H is not fixed by Γ−I, then the sequenceTkw would contradict the proper discontinuity of the action of Γ. Thus Fuchsiangroups are precisely the subgroups of PSL(2,R) acting properly discontinuously onH.


Figure 6. The orbit of a point under a Fuchsian group Γ.

F

Figure 7. A fundamental domain and its corresponding tessellation.

The quotient Γ\H is an orbifold whose points correspond to the orbits of Γ inH. Figure 6 shows an example of an orbit. Note the accumulation on ∂H; this doesnot contradict the properly discontinuity of the action because the accumulationoccurs only in the Riemann sphere topology, not in the topology of H. The quotientis a smooth surface if and only if Γ has no elliptic elements. The term “surface” isfrequently used for Γ\H, even in the non-smooth case; this usage presumably comesfrom the interpretation of Γ\H as an algebraic variety. Since Γ acts by hyperbolicisometries, the quotient inherits a hyperbolic metric from H.

A fundamental domain for Γ is a closed region F such that the translates of Funder Γ tesselate H. An example is shown in Figure 7. The fact that Γ is discreteimplies that this tesselation will be locally finite (a compact set meets only finitelymany translates of F).

2.1. Automorphic forms and functions. A function f on the quotient Γ\His equivalent to a function on H satisfying

f(γz) = f(z) for γ ∈ Γ.

The latter is called an automorphic function for Γ. In number theory applicationsone generally considers vector valued-functions H → V along with a unitary rep-resentation ρ : Γ → GL(V ). The condition for an automorphic function is thenf(γz) = ρ(γ)f(z). For simplicity, we will consider only the trivial representation inthese notes.

Another possible generalization is an automorphic form of weight k, for whichthe transformation rule is

f(γz) = (cz + d)kf(z).

Here c and d are matrix elements of γ as in (1.1). Since γ′(z) = (cz + d)−2, theautomorphic of even weight are sections of Lk/2, where L is the holomorphic tangentbundle over Γ\H. In some contexts, the term automorphic form carries with it arestriction to meromorphic or holomorphic functions. Indeed, this is the classicalusage of the term.

6 D. BORTHWICK

2.2. Uniformization. The Uniformization Theorem for hyperbolic surfacesessentially has two parts. The first says that any metric on a surface is conformallyrelated to a hyperbolic metric.

Theorem 2.1 (Koebe, Poincare). For any smooth complete Riemannian metricon a surface, there is a conformally related metric of constant curvature.

Given a general Riemannian surface (M, g), the equation for the curvature ofg := e2ug, with u ∈ C∞(M) is

Kg = e−2u(Kg + ∆u),

where ∆ is the (positive) Laplacian operator,

∆ := −div grad .

The proof of Theorem 2.1 amounts to finding a solution u for which Kg is constant.There may be restrictions on the value of this constant coming from the Gauss-Bonnet theorem, depending on the topology, but provided those conditions are metsolutions always exist.

The second part of uniformization is the characterization of surfaces of constantcurvature by their universal covering spaces. We can use a global rescaling torestrict our attention to K = 1, 0, or −1, and then there is only one possibility foreach case.

Theorem 2.2 (Hopf). Up to isometry and global rescaling, the only smooth,complete, simply connected surfaces of constant curvature are the sphere S2, theEuclidean plane R2, and the hyperbolic plane H2.

The local part of the proof is fairly straightforward: in geodesic polar coordi-nates any metric takes the form dr2 + φ2dθ2, with φ ∼ r as r → 0. The curvatureis given by K = −∂2

rφ/φ in these coordinates. Setting K = 1, 0,−1 results in a 2ndorder ODE with unique solutions, φ = sin r, r or sinh r, respectively. The globalresult requires the fact that a local isometry of complete Riemannian manifoldsmust be a covering map, which is closely related to the Hopf-Rinow theorem.

As a corollary of these theorems, we find that any smooth surface with χ < 0is conformally related to a hyperbolic quotient Γ\H. It is thus fair to say thathyperbolic surfaces provide the uniformizing models for ‘most’ surfaces.

Riemannian orbifolds are considered good if they arise as quotients of a smoothRiemannian manifold under a properly discontinuous group action, and bad if not.There is a uniformization theorem for good 2-dimensional orbifolds without bound-ary which says that they are isomorphic as orbifolds to the quotient of S2, R2, orH2 by some discrete group of isometries (see Thurston [47]).

As we noted above, the term hyperbolic refers specifically to curvature. How-ever, because the hyperbolic isometries of H are precisely the conformal auto-morphisms, any quotient Γ\H inherits a natural complex structure from H. Inother words, a hyperbolic surface is also a Riemann surface, which means a one-dimensional complex manifold. The terms hyperbolic surface and Riemann surfaceare sometimes used interchangeably, especially for smooth compact surfaces, wherethe only non-hyperbolic Riemann surfaces are the sphere and tori.

We have described the geometric Uniformization Theorem, but there is alsoa complex version in the context of Riemann surfaces This says that any simplyconnected Riemann surface is holomorphically equivalent to the either the Riemannsphere C ∪ ∞, the complex plane C, or the upper half-plane H.


Figure 8. Orbits of parabolic and hyperbolic cyclic groups

2.3. Limit set. The limit set Λ(Γ) of Γ ⊂ PSL(2,R) is the collection of limitpoints of orbits of Γ, in the Riemann sphere topology. Since Γ acts properly discon-tinuously, orbit points can only accumulate in ∂H. One can see this accumulationin Figure 6. As long as w ∈ H is not an elliptic fixed point, it suffices to take the setof limit points of the single orbit Γw. It follows that Λ(Γ) is closed and Γ-invariant.

Limit sets provide the standard classification of Fuchsian groups, according tothe following:

Theorem 2.3 (Poincare, Fricke-Klein). Any Fuchsian group is of one of thefollowing types:

(1) Elementary: Λ(Γ) contains 0, 1, or 2 points;(2) First Kind: Λ(Γ) = ∂H;(3) Second Kind: Λ(Γ) is a perfect, nowhere-dense subset of ∂H.

The orbit of a parabolic cyclic group accumulates (very slowly!) at the fixedpoint of the generator of the group, as shown on the left in Figure 8. For theparticular case of Γ∞ = 〈z 7→ z+ 1〉, we have Λ(Γ) = ∞. Similarly, the orbit of ahyperbolic cyclic group accumulates at both of the fixed points. These groups arethe only possibilities where Λ(Γ) has 1 or 2 points. All other elementary groups arefinite groups with only elliptic elements, for which Λ(Γ) is empty. See Katok [23]for details.

The group Γ is said to be cofinite if Area(Γ\H) < ∞, and in fact these areprecisely the Fuchsian groups of the first kind. All arithmetic surfaces are in thisclass.

2.4. Geometric finiteness. A Fuchsian group is geometrically finite if Γ ad-mits a fundamental domain that is a finite-sided convex polygon. For such a group,the Dirichlet domain,

Dw := z ∈ H : d(z, w) ≤ d(z, γw) for all γ ∈ Γ,where w is not an elliptic fixed point, will always furnish a fundamental domainwith finitely many sides. The fundamental domain shown in Figure 7 is actually aDirichlet domain.

Theorem 2.4. For a Fuchsian group the following are equivalent:(1) Γ is geometrically finite.(2) The surface Γ\H is topologically finite (meaning homeomorphic to a com-

pact surface with finitely many punctures).

8 D. BORTHWICK

zγz

`(γ)

π(z)

Figure 9. The axis of a hyperbolic transformation descends to aclosed geodesic under π : H→ Γ\H.

(3) Γ is finitely generated.

See, e.g., Borthwick [3, Thm. 2.10] for a proof. The spectral theory of hyper-bolic surfaces is only tractable in general for geometrically finite Γ. The reason isthat we lose control over the geometry “at infinity” for geometrically infinite sur-faces. A theorem of Siegel says that all groups of the first kind are geometricallyfinite (see Katok [23, Thm. 4.1.1]). But for groups of the second kind (i.e. infinitearea surfaces) we must insist on this condition.

2.5. Geometric features. One of the most appealing aspects of the theoryof hyperbolic surfaces is the fact that we can associate distinct geometric featuresto each class of elements of the group Γ.

For example, if γ ∈ Γ is hyperbolic, then there is a unique geodesic connectingthe two fixed points of γ, called the axis of γ. (In Figure 5, the axes were highlightedin gray.) We can see this easily by conjugating the repelling fixed point to the origin,and the attracting fixed point to infinity. Then γ : z 7→ e`z for some ` > 0. Thenthe vertical geodesic from 0 is the axis, and on this geodesic g acts as translation by`. This value `(γ) is called the translation length of γ. For general γ ∈ PSL(2,R)we have

(2.1) `(γ) = 2 cosh−1(| tr γ|/2)

In the quotient Γ\H, the axis of a hyperbolic element γ descends to a closed geodesicof length `(γ), as illustrated in Figure 9. This gives a 1-1 correspondence:

closed geodesics of Γ\H ←→ conjugacy classes ofhyperbolic elements of Γ.

The parabolic elements of Γ create cusps in Γ\H. Any parabolic element T fixessome point p ∈ ∂H. This element will also fix the interior O of a horocycle, a circletangent to ∂H at p (shown on the left in Figure 10). A cusp is defined as a quotientof the form 〈T 〉\O. A portion of the cusp is shown embedded in R3 on the rightin Figure 10; the full cusp is infinitely long, although of finite area. Because anyparabolic element is conjugate to the translation z 7→ z + 1, all cusps are isometricto each other in some neighborhood of ∞. There is a 1-1 correspondence:

cusps←→ orbits of parabolic fixed points of Γ.

Equivalently, we could associate cusps to conjugacy classes of maximal cyclic par-abolic subgroups of Γ.

An elliptic element of Γ fixes a point in Γ\H and so gives rise to a conicalsingularity in Γ\H. This is just as in the definition of an orbifold, except that inour case the conical point is always generated by a global symmetry, not just a


Figure 10. A parabolic fixed point generates a cusp.

Figure 11. An elliptic fixed point corresponds to a conical point.

local symmetry. (That is, we only see good orbifolds.) A sample conical point isshown embedded in R3 in Figure 11. There is a 1-1 correspondence:

conical points←→ orbits of elliptic fixed points of Γ.

2.6. Hyperbolic ends. If a geometrically finite hyperbolic surface Γ\H is notcompact, then it will have ends. Under the homeomorphism that identifies Γ\Hwith a compact surface with punctures, the ends are defined as neighborhoods ofthe punctures. Since Γ\H carries a complete Riemannian metric, the puncturesare necessarily moved out to infinity. The main reason that spectral theory isso tractable for geometrically finite hyperbolic surfaces is that the ends are easilyclassified geometrically. There are essentially only two types.

The first possibility is the cusp end, as seen above in Figure 10. In normalcoordinates the metric for a cusp end is

(2.2) ds2 = dr2 + e−2r dθ2

(2π)2,

where r ≥ 0, θ ∈ R/2πZ. There is no canonical location for the horocycle boundaryof the cusp, but it is always possible to take the boundary length equal to at least1 (see [3, Lemma 2.12]). So in (2.2) we can make r ≥ 0 the standard choice ofdomain.

The second type of hyperbolic end is the funnel. Consider a hyperbolic elementγ ∈ PSL(2,R). The quotient 〈γ〉\H is called a hyperbolic cylinder. The axis of γ

10 D. BORTHWICK

Figure 12. A portion of the fundamental domain intersecting ∂Hin an interval descends to a funnel.

cusps

Nielsen region = convex core

K

funnels

Figure 13. Decomposition into core plus cusps and funnels.

corresponds to a single closed geodesic at the neck of the cylinder. A funnel is halfof a hyperbolic cylinder, bounded by this closed geodesic. The canonical funnelmetric is

(2.3) ds2 = dr2 + `2 cosh2 rdθ2

(2π)2,

where r ≥ 0, θ ∈ R/2πZ and the parameter ` gives the length of the boundinggeodesic at r = 0. Figure 12 shows a portion of a funnel embedded in R3. The fullfunnel continues to flare out exponentially and has infinite area.

Theorem 2.5. Any non-elementary geometrically finite hyperbolic surface X =Γ\H admits a decomposition

X = K ∪ C1 ∪ · · · ∪ Cnc ∪ F1 ∪ · · · ∪ Fnf ,

where K is a compact orbifold with boundary, the Ci’s are cusps with boundarylength 1, and the Fj’s are funnels with boundary lengths `1, . . . , `nf .

The subset K would be called the compact core of X, while K ∪C1 ∪ · · · ∪Cnc

is called the Nielsen region, as shown in Figure 13. The Nielsen region is also theconvex core, meaning the smallest geodesically convex subset of X. If X is a non-compact surface with only funnel ends, then the group Γ is called convex-cocompact,which refers to the compactness of the convex core of Γ\H.

Theorem 2.5 is proven by taking a finite-sided domain and carefully piecingtogether those portions of the domain that meet ∂H. The details are essentiallygiven in Fenchel-Nielsen [13], or see the proof in [3, Thm. 2.13].


The elementary cases not covered by Theorem 2.5 are easily understood. Thereis H itself, of course, whose end carries the metric

(2.4) dr2 + sinh2 r dθ2.

We could also have quotients of H by finite elliptic groups, whose metrics would begiven again by (2.4), but with the period of θ being some integer fraction of 2π.

The other two non-compact elementary surfaces are quotients of H by hyper-bolic or parabolic cyclic groups. The former case gives a hyperbolic cylinder, whichis the union of two funnels and so fits the framework of Theorem 2.5 with K givenby a circle.

The parabolic cylinder is a special case. Its small end is a cusp, while the“horn” end carries the metric

dr2 + e2r dθ2

(2π)2,

for r > 0. The hyperbolic plane, funnel, and horn all have the same exponentialasymptotic behavior, but they are distinct as isometry types. The planar and hornends do not occur in any other hyperbolic surfaces.

Notes. Most of the material in this section was adapted from Borthwick [3,Ch. 2]. For the basics of Fuchsian groups, Katok [23] provides an excellent andconcise reference. See also Beardon [2], Fenchel-Nielsen [13], and Ratcliffe [35].

3. Spectrum and resolvent

Influenced by prior work of Maass, Selberg pioneered the study of the spec-tral theory of hyperbolic surfaces in the 1950’s. The essential idea was to bringtechniques from harmonic analysis into the study of automorphic forms. Spectraltheory was of course already a well-established topic in physics at that point, but itwas studied mostly in the Euclidean context (i.e. domains, obstacles, or potentialsin Rn). It turns out that the relation between hyperbolic surfaces and Fuchsiangroups makes their spectral theory much easier to analyze than some of these morestandard physical situations. Moreover, hyperbolic surfaces proved to be of greatphysical interest as relatively simple models for which the underlying classical dy-namics is chaotic. The Selberg trace formula became something of a beacon toquantum physicists, who could see in it an exact version of the correspondenceprinciple of quantum mechanics.

3.1. The Laplacian. The Laplacian operator associated to the hyperbolicmetric on H (also commonly called the Laplace-Beltrami operator in a geometriccontext) is

(3.1) ∆ := −y2

(∂2

∂x2+

∂2

∂y2

).

The Laplacian’s essential property is invariance under local isometry, so for H wehave

∆ γ = γ ∆, for γ ∈ PSL(2,R).

This means that ∆ descends to an operator on the quotient Γ\H, even when thequotient is not smooth. At least, the definition of ∆ acting on smooth functions onΓ\H is unambigious. In order to do spectral theory, however, we need to interpret

12 D. BORTHWICK

∆ as an (unbounded) self-adjoint operator on L2(Γ\H, dA), where dA is the hyper-bolic measure introduced above. This means that we need to choose a appropriatedomain within L2(Γ\H, dA).

There is a natural procedure for this, called the Friedrichs extension. We startfrom the domain

D :=f ∈ C∞0 (Γ\H) : f and ∆f ∈ L2(Γ\H, dA)

,

and then the Friedrichs extension produces a larger domain in L2(Γ\H, dA) onwhich ∆ is self-adjoint. If Γ\H is smooth then ∆ is essentially self-adjoint on D,meaning that the self-adjoint extension is unique. The resulting domain is just theH2 Sobolev space.

If Γ\H has conical points, then the Friedrichs extension is just one of a rangeof possible self-adjoint extensions. The choice of extension has a strong effect onthe spectral theory in general, so it is important to specify in these cases. In thearithmetic context the Friedrichs extension seems to be the standard choice.

Note that our sign convention for ∆, with the minus sign included in (3.1),makes ∆ a positive operator on L2. Physicists generally don’t include the minus,but instead they write −∆ in spectral formulas. Thus, wherever one puts the minussign, the common convention is to study a positive spectrum.

3.2. Eigenvalues. We say that λ is an eigenvalue of ∆ on Γ\H, with eigen-vector φ, if φ ∈ L2(Γ\H, dA) and

(3.2) ∆φ = λφ.

The L2 restriction is irrelevant in the compact case, because any solutions of theeigenvalue equation will be smooth by elliptic regularity. But in the non-compactcase we can easily have smooth solutions of (3.2) that are not eigenfunctions becausethey grow too rapidly at infinity.

Note that the action of ∆ could be extended to automorphic forms as well asfunctions. In the context of automorphic forms, eigenvectors of the Laplacian onΓ\H are called Maass forms. This terminology is used primarily in the arithmeticcontext.

If Γ\H is compact, then the eigenvalues fill out the spectrum, by the followinggeneral result.

Theorem 3.1 (Spectral theorem for compact manifolds). If (M, g) is a com-pact smooth Riemannian manifold, with Laplacian ∆g defined by the Friedrichsextension, then there exists a complete orthonormal basis for L2(M,dVg) such that

∆gφj = λjφj ,

with0 = λ0 < λ1 ≤ λ2 ≤ · · · → ∞.

The proof is not exactly elementary, but is straightforward once a little func-tional analysis background is established. The key facts are that the exact domainof ∆g is the Sobolev space H2(M,dVg), and that H2(M,dVg) is a compact subspaceof L2(M,dVg). This implies that (∆g + 1)−1 is a compact self-adjoint operator onL2(X, dVg). The theorem then follows from the spectral theorem for compact self-adjoint operators, which is covered in any basic functional analysis course. Onecould also use the heat kernel in place of the resolvent in this argument, see Buser[7]


0

λ+ iε

λ− iε

Figure 14. Limits of the resolvent in Stone’s formula

For compact Riemannian manifolds we have some general theorems about theeigenvalue spectrum. For example, we have the Weyl asymptotic formula

λk ∼ (4π)−d/2vol(M, g)Γ(n2 + 1)

k2/d,

as k → ∞, where d = dimM . The standard proof is by analyzing the small-timebehavior of the trace of heat kernel of ∆.

3.3. Resolvent. If Γ\H is non-compact, then the spectral theory of ∆ is nec-essarily more complicated. By definition, the spectrum σ(∆) consists of those val-ues of λ for which the resolvent (∆− λ)−1 fails to exist as a bounded operator onL2(Γ\H, dA). Since ∆ is self-adjoint and positive, we always have σ(∆) ⊂ [0,∞).

Theorem 3.2 (Spectral Theorem). For a self-adjoint operator A on a separableHilbert space H, there exists a measure space (Ω, µ), where Ω is a union of copiesof R, a map W : L2(Ω, µ)→ H, and a real-valued measurable function a, such thatW conjugates A to multiplication by a, i.e.

W−1AWf = af

for f ∈ L2(Ω, dµ).

The Spectral Theorem defines a functional calculus for the operator A, meaningwe can obtain functions of the operator A by setting

h(A) := W (h a)W−1,

for any Borel-measurable function h. This is analogous to defining h(A) for a finite-dimensional self-adjoint matrix A by letting h(A) act as h(λ) on the λ-eigenspace.

Continuing this analogy to the finite dimensional case, we know from linearalgebra that it is useful to introduce spectral projectors onto the eigenspaces ofa matrix. For self-adjoint operators on a Hilbert space, we can use the spectraltheorem to define such projectors, essentially by taking the limit of h(A) as happroaches the characteristic function of an interval [α, β] ⊂ R. The result is thefollowing:

Theorem 3.3 (Stone’s formula). For a self-adjoint operator A, the spectralprojector onto [α, β] is given by

Pα,β = limε→0+

12πi

∫ β

α

[(A− λ− iε)−1 − (A− λ+ iε)−1

]dλ.

Figure 14 shows that limits of the resolvent that appear in the integrand ofStone’s formula. The projector Pα,β is actually defined as the average of the spectralprojectors onto [α, β] and (α, β). This makes a difference only if there is pointspectrum at one of the endpoints.

From the abstract statement, it is a little hard to see how Stone’s formula couldbe useful. And indeed, for the general complete Riemannian manifold, we can’t get

14 D. BORTHWICK

much from this result without some fairly strong extra assumptions. However, ifwe impose some asymptotic structure on the metric near infinity, then in manyinteresting situations we can develop a very good understanding of the limits of(∆− λ± iε)−1.

Let us consider first the case of the hyperbolic plane H. The fact that

∆ys = s(1− s)ys

gives a strong hint that λ = s(1− s) will be a natural substitution for the spectralparameter. Accordingly, we start from the definition

RH(s) := (∆− s(1− s))−1.

Since the map s→ λ is a essentially a double cover of C, we must pick a half-plane,say Re s > 1

2 , λ 6= [ 12 , 1], to correspond to the region λ /∈ [0,∞) where (∆ − λ)−1

was originally defined. This is called the physical half-plane.We can calculate the kernel of the resolvent (which physicists would call the

Green’s function) by solving the PDE

(∆− s(1− s))RH(s; z, w) = δ(z − w).

After switching to polar coordinates we can separate variables, and the radial equa-tion is of hypergeometric type. The resulting formula for the Green’s function is

(3.3) RH(s; z, z′) =1

4πΓ(s)2

Γ(2s)σ−s F

(s, s; 2s;σ−1

).

where

σ := cosh(d(z, z′)/2) =(x− x′)2 + (y + y′)2

4yy′,

and F is the Gauss hypergeometric function. See [3, §4.1] for the details.We can see, in the fact that RH(s; z, z′) extends to meromorphic of s ∈ C,

confirmation that the spectral parameter was chosen wisely. Note that this contin-uation does not contradict the fact that (∆− λ)−1 should fail to exist when we hitσ(∆), because our formula for RH(s) gives an unbounded operator for Re s ≤ 1

2 .The same picture holds for Γ\H for any geometrically finite Γ, although mero-

morphic continuation is not so easily proven.

Theorem 3.4. The resolvent R(s) = (∆− s(1− s))−1 admits a meromorphiccontinuation to s ∈ C as an operator on C∞0 (X).

The idea of approaching the spectral theory of hyperbolic surfaces throughmeromorphic continuation of the resolvent was pioneered by Faddeev [12], whoproved the result for finite-area hyperbolic surfaces. Selberg’s approach to thespectrum was based on the continuation of Eisenstein series, which we’ll introducebelow. That method is restriced to the case of hyperbolic quotients, whereas theresolvent approach exemplifies the more general methods of spectral theory.

In the full case where Γ\H possibly has cusps and also infinite area, Theorem 3.4was proven by Guillope-Zworski [15], using a parametrix construction inspired byMazzeo-Melrose [28]. Here is an outline of the proof:

(1) In the interior, we can use R(s0) for some fixed large Re s0 to approximateR(s).

(2) In cusps and funnels, use the resolvents for cylindrical hyperbolic andparabolic quotients, which can be constructed explicitly in terms of hy-pergeometric functions, to construct a model resolvent R0(s).


0 1

sλ

14

12

Figure 15. The transformation from λ to s. The arrows showthe limits taken in Stone’s formula.

(3) Paste the model resolvents together using nested cutoffs χj ∈ C∞0 (Γ\H),with χj+1 = 1 on supp χj , to get the parametrix

M(s) := χ2R(s0)χ1 + (1− χ0)R0(s)(1− χ1).

(4) Compute the error

(∆− s(1− s))M(s) = I −K(s),

and show that K(s) is compact on a weighted L2 space.(5) Use the Analytic Fredholm Theorem to invert

R(s) = M(s)(I −K(s))−1.

Note that the curvature of the metric enters only through the classification ofhyperbolic ends. Indeed, the result of [15] allows the metric to be arbitrary withina compact set.

3.4. Spectrum of ∆. As already noted in Theorem 3.1, in the compact casethe eigenvalue spectrum is the whole spectrum. In this case the meromorphiccontinuation of the resolvent has poles only in the set Re s = 1

2 ∪ [ 12 , 1], at

the points where s(1 − s) ∈ σ(∆). Away from these poles, R(s) is bounded onL2(Γ\H, dA) even for Re s ≤ 1

2 .In the non-compact case, we can try to understand the spectrum via Stone’s

formula (Theorem 3.3). The set λ ∈ [0,∞), where we need to take limits of theresolvent, corresponds to s ∈ [ 1

2 , 1] ∪ Re s = 12, as shown in Figure 15. By

analyzing the behavior of R(s) as s approaches the critical line Re s = 12 , use the

structure of the resolvent obtained through the parametrix construction, we candeduce the properties of the spectral projectors.

For (α, β) ⊂ [ 14 ,∞), the substitution λ = s(1 − s) transforms Stone’s formula

into

(3.4) Pα,β =1

2πi

∫ √β−1/4

√α−1/4

[R( 1

2 − iξ)−R( 12 + iξ)

]2ξ dξ.

In the non-compact case, we can deduce from this formula that the projectorsPα,β will have infinite rank for (α, β) ⊂ [ 1

4 ,∞). This is the defining condition forcontinuous spectrum. Moreover, we can see easily that the spectrum is absolutelycontinuous, meaning that the spectral projection onto a set of Lebesgue measurezero that contains no eigenvalue will vanish. As in the compact case, the continuedresolvent R(s) will have poles in Re s = 1

2 ∪ [ 12 , 1] corresponding to points where

s(1−s) lies in the discrete spectrum. If these poles lie on the critical line Re s = 12 ,

16 D. BORTHWICK

12

4

8

−4

−8

Figure 16. A portion of the resonance set for a hyperbolic surfaceof genus zero with 3 funnel ends.

then the corresponding eigenvalues lie inside the continuous spectrum and are calledembedded eigenvalues. It turns out that embedded eigenvalues can be ruled out ifΓ\H has at least one funnel, so they occur only for Fuchsian groups of the firstkind.

Let us summarize this basic spectral information.

Theorem 3.5. For Γ geometrically finite, the spectrum of ∆ is as follows:(1) For Γ\H compact, ∆ has purely discrete spectrum in [0,∞).(2) For Γ\H non-compact, ∆ has absolutely continuous spectrum [ 1

4 ,∞) anddiscrete spectrum contained in [0,∞).

(3) If Γ\H has infinite-area, then there are no embedded eigenvalues, i.e. thediscrete spectrum is finite and contained in (0, 1

4 ).

We had already noted (1) as a general result for compact manifolds in Theo-rem 3.1. Selberg proved (2), and (3) is due to Lax-Phillips [24].

There is one more interesting feature that shows up in the meromorphic contin-uation. In the non-compact case the resolvent may have poles for Re s < 1

2 that donot correspond to eigenvalues. The full collection of poles of R(s) (including thosecoming from eigenvalues) is the set RΓ of resonances of Γ\H. These are countedwith a multiplicity defined as the rank of the residue operator at the pole. Figure 16shows a numerical calculation of the resonance set for a simple hyperbolic surface.

Notes. Reed-Simon [36] is a standard reference for the basic spectral theoryused here. Another good source is Taylor [45].

4. Spectral theory: finite-area case

4.1. Cusp forms. For the moment, we focus on the non-compact finite-areacase, where Γ\H has at least one cusp but no funnels. Arithmetic surfaces, whichwe will discuss later, fall into this category, so this is the case of greatest importancein number theory.

For simplicity, let us start by assume that Γ\H has a single cusp. By conjugatingΓ if necessary, we can assume that this cusp corresponds to a parabolic fixed point


12− 1

2

Figure 17. Standard fundamental domain position for a surfacewith one cusp at ∞.

at ∞ ∈ H, with stabilizer Γ∞ := 〈z 7→ z + 1〉 ⊂ Γ. The fundamental domain F isassumed to be bounded by |Re s| = 1

2 , as illustrated in Figure 17.We can identify L2(Γ\H, dA) with L2(F , dA). The cusp forms then constitute

the subspace

Hcusp :=f ∈ L2(F , dA) :

∫ 1

0

f(x, y) dx = 0 for a.e. y > 0.

Note that we’re still talking about functions here; the terminology “cusp form” isstandard even for this special case. (The definition does of course extend to auto-morphic forms, but we’ll focus on the function case.) If we expand f ∈ L2(F , dA)as a Fourier series in the x-variable,

(4.1) f(z) =∑n∈Z

cn(y)e2πinx,

then the defining condition for a cusp form is equivalent to the vanishing of thezero mode

f ∈ Hcusp ⇐⇒ c0(y) = 0.For surfaces with multiple cusps, cusp forms are required to satisfy the vanishingcondition separately with respect to each cusp.

Consider the Fourier decomposition of an f ∈ L2(F , dA) that satisfies theeigenvalue equation

(∆− s(1− s))f = 0.The coefficients from (4.1) then must satisfy

(4.2)[−y2∂2

y + 4π2n2y2 − s(1− s)]cn(y) = 0.

For n = 0 the two independent solutions are obvious:

(4.3) c0(y) = A0y1−s +B0y

s.

The non-zero mode equation is of Bessel type. The solutions for n 6= 0 are

cn(y) = An√yIs−1/2(2π|n|y) +Bn

√yKs−1/2(2π|n|y).

These Bessel I and K modes either grow or decay exponentially as y →∞, respec-tively. Since only the latter behavior is allowed for f ∈ L2, the coefficients An allvanish for n 6= 0.

We conclude that cusp forms satisfying the eigenvalue equation must decayexponentially as y →∞. This enforced exponential decay can be used to show that

18 D. BORTHWICK

the restriction of ∆ to Hcusp has purely discrete spectrum. The eigenfunctions ofthe restriction of ∆ to Hcusp are called Maass cusp forms.

Embedded eigenvalues (for which λ ≥ 14 ) must come from Maass cusp forms,

because a zero mode of the form (4.3) with Re s = 12 could not be L2. Below the

continuous spectrum (λ < 14 ), eigenvalues may or may not come from cusp forms.

Selberg’s trace formula shows that certain arithmetic surfaces have an abun-dance of Maass cusp forms. But Phillips and Sarnak showed that these disappearwhen the arithmetic surface is deformed to an ordinary hyperbolic surface [33].They conjectured that Hcusp is small or empty for a generic cofinite Fuchsian group.Wolpert [49] was able to prove this conjecture under the hypothesis that the cuspi-dal spectrum is simple. The general belief now seems to be that Hcusp is non-trivialonly in the arithmetic case, but this issue is as yet unresolved.

4.2. Eisenstein series. Let us continue our discussion under the assumptionof a single cusp at ∞ with stabilizer Γ∞ ⊂ Γ. From (4.3) we can see that ys isa zero-mode solution of the eigenvalue equation in Γ∞\H, but of course it is notinvariant under the full group Γ. We can try to remedy this by averaging over therest of the group, setting

E(s; z) :=∑

γ∈Γ∞\Γ

(Im γz)s

=∑

γ∈Γ∞\Γ

ys

|cz + d|2s.

(4.4)

This construction is called an Eisenstein series. It is not too hard to argue thatthe series converges for Re s > 1 (this follows from (6.2)), but continuation beyondthat is not at all obvious.

Meromorphic continuation can be established by various routes; see, for ex-ample, the elegant proof given by Colin de Verdiere [9]. From a spectral theoryviewpoint, the natural method is to connect the Eisenstein series to the resolvent,whose meromorphic continuation is already understood. (As we noted above, thisapproach was introduced by Faddeev [12].) We can also write the kernel of R(s)as an average over Γ∞\Γ,

R(s; z, w) =∑γ∈Γ

RH(s; γz, w)

=∑

γ∈Γ∞\Γ

RΓ∞\H(s; γz, w).

The resolvent RΓ∞\H(s) can be computed explicitly as a Fourier series in the x-variable [3, Prop. 5.8], and this calculation shows that

RΓ∞\H(s; z, z′) =ysy′

1−s

2s− 1+O(y′−∞),

as y′ →∞ with y held fixed. Thus we have the relation

(4.5) limy′→∞

y′s−1

R(s; z, z′) =1

2s− 1E(s; z).

The parametrix construction can be used to insure that the limit on left-hand side iswell-defined, and the meromorphic continuation of E(s; z) then follows immediatelyfrom that of R(s).


4.3. Spectral decomposition: finite-area case. For finite area Γ\H, wemust distinguish two forms of discrete spectrum. The cusp form spectrum, whichincludes all embedded eigenvalues, has already been introduced above. The rest ofthe discrete spectrum, consisting of eigenvalues below 1

4 which do not come fromcusp forms, is called the residual spectrum. The span of the residual eigenfunctionsis denoted Hres.

We are now prepared to parametrize the full spectrum. On Hcusp ⊕ Hres wehave an complete eigenbasis φj , λj for ∆. To complete our decomposition, let usdenote the remaining part of the Hilbert space as the continuous Hilbert space,

Hcont := (Hcusp ⊕Hres)⊥,

so thatL2(Γ\H, dA) = Hcusp ⊕Hres ⊕Hcont.

The spectrum of ∆ on Hcont is purely continuous, and we can use Eisenstein seriesas continuous analog of the eigenbasis on this subspace.

The full spectral decomposition result is the following:

Theorem 4.1 (Roelcke-Selberg). A function f ∈ L2(Γ\H, dA) can be writen

f(z) =∫ ∞−∞

E( 12 + ir; z)a(r) dr +

∑bjφj(z),

wherea(r) :=

∫FE( 1

2 + ir; z)f(z) dA(z),

bj :=∫Fφj(z)f(z) dA(z).

See [46, Thm. III.7.1] for a proof. This spectral decomposition theorem isanalogous to the Fourier transform in Rn, in that the function f is expressed as asuperposition of oscillatory states of various frequencies. The Eisenstein series playthe role of Euclidean plane waves in this analogy.

4.4. Scattering matrix: finite-area case. We continue to work with theexample of a finite-area quotient Γ\H with a single cusp. We’ve noted that E(s; z) isthe analog of a plane wave, but it is more accurate to think of it as the superpositionof incoming and outgoing waves. As y →∞,

(4.6) E(s; z) = ys + ϕ(s)y1−s +O(y−∞),

for a meromorphic function ϕ(s). The ys term corresponds to an incoming solutionof the wave equation and y1−s to an outgoing term. That is, if we use E(s; z) tobuild a solution of the wave equation, ∂2

t u = ∆u, via separation of variables, we’llfind

u(t, z) = ei√λtE(s; z),

where λ = s(1− s). For s = 12 + iν with ν > 0,

ei√λtys =

√y exp

[i√λt+ iν ln y

],

exhibits a wavefront moving inward. Likewise, the y1−s term yields an wavefrontmoving outward.

If there are multiple cusps, then for each cusp j we define a coordinate yj asthe y coordinate when the corresponding parabolic fixed point is moved to ∞ and

20 D. BORTHWICK

stablized by Γ∞. In this case a different Eisenstein series Ei(s; z) is assigned toeach cusp i. As above, Ei(s; z) is defined as an average of ysi over Γ∞\Γ. From theasymptotic expansion of Ei(s; z) in the j-th cusp,

Ei(s; z) ∼ δijysj + ϕij(s)y1−sj ,

for a set of meromorphic functions ϕij(s). The physical interpretation is that Ei(s)contains a single incoming waveform, in cusp i, and ϕij(s) gives the coefficients ofthe resulting outgoing solution in cusp j.

These coefficients collectively define the scattering matrix

S(s) := [ϕij(s)].

The asympotic construction implies certain basic relations for the scattering matrix,including the inversion formula

S(s)−1 = S(1− s).Taking the determinant yields a meromorphic function,

ϕ(s) := detS(s),

called the scattering determinant.The poles of ϕ(s) are the scattering poles. These include points where λ =

s(1 − s) is in the residual spectrum. From the relation (4.5), we can deduce thatthe resonances will give rise to scattering poles provided the coefficient of y′s inthe residue of R(s; z, z′) at the pole does not vanish. It turns out that this willalways be the case unless the resonance is caused by a cusp form. Cusp formsdecay exponentially in the cusps and thus do not give rise to scattering resonances.

We thus essentially have the following relationship between resonances andscattering poles:

resonances = scattering poles + cusp resonances,

but this simple statement is not quite accurate. Residual eigenvalues generallygive rise to scattering poles, but there is a possible exception in the case of residualspectrum. Because of the symmetry ϕ(1−s) = 1/ϕ(s), a resonance at ζ correspondsalso to a zero of ϕ(s) at 1 − ζ. Since the only resonances with Re s > 1

2 lie in theinterval ( 1

2 , 1], this cancellation could only occur when s ∈ [0, 1].Figure 18 shows a portion of the resonance plot for the modular surface, which

we’ll introduce properly later. The resonance at s = 1 corresponds to the eigenvalueΛ = 0, which appears by default for any finite-area surface.

Notes. There are many references for the spectral theory of cofinite Fuchsiangroups, including Selberg [40], Hejhal [19], Iwaniec [20], Terras [46], and Venkov[48].

5. Spectral theory: infinite-area case

If the quotient surface Γ\H has at least one funnel end, then its area is infinite.The presence of a funnel has a significant effect on the spectral theory. For example,suppose we have an eigenfunction u ∈ L2(Γ\H, dA) satisfying (∆− s(1− s))u = 0.Using the structure of the resolvent, we can argue that in any funnel end, such afunction must an asymptotic expansion

u ∼∑k

ρs+kfk,


scattering

10

20

cusp form

12

1

Figure 18. A portion of the resonance plot for PSL(2,Z)\H

for fk ∈ C∞(R/2πZ), with ρ = 2e−r with respect to the normal coordinates (2.3).(This ρ serves as a defining function for a compactification of the funnel to cylinderwith circular boundary.) If Re s = 1

2 , then u can be in L2 only if f0 = 0. But then,by applying ∆ to the expansion of u at infinity, one can see by induction that allthe fk must vanish. In the compactified picture, this means that u must vanish toinfinite order at the boundary ρ = 0. Then a unique continuation argument canbe applied to show that u = 0. (See Mazzeo [27] for the original version of thisargument, which gives a more general result than we have stated here, or [3, Ch. 7]for the details in this special case.)

This argument shows in particular that there are no embedded eigenvalues inthe infinite-area case. Even if such a surface has cusps, there can be no cusp forms.We had already noted this result in Theorem 3.5, which was originally proven inthe context of hyperbolic manifolds by Lax-Phillips [24].

For simplicity of notation, let us assume that X = Γ\H has a single funnel F .The function ρ = 2e−r introduced above defines a compactification X. The analogof the Eisenstein series is an operator called the Poisson operator,

E(s) : C∞(∂X)→ C∞(X),

with integral kernel defined by a boundary limit of the Green’s function,

(5.1) E(s; z, θ′) := limρ′→0

ρ′−sR(s; z, z′),

where z′ = (r′, θ′) are the standard funnel coordinates and ρ′ = 2e−r′. The justifi-

cation for term “Poisson operator” becomes clear if we examine the correspondingsituation for H. As long as we stay away from∞, the coordinate y makes a suitableboundary-defining coordinate ρ. With this choice, we see from (3.3) that

EH(s; z, y) := limy→0

y−sRH(s; z, z′)

=1

4πΓ(s)2

Γ(2s)

[y

(x− x′)2 + y2

]s.

22 D. BORTHWICK

For s = 1 this is indeed the classical Poisson kernel associated to the boundaryvalue problem for harmonic functions in the upper half-plane. For s 6= 1 the oper-ator EH(s) maps compactly support functions on R to solutions of the eigenvalueequation (∆− s(1− s))u = 0 in H. This follows from (∆− s(1− s))R(s) = I, butwe could also check it explicitly by differentiating EH(s; ·, y).

Likewise, for our surface X the Poisson operator associates to a function f ∈C∞(∂X) the solution u = E(s)f of the eigenvalue equation (∆− s(1− s))u = 0 inX. For s 6= 1 this is not the solution to a simple boundary value problem, however.The boundary expansion of u at ρ = 0 has two parts, with leading terms given by

(2s− 1)u ∼ ρ1−sf + ρsf ′,

with f the original boundary function and some other f ′ ∈ C∞(∂X). In the contextof a funnel end, it is the map

(5.2) S(s) : f 7→ f ′

that is called the scattering matrix. This time we’re really not talking about amatrix—S(s) is in fact a pseudifferential operator on ∂X—but the term “matrix”is still standard usage for historical reasons.

The formula (5.1) for the kernel of the Poisson operator is analogous to theexpression (4.5) for an Eisenstein series. And, just as in the finite-area case, forRe s > 1 we can express the kernel E(s) as an average over translates of EH(s; z, y)by Γ. Hence the terms Poisson kernel and Eisenstein series are often used inter-changeably in this context. The only possible source of confusion is the fact thatthe natural normalizations for the two objects differ by a factor of (2s− 1).

The Poisson operator defines a parametrization of the continuous spectrum,similar to what we saw for Eisenstein series in the finite-area case. Using thestructure of the resolvent and a clever integration by parts, we can show that

R(s; z, w)−R(1− s; z, w) = −(2s− 1)∫ 2π

0

E(s; z, θ)E(1− s;w, θ) dθ.

Plugging this into Stone’s formula, as written in (3.4), expresses the spectral pro-jections in terms of the Poisson operators:

Pα,β =1

2πi

∫ √β−1/4

√α−1/4

E( 12 + iξ)E( 1

2 − iξ)t4iξ2 dξ.

In the general case of a surface with multiple funnel and cusp boundaries,the boundary ∂X becomes a disjoint union of circles (one for each funnel) andpoints (one for each cusp). The Poisson operator is defined just as above, by anintegral kernel given as a boundary limit of the Green’s function. It just becomesa bit more notationally complicated, with one component of the operator for eachboundary element. Likewise, the scattering matrix is obtained as in (5.2), butit has a separate component for each pair of boundary elements. The diagonalfunnel-funnel terms are pseudodifferential operators, while the off-diagonal termsare smoothing operators. See Borthwick [3, Ch. 7] for a complete description ofthis structure.

Notes. For more details on the material in this section, see Borthwick [3].


6. Selberg trace formula

The central result in the spectral theory of hyperbolic surfaces is the Selbergtrace formula. This connects the spectral data associated to the Laplacian to thegeometric structure of the quotient. In physical terms, we could view this as aconnection between quantum and classical mechanics, with the quantum side rep-resented by the spectral theory and classical mechanics by the geometry of geodesicson the surface. Indeed, the trace formula can be given a heuristic formulation interms of Feynman path integrals (see Gutzwiller [18]). There exist more generaltrace formulas connecting quantum physics with classical mechanics, but these aretypically asymptotic expansions as Planck’s constant goes to zero. The theory ofsuch expansions is called semiclassical analysis. The Selberg trace formula is notonly a prototype for these expansions, but also a special case where the expansionis exact rather than asymptotic. The crucial extra ingredient in the hyperbolic caseis the algebraic structure that serves as an intermediary between the analytic andgeometric sides of the formula.

6.1. Smooth compact case. For ease of exposition, we start by consideringthe case where Γ\H is smooth and compact. The Laplacian thus has spectrum λjcorresponding to a complete orthonormal basis of eigenvectors φj.

To any f ∈ C∞[0,∞) we can try to define an operator Kf with integral kernel

(6.1) Kf (z, w) :=∑γ∈Γ

f(d(z, γw)).

For u ∈ L2(Γ\H), the action of this operator is

Kfu(w) =∫FKf (d(z, γw))u(w) dA(w)

=∫

Hf(d(z, w))u(w) dA(w).

Of course, we need to insist on some decay of f at ∞ for these expressions toconverge.

It is not too hard to prove for any geometrically finite Γ that

(6.2) #γ ∈ Γ : d(z, γw) ≤ t

= O(et).

Note that the left-hand side counts the number of orbit points in Γw inside the diskB(z; t). We can assume z, w ∈ F and choose a compact subset F0 ⊂ F containingboth points. Each orbit point of w sits inside a separate translate of F0 by thegroup. Whenever d(z, γw) ≤ t, the image γF0 is contained in a disk of radius t+ dcentered at z, where d = diam(F0). Since the images of F0 must be disjoint, wehave an estimate

#T ∈ Γ : d(z, Tw) ≤ t

≤ Area(B(z; t+ d))

Area(F0)≤ C cosh(t+ d),

which implies (6.2). It follows that the sum (6.1) defining Kf will converge uni-formly provided we impose the condition

|f(u)| = O(u−1−ε).

for some ε > 0. Since the resulting kernel is smooth and Γ\H is compact, thismakes Kf a smoothing operator L2(Γ\H, dA)→ C∞(Γ\H).

24 D. BORTHWICK

The Selberg trace formula computes the trace of Kf in two different ways. Onthe one hand, since Kf is trace class and has a continuous kernel, the trace couldbe written as

(6.3) trKf =∫

Γ\HKf (z, z) dA(z).

On the other hand, if we let κj be the eigenvalue spectrum of Kf , then the traceis simply

(6.4) trKf =∑j

κj .

These two expressions for the trace would be valid for any operator on a compactsurface defined by a smooth kernel. The novelty in our case is that both expressionsare explicitly computable in terms of f .

For the spectral side (6.4), we start with the observation that Kf is self-adjointand commutes with the Laplacian, which follows directly from the symmetry

∆zd(z, w) = ∆wd(z, w).

Then, since Kf and ∆ are commuting self-adjoint operators, they can be simulta-neously diagonalized by a basic result of functional analysis. In particular, we canassume that the eigenfunctions φj of ∆ also satisfy

Kfφj = κjφj ,

for some κj . To compute κj , we write this eigenvalue equation in integral form,

(6.5) κjφj(w) =∫

Hf(d(w, z)) φj(z) dA(z).

Now specialize to w = i and let sj =√λj − 1/4. Note that ysj satisfies the same

eigenvalue equation as φj ,∆ysj = λjy

sj .

If C(u) denotes the circular average of u with respect to elliptic rotations centeredat i, then both C(φj) and C(ysj ) satisfy the radial version of the eigenvalue equation

(−∂2r + coth r ∂r)u = λju,

where r = d(i, ·) here. Matching the boundary conditions at r = 0 gives

C(φj) = φj(i)C(ysj )

Inserting this into the integral (6.5), and noting the invariance of d(i, z) underrotations about i, gives

κjφj(i) = φj(i)∫

Hf(d(w, z)) ysj dA(z).

This gives the crucial formula expressing κj in terms of f ,

κj =∫

Hf(d(i, z)) ysj dA(z)

Using this result we can write the spectral side of the trace formula as

(6.6) trKf =∞∑j=0

h(√

λj − 14

),


where

(6.7) h(r) :=∫

Hf(d(i, z)) yr dA(z).

The restriction on the growth of f translates to the assumption that h extends toan analytic function on | Im z| < 1

2 + ε, satisfying the bound

h(z) = O((1 + |z|2)−1−ε).

A standard choice is to take h(z) = e−t(z2+1/4), with t > 0, for which Kf is the

heat operator e−t∆ and (6.4) becomes

tr e−t∆ =∑j

e−tλj .

Now let’s turn to the geometric side (6.3) of the trace formula. Our assumptionthat Γ\H is smooth and compact implies that Γ contains only hyperbolic elements.The length trace computation starts from

trKf =∫FKf (z, z) dA(z)

=∑γ∈Γ

∫Ff(d(z, γz)) dA(z).

The trick is to organize the sum over Γ as a sum over conjugacy classes, and thenexpress these in terms of lengths of closed geodesics. Let Π be a complete list ofrepresentatives of conjugacy classes of primitive elements of Γ. Then we can write

(6.8) Γ− I =⋃γ∈Π

⋃k∈N

⋃σ∈Γ/〈g〉

σγkσ−1.

The innermost union over σ gives the conjugacy class of γk in Γ, and by the defi-nition of Π, each non-trivial conjugacy class in Γ corresponds to exactly one γ ∈ Πand k ∈ N.

Associated to each γ ∈ Π is a closed geodesic of Γ\H which is also primitive.(For a geodesic, primitive means traversed in only a single iteration.) The corre-sponding set of lengths (as defined by (2.1)) forms the primitive length spectrum ofΓ\H:

L(Γ) := `(γ) : γ ∈ Π.Note this is the ‘oriented’ length spectrum; since γ and γ−1 are listed separately inΠ, each length will appear twice in L(Γ).

Using the conjugacy class decomposition (6.8) of Γ, we write the trace as

trKf = f(0) Area(Γ\H) +∑γ∈Π

∑k∈N

∑η∈Γ/〈g〉

∫Ff(d(z, ηγkη−1z)) dA(z)

A change of variables in the integral gives∫Ff(d(z, ηγkη−1z)) dA(z) =

∫ηFf(d(z, γkz)) dA(z)

We then observe that the union of ηF over η ∈ Γ/〈γ〉 forms a fundamental domainfor the cyclic group 〈γ〉. We could replace this by any other fundamental domain

26 D. BORTHWICK

F〈γ〉, so that∑η∈Γ/〈γ〉

∫Ff(d(z, ηγkη−1z)) dA(z) =

∫F〈γ〉

f(d(z, γkz)) dA(z).

In particular, if we conjugate γ to z 7→ e`z, where ` = `(γ) then the convenientchoice is F〈γ〉 = 1 ≤ y ≤ e`. We then compute∑η∈Γ/〈g〉

∫Ff(d(z, ηγkη−1z)) dA(z) =

∫1≤y≤e`

f(d(z, ek`z)) dA(z)

=`

sinh(k`/2)

∫ ∞k`

f(cosh t)√2 cosh t− 2 cosh k`

sinh t dt.

This essentially completes the evaluation of the length side of the trace formula:

trKf = f(0) Area(Γ\H)

+∑

`∈L(Γ)

∑k∈N

`

sinh(k`/2)

∫ ∞k`


sinh t dt.

It is, however, customary to express both sides in terms of the function h(r)that appeared on the spectral side of the trace formula (6.7). Through some directbut not so simple calculations (see Buser [7, §7.3]), we find

f(0) =∫ ∞−∞

rh(r) tanhπr dr,

and ∫ ∞k`


sinh t dt = h(k`)

Theorem 6.1 (Selberg Trace Formula - smooth compact case). For h satisfyingthe conditions given above,

∞∑j=0

h(√

λj − 14

)=

Area(Γ\H)4π

∫ ∞−∞

rh(r) tanhπr dr

+∑

`∈L(Γ)

∑k∈N

`

sinh(k`/2)h(k`).

For smooth hyperbolic surfaces the Gauss-Bonnet theorem gives

Area(Γ\H) = −2πχ,

where χ is the Euler characteristic of Γ\H. So the area term is really a topologicalterm proportional to χ.

As noted above, the choice h(z) = e−t(z2+1/4) yields the heat trace formula,

∞∑j=0

e−tλj = Area(Γ\H)e−t/4

(4πt)3/2

∫ ∞0

re−r2/4t

sinh(r/2)dr

+12

e−t/4

(4πt)1/2

∑`∈L(Γ)

∞∑k=1

`

sinh(k`/2)e−`

2/4t.


By studying the divergence of both sides as t→ 0 we can use a Tauberian argumentto prove the Weyl asymptotic formula,

#λj ≤ r

∼ Area(Γ\H)

4πr.

The behavior of the heat trace as t→ 0 is actually quite well understood in muchgreater generality, and this type of Weyl asymptotic for the spectrum holds for anycompact manifold, with the power of r given by the dimension.

By considering a full asymptotic expansion of the heat trace as t→ 0, we canrecover the full length spectrum Λ(Γ) from that side of the heat trace formula. Onthe other hand, the eigenvalue spectrum is easily recovered from the asymptoticexpansion of the heat trace as t → ∞. These facts yield a direct proof of thefollowing:

Corollary 6.2 (Huber’s Theorem). For Γ\H smooth and compact the eigen-value spectrum and length spectrum (both with multiplicities) determine each other,as well as the Euler chacteristic χ.

Because leading term on the spectral side of the heat trace as t→∞ is 1+o(1),corresponding to the default eigenvalue λ0 = 0, we can deduce from this limit that∑

`∈L(Γ)

∞∑k=1

`

sinh(k`/2)e−`

2/4t = 2(4πt)1/2et/4(1 + o(1)),

as t → ∞. Applying a Tauberian argument to this asymptotic yields anotherfamous corollary of the Selberg trace formula, also due to Huber:

Corollary 6.3 (Prime geodesic theorem). As x→∞,

#` ∈ Λ(Γ) : e` ≤ x

= Lix+

m∑j=1

Li(xsj ) +O(x7/8+ε/ log x).

where the values sj ∈ ( 12 , 1] correspond to the eigenvalues λ1, . . . , λm of ∆ in

[0, 14 ) according to the relation λj = sj(1− sj).

Here Lix denotes the log integral function,

Lix :=∫ x

2

dt

log t,

playing the same role here as in the Prime Number Theorem.Many applications of the trace formula come through the Selberg zeta function,

(6.9) Z(s) :=∏

`∈L(Γ)

∞∏k=1

(1− e−(s+k)`

).

The product converges for Re s > 1 by the estimate (6.2). And in the compact caseit extends to an entire function of s ∈ C.

We cannot compute Z(s) directly using the trace formula. However, with thechoice

h(z) =1

z2 + (s− 1/2)2− 1z2 + (a− 1/2)2

,

28 D. BORTHWICK

the trace formula connects the zeta function to a regularized trace of the resolventon the spectral side,

tr[R(s)−R(a)] =1

2s− 1Z ′

Z(s)− 1

2a− 1Z ′

Z(a)− χ

∑k=0

[1

s+ k− 1a+ k

].

The resolvent itself is not trace class, but the difference R(s) − R(a) will be traceclass in the compact case.

We defined resonances to be the poles of the resolvent, so this version relatesthe resonance set to the divisor of the zeta function. Indeed, it reveals the beautifulfact that the zeros of Z(s) occur at the resonances, with some extra zeros at −N0

coming from the topological term on the length side). To summarize these results:

Corollary 6.4. For Γ\H smooth and compact, Selberg’s zeta function Z(s)is an entire function of order 2. Its zero set consists of the resonance set of ∆(points s ∈ C for which s(1−s) is an eigenvalue) and a set of topological zeros withmultiplicities proportional to χ.

6.2. General finite area case. The argument for the trace formula doesn’tchange much in the compact orbifold case, where we allow elliptic elements in thegroup. The only new feature is that in the decomposition (6.8) of Γ into conjugacyclasses, we must include a sum over representatives of primitive elliptic conjugacyclasses. For each such representative η we’ll have a contribution to the trace formulaof the form

m−1∑k=1

∑σ∈Γ/〈T 〉

∫Ff(d(z, σηkσ−1z)) dA(z) =

m−1∑k=1

∫F〈η〉

f(d(z, ηkz)) dA(z),

where m is the order of η. Here 〈η〉 is the cyclic group generated by η, with F〈η〉the corresponding fundamental domain. By choosing F〈η〉 wisely we can computethese integrals and then express them in terms of h,∫

F〈η〉f(d(z, ηkz)) dA(z) =

1m sin(πk/m)

∫ ∞−∞

e−2πkr/m

1− e−2πrh(r) dr

Thus in the compact orbifold case the only change to the trace formula is a someover elliptic conjugacy classes,

nc∑j=1

1mj sin(πk/mj)

∫ ∞−∞

e−2πkr/mj

1− e−2πrh(r) dr,

that appears on the length side.The trace formula is substantially harder to prove for non-compact Γ\H, be-

cause the operator Kf is not trace class. Moreover, the integral∫Γ\H

Kf (z, z) dA(z)

diverges! So at first glance, neither side of the trace formula makes sense.To extend the trace formula to this case, we impose cutoffs, restricting the

kernel of the operator to the range yj ≤ N within each cusp. On both sides of thetrace formula terms appear that will diverge like logN as N → ∞. The formulais obtained by carefully canceling these terms on both sides before taking takingN →∞.


The details of this argument are a little too technical for us to get into, butwe will explain what happens to the trace formula. On the spectral side, extra‘scattering’ terms appear to account for the continuous spectrum, of the form

− 12π

∫ ∞−∞

ϕ′

ϕ( 1

2 + ir)h(r) dr + 12h(0) tr[ϕij( 1

2 )],

where [ϕij(s)] is the scattering matrix and ϕ(s) the scattering determinant.On the length side, our decomposition of Γ must include conjugacy classes of

primitive parabolic elements, as well as hyperbolic and elliptic. For each cusp, weadd a term

− 1π

∫ ∞−∞

Γ′

Γ(1 + ir)h(r) dr +

12h(0)− 1

πh(0) log 2

Theorem 6.5 (Selberg trace formula - finite area version). Let X be a non-compact finite-area hyperbolic surface. If that g ∈ C∞0 (R) is even then∑

j

h(√

λj − 1/4)− 1

2π

∫ ∞−∞

ϕ′

ϕ( 1

2 + ir)h(r) dr

=Area(Γ\H)

4π

∫ ∞−∞

r tanh(πr)h(r) dr +∑

`∈L(Γ)

∑k∈N

`

sinh(k`/2)h(k`)

−nc

π

∫ ∞−∞

Ψ(1 + ir)h(r) dr +12

(nc − tr[ϕij( 12 )])h(0)− nc

πlog 2 h(0),

where λj are the eigenvalues of ∆X , nc is the number of cusps, and Ψ(z) is thedigamma function Γ′/Γ(z).

The trace formula leads to applications analogous to those in the compact case,including:

(1) For finite-area hyperbolic surfaces the resonance set and the length spec-trum determine each other, and also χ and number of cusps.

(2) Prime Geodesic Theorem: for finite-area hyperbolic surfaces

#e` ≤ x

∼ Lix+

∑Li(xsj ),

where λj = sj(1− sj) are the eigenvalues in (0, 14 ).

(3) Weyl-Selberg asymptotic formula for finite-area hyperbolic surfaces:

#|λj | ≤ r

− 1

4π

∫ √r−1/4

−√r−1/4

ϕ′

ϕ( 1

2 + it) dt ∼ Area(Γ\H)4π

r.

The zeta function is still defined by the formula (6.9) in the non-compact case,as a product over the primitive length spectrum (or primitive hyperbolic conjugacyclasses). If Γ\H has cusps, then Z(s) will extend to a meromorphic function of s ∈C, with poles at negative half-integer points whose multiplicities are proportionalto the number of cusps.

6.3. Infinite-area surfaces. In the infinite-area case, convergence of thetrace is even more of an issue, and we cannot develop a trace formula for thefull class of operators Kf . Analytically, it proves most convenient to focus on the

30 D. BORTHWICK

trace of the wave operator, cos(t√

∆− 1/4), whose kernel is one of the fundamentalsolutions of the wave equation

∂2t u+ (∆− 1

4 )u = 0.

Even in the compact case, the wave operator is not trace class and its trace must beunderstood as a distribution in t rather than a function. But additional regulariza-tion is required for an infinite-area surface. Using the boundary defining functionρ introduced above, we can define a formal trace called the 0-trace. For a smoothkernel K(z, z′) with a polyhomogeneous asymptotic expansion in powers of ρ, ρ′ atinfinity, we define

0-trK := FPε→0

∫ρ≥ε

K(z, z) dA(z),

where FP denotes the Hadamard finite part. Then we define the wave 0-trace

Θ(t) := 0-tr cos(t√

∆− 1/4),

as a distribution on R. This means that for a test function ϕ ∈ S, we first integrateover t, which produces a smoothing operator, and then take the 0-trace:

(Θ, ϕ) := 0-tr∫

cos(t√

∆− 1/4)ϕ(t) dt.

The following result is generally called the Poisson formula, by analogy withthe classical summation formula.

Theorem 6.6 (Guillope-Zworski). For a smooth geometrically finite hyperbolicsurface,

Θ(t) =12

∑ζ∈R

e(ζ− 12 )t +

nc

4,

as a distribution on R+.

The theorem proven in [16] is actually more general, applying to surfaces withhyperbolic ends but with an arbitrary metric inside some compact set. See [3] fora detailed exposition of the proof of the version stated here. The Poisson formulagives a realization of the spectral side of the trace formula, with resonances takingup the role played by eigenvalues in the compact case.

It’s worth noting that if we had used the heat operator, the putative spectraltrace would be a sum of e−tζ(1−ζ) over ζ ∈ R. In the infinite-area case, the res-onances are spread out over the half-plane Re s ≤ 1

2 , so that the values ζ(1 − ζ)are distributed throughout the complex plain. Thus, without better information onthe distribution of resonances than we currently have, there is no way to regularizethis spectral heat trace.

For the length side of the wave trace formula, we can apply the same methodused in the compact case, breaking up the group into a sum over primitive conjugacyclasses and then computing the 0-traces explicitly for each class. The result [17] is:

Theorem 6.7 (Guillope-Zworski). For a smooth, geometrically finite, non-elementary hyperbolic surface Γ\H,

Θ(t) =∑

`∈L(Γ)

∞∑k=1

`

4 sinh(k`/2)δ(|t| − k`) +

χ

4cosh(t/2)sinh2(t/2)

+nc

4coth(|t|/2) + ncγδ(t),


t

# = N(t)

Figure 19. The resonance counting function.

as a distribution on R, where γ is Euler’s constant.

Note that the length side of the wave trace formula does extend through t = 0,in the distributional sense. The characterization of the singular support of the wavetrace that one obtains from Theorem 6.7 holds more generally, but one usually mustinclude a smooth error term. This is a famous result of Duistermaat-Guillemin [11]in the compact case, extended by Joshi-Sa Barreto [22] to asymptotically hyperbolicmanifolds.

The combination of the two wave trace results gives a distributional version ofthe Selberg trace formula for infinite-area surfaces:

Corollary 6.8. For smooth, geometrically finite hyperbolic surfaces,∑ζ∈R

e(ζ− 12 )t =

∑`∈L(Γ)

∞∑k=1

`

2 sinh(k`/2)δ(|t| − k`)

+χ

2cosh(t/2)sinh2(t/2)

+nc

2[coth(|t|/2)− 1],

(6.10)

as a distribution on R+.

Guillope-Zworski [15, 16, 17] also established bounds on the distribution ofresonances, including the resonance counting function,

N(t) := #ζ ∈ R : |ζ − 12 | ≤ t t

2,

which counts resonances in the region shown in Figure 19. They proved upper andlower bounds of the optimal order, i.e.,

N(t) t2,The upper bound is proved by estimating a Fredholm determinant constructed fromthe resolvent and is needed in the proof of the trace formula. The lower bound thenis deduced from the big singularity of the wave trace at t = 0.

No Weyl law for the resonance counting is known (in the infinite-area case), butthe trace formula does lead to a Weyl law for the scattering phase [16], which couldbe viewed as a substitute. In fact, we can use this scattering phase asymptotic torefine the upper bound on resonances, see Borthwick [4]. The resulting bound is∫ a

0

2Ng(t)t2

dt ≤

|χ|+ nf∑j=1

`j4

+ o(1)

a2,

32 D. BORTHWICK

− 12

12

−1

1

δ ≈ 0.2337

Figure 20. Location of the first resonance at the exponent ofconvergence δ.

where the `j ’s are the lengths of the closed geodesics bounding the funnels. Thisestimate is sharp in the finite-volume case, where Parnovski [30] showed thatN(t) ∼ |χ| t2. (This does agree with the Weyl asymptotic in the compact case—remember that we are counting resonances rather than eigenvalues, so the power isdoubled).

Another fairly direct consequence of Corollary 6.8 is the analog of Huber’stheorem, that the resonance set and length spectrum determine each other up tofinitely many possibilitity [5]. This fact was used to prove that the resonance setdetermines an infinite-area hyperbolic surface up to finitely many possibilities inBorthwick-Judge-Perry [6].

We can also consider the Selberg zeta function Z(s) in the infinite-area case.The same definition (6.9) applies, and indeed the convergence is slightly betterthan in the finite-area case. The product formula for the zeta function convergesfor Re s > δ, where δ is the exponent of convergence, usually defined as

δ := infs ≥ 0 :

∑T∈Γ

e−sd(z,Tw) <∞,

which is independent of the choice of z, w ∈ H, serves also as the abscissa ofconvergence for the zeta function. For quotients Γ\H of infinite area we have δ < 1,with δ = 0 for the elementary groups, and δ = 1 precisely for the quotients of finitearea.

The exponent δ is quite interesting in its own right. On the one hand, thePatterson-Sullivan theory of measures on the limit set [31, 44] shows that δ is theHausdorff dimension of Λ(Γ). On the other hand, Patterson also proved [31, 32]that it gives the location of the first resonance of ∆, which corresponds to aneigenvalue if and only if δ > 1

2 . For the 3-funnel surface shown in Figure 16, thisvalue is δ ≈ 0.2337, as illustrated in the resonance plot in Figure 20.

The meromorphic extension of the zeta function to s ∈ C was proven beforethe trace formula in this context, by Guillope [14]. But, as in the compact case,the trace formula gives additional information about its divisor. From it we canderive a factorization of Z(s) in terms of the resonance set. Define the Hadamardproduct,

H(s) :=∏ζ∈R

(1− s

ζ

)es/ζ+s

2/2ζ2 ,


Figure 21. The standard fundamental domain for the modulargroup, with the corresponding tessellation of H.

and let

G∞(s) = (2π)−1Γ(s)G(s)2,

where G(s) is the Barnes G-function. It was proven in Borthwick-Judge-Perry [6]that for a geometrically finite hyperbolic surface of infinite area,

Z(s) = eq(s)G∞(s)−χΓ(s− 12 )ncH(s),

where q(s) is a polynomial of degree at most 2. In particular, the divisor of Z(s)consists of the resonance set plus topological contributions determined by χ andnc.

Notes. The material on the trace formula in the compact case is adapted fromMcKean [29] and Buser [7]. For the finite-area case see, e.g. Hejhal [19], Iwaniec[20], Terras [46], or Venkov [48]. The infinite-area case is covered in Borthwick [3].

7. Arithmetic surfaces

The term arithmetic implies a restriction to integers, and in the case of arith-metic surfaces the idea is this restriction is applied to the entries of the matricesforming the Fuchsian group. If we do this directly in PSL(2,R), the result is themodular group

ΓZ := PSL(2,Z),

the fundamental example of an arithmetic group.In greater generality, we might start with a finite dimensional representation

ρ : PSL(2,R)→ GL(n,R) and then restrict to matrices with integer entries withinthat representation. The resulting subgroup is discrete and thus defines a Fuchsiangroup,

Γ :=γ ∈ PSL(2,R) : ρ(γ) ∈ GL(n,Z)

,

The full definition of arithmetic Fuchsian group includes all the groups obtained bythis construction, as well as any subgroups of finite index.

34 D. BORTHWICK

F

i p = eπi/3 i p

X = ΓZ\H

Figure 22. Constructing the modular surface by folding the fun-damental domain and identifying edges.

7.1. Modular surfaces. We will focus mainly on the modular group ΓZ inthis exposition. This group is generated by the elements

τ =(

1 10 1

), σ =

(0 1−1 0

),

corresponding to the maps,

τ : z 7→ z + 1, σ : z 7→ −1z.

The map τ is of course the standard parabolic translation fixing ∞, the generatorof Γ∞. The element σ is a rotation of order 2 about the fixed point i. The modulargroup also has an elliptic fixed point of order 3: the point p = eiπ/3 is fixed byστ−1.

The standard fundamental domain for ΓZ is

F =z ∈ H : |Re z| ≤ 1

2 , |z| ≥ 1,

as shown in Figure 21.From this domain the quotient surface X := ΓZ\H, called the modular surface,

can be constructed by folding along the central axis Re z = 0. This is illustratedin Figure 22. Note that since the fundamental domain is a geodesic triangle withangles π/3, π/3, and 0, the Gauss-Bonnet theorem gives Area(X) = π

3 .Some of the most important other examples of arithmetic Fuchsian groups are

finite-index subgroups of ΓZ. For N > 1 the principal congruence subgroup of levelN is defined to be

ΓZ(N) :=g ∈ ΓZ : g ≡ I mod N

.

For example, the level 2 subgroup is just

ΓZ(2) =(

odd eveneven odd

)⊂ PSL(2,Z)

.

Figure 23 shows the standard fundamental domain for this group, which is a geo-desic triangle all of whose vertex angles are 0.

The quotients of H by principal congruence subgroups are denoted by

X(N) := ΓZ(N)\H.These are also called called modular surfaces, but “the modular surface” alwaysrefers to X = X(1). The surface X(2) has genus zero with 3 cusps and no ellipticfixed points, and so is geometrically quite simple. The geometry of X(N) gets more


Figure 23. The fundamental domain and tessellation for ΓZ(2).

complicated as N increases, with the genus of X(N) approximately equal to N3

for N large.There are other arithmetic Fuchsian groups and even other kinds of congruence

subgroups of ΓZ. But number theorists are particularly interested in Maass cuspforms on X(N).

7.2. Spectral theory of the modular surface. For the rest of this sectionwe’ll focus on the modular surface X. First we consider the scattering matrix,which is just a function since X has only a single cusp. The modular group fits theframework of §4.2, and the Eisenstein series is defined as in (4.2) by

E(s; z) :=∑

Γ∞\ΓZ

ys

|cz + d|2s.

To compute the scattering matrix we need to work out the asymptotic expan-sion of E(s; z) as y →∞. We first note that for

γ =(a bc d

)∈ ΓZ,

the coset of Γ∞ takes the form

(7.1) Γ∞γ =(

a+ cZ b+ dZc d

).

Note that determinant condition ab−cd = 1 implies gcd(c, d) = 1. To each relativelyprime pair (c, d) ∈ Z2 we can associate an element of ΓZ. If we assume that c ≥ 0,to remove the sign ambiguity, then (7.1) shows that the corresponding coset inΓ∞\ΓZ is fixed by the choice of (c, d). We conclude that

(7.2) E(s; z) = ys +∞∑c=1

∑d∈Z

gcd(c,d)=1

ys

|cz + d|2s.

Recall that the non-zero Fourier modes of E(s; z), with respect to the periodicx variable, vanish exponentially as y → ∞. So to find the scattering matrix, weneed only to consider the 0-mode,

c0(s; y) :=∫ 1

0

E(s; z) dx.

36 D. BORTHWICK

In the sum in (7.2), we write d = cn + r, where n ∈ Z and 1 ≤ r < c andgcd(r, c) = 1. For fixed c ≥ 1 the contribution to the zero-mode is∑

1≤r<cgcd(r,c)=1

∑n∈Z

∫ 1

0

ys

|c(z + n) + r|2sdx =

∑1≤r<c

gcd(r,c)=1

ys

c2s

∫ ∞−∞

[x2 + y2]−s dx

=∑

1≤r<cgcd(r,c)=1

y1−s

c2sΓ( 1

2 )Γ(s− 12 )

Γ(s).

Since the r dependence has been integrated out, the sum over r just yields a factor ofthe totient function φ(c) := #1 ≤ r < c : gcd(r, c) = 1. With these calculationsthe formula for the zero mode becomes

c0(s; y) = ys + y1−sΓ( 12 )Γ(s− 1

2 )Γ(s)

∞∑c=1

φ(c)c2s

.

The sum can now be evaluated in terms of the Riemann zeta function by theDirichlet series identity,

∞∑n=1

φ(n)nz

=ζ(z − 1)ζ(z)

.

This yields

(7.3) c0(s; y) = ys + y1−sΓ( 12 )Γ(s− 1

2 )Γ(s)

ζ(2s− 1)ζ(2s)

From the asymptotic (7.3) and the definition (4.6) we instantly read off thescattering matrix

ϕ(s) =Γ( 1

2 )Γ(s− 12 )

Γ(s)ζ(2s− 1)ζ(2s)

.

Of course, our computations were valid only in the realm of absolute convergenceof these series, namely Re s > 1. But the result clearly extends meromorphically tos ∈ C. We conclude that the scattering poles of X correspond precisely to the non-trival zeros of ζ(2s), plus the pole at s = 1 coming from ζ(2s−1) (which correspondsto the default eigenvalue λ = 0). The resulting resonance picture is shown inFigure 24 (with the locations of cusp forms taken from numerical computations ofHejhal [19]).

The Selberg trace formula easily shows that the the modular surface must havemany cusp forms that do not appear in the scattering matrix. The Weyl-Selbergasymptotic formula cited in §6.2 implies that

#

resonances with |s− 12 | ≤ t

∼ t2

6.

By Riemann’s asymptotic formula for the number of zeros of the zeta function(proven by Mangoldt in 1905), we also know that

#

scattering poles with |s− 12 | ≤ t

∼ 2t log t

π.

Therefore almost all of the resonances are actually cusp forms; the cusp formsdominate the spectrum.


10

−10

12 1

Figure 24. The resonances of the modular surface. The cuspforms lie on Re s = 1. The resonances with Re s = 1

4 come fromRiemann zeros.

7.3. Hecke operators. The abundance of cusp forms in the modular surfacenoted at the end of the last section holds for other arithmetic hyperbolic surfaces aswell. This contrasts sharply with the general hyperbolic case, where cusp forms arethought to be very rare and perhaps do not occur at all. The natural question toask at this point is what makes the spectral theory of arithmetic surfaces so special.Why shouldn’t these surfaces behave just like surfaces associated to general cofiniteFuchsian groups?

The answer is that the arithmetic surfaces posses extra, hidden symmetries, inthe form of a family of transformations called Hecke operators. For the modulargroup the definition of a Hecke operator is fairly simple. For each n ∈ N we definean operator Tn on L2(ΓZ\H, dA) by

(7.4) Tn f(z) :=1√n

∑ad=n

d−1∑b=0

f

(az + b

d

).

Since it’s built as an average over isometries of H, it’s clear that Tn commuteswith the Laplacian ∆H. What is not immediately obvious is that Tnf retains itsinvariance under ΓZ so that its action descends to the quotient ΓZ\H.

We’ll give the algebraic proof of this (which is not very hard) in a moment.But to understand how special these operators are from a geometric point of view,it is useful consider an example first. Consider

(7.5) T3 f(z) =1√3

[f(3z) + f

(z3

)+ f

(z + 1

3

)+ f

(z + 2

3

)].

Note that invariance under τ : z 7→ z + 1 is immediately obvious, so it suffices toconsider the other generator, σ : z 7→ −1/z. Under this symmetry, we have

T3 f

(−1z

)=

1√3

[f

(−3z

)+ f

(− 1

3z

)+ f

(z − 1

3z

)+ f

(2z − 1

3z

)].

Using the invariance of f under ΓZ, we see that the first 2 terms just switchedplaces:

f

(−3z

)= f

(z3

), f

(− 1

3z

)= f(3z).

38 D. BORTHWICK

For the other terms we can use the matrix identity,(1 −13 0

)=(

1 −13 −2

)(1 20 3

)(note that the first matrix on the right has determinant 1, and so lies in ΓZ), to seethat

f

(z − 1

3z

)= f

(z + 2

3

).

Similarly,

f

(2z − 1

3z

)= f

(z + 1

3

).

With these identications, we have shown

T3 f

(−1z

)= T3f(z),

and thus T3f is again an automorphic function for ΓZ.We have of course gone through this argument in a rather naive way, just

to emphasize the point that the algebraic structure that gives rise to the Heckesymmetries is very special and delicate. From the correct point of view, the generalproof is much easier. The trick is to define

Υn := A ∈ GL(2,Z) : detA = n,

and consider the coset space ΓZ\Υn.

Lemma 7.1. The set,

An =(

a b0 d

): ad = n, 0 ≤ b ≤ d− 1

(i.e., the set of matrices appearing in (7.4)), contains a complete set of distinctrepresentatives for ΓZ\Υn.

Proof. To prove the lemma we start from

α =(a bc d

)∈ Υn,

and try to find a representative of ΓZα in An. If q = gcd(a, c), then there is anelement of ΓZ with bottom row given by (−c/q, a/q) and we have(

∗ ∗−c/q a/q

)(a bc d

)=(∗ ∗0 ∗

).

So we can always choose a representative with c = 0. Once we assume that c = 0,we must have ad = n, and the only remaining freedom is to apply an element ofΓ∞, (

1 k0 1

)(a b0 d

)=(a b+ kd0 d

).

We can thus fix the representative uniquely by restricting b to the range 0, . . . , d−1.

Corollary 7.2. If f(·) is invariant under ΓZ, then Tnf(·) is also, so Tn mapsL2(Γ\H, dA) to itself.


Proof. Suppose γ ∈ ΓZ and α ∈ An. Then det(αγ) = n, so αγ ∈ Υn.Lemma 7.1 then says that

αγ ∈ ΓZα′,

for some α′ ∈ An. Then, assuming that f is automorphic with respect to ΓZ, wehave

f(αγz) = f(α′z).The map α 7→ α′ induced by γ is bijective, so that

Tn f(γz) =1√n

∑α∈An

f(αγz) =1√n

∑α′∈An

f(α′z) = Tnf(z).

In view of Lemma 7.1, we could rewrite the definition of Tn as

(7.6) Tnf(z) =1√n

∑α∈ΓZ\Υn

f(αz),

since f(αz) clearly depends only on the coset ΓZα generated by α.Another significant feature of the Hecke operators is that they form a commu-

tative ring. Indeed we have a nice composition formula (which explains why thenormalization 1/

√n is used in the definition of Tn):

Lemma 7.3.TmTn =

∑d | gcd(m,n)

Tmn/d2 .

For gcd(m,n) = 1, this follows pretty easily from the definition. For the generalcase, one can first prove that

TpkTp = Tpk+1 + Tpk−1 ,

for p prime and k ∈ N, and then argue by induction. (See [46, §III.3.6] for thedetails.)

One final property is also rather significant from a spectral point of view: self-adjointness. As with the other properties of Hecke operators detailed above, this isnot terribly difficult to prove, but not immediately obvious either. This version ofthe proof is adapted from Shimura [41].

Lemma 7.4. As an operator on L2(ΓZ\H, dA), each Tn is self adjoint.

Proof. By Lemma 7.3 it suffices to consider Tp for p prime. In that case wecan write

Υp = ΓZβΓZ,

for

β =(

1 00 p

).

Let Γβ := ΓZ∩β−1ΓZβ, and choose a set of representatives ηj for Γβ\ΓZ, so that

(7.7) ΓZ =⋃j

Γβηj

is a disjoint union. It’s then easy to deduce a corresponding decomposition

ΓZβΓZ =⋃j

ΓZβηj ,

40 D. BORTHWICK

so that βηj serves as a set representatives for ΓZ\Υp.By (7.6) we then have, for f, g ∈ L2(Γ\H, dA),

〈f, Tpg〉 =1√p

∑j

∫Ff(z)g(βηjz) dA(z)

=1√p

∫∪ηjF

f(z)g(βz) dA(z).

where F is a fundamental domain for ΓZ. By (7.7), the set ∪jηjF is a fundamentaldomain for Γβ , to the expression becomes

〈f, Tpg〉 =1√p

∫Γβ\H

f(z)g(βz) dA(z).

On the other hand, we could make exactly the same argument starting fromβ := pβ−1, since Υp = ΓZβΓZ also. Apply this reasoning gives

〈Tpf, g〉 =1√p

∫Γβ\H

f(βz)g(z) dA(z).

Then, if we note that Γβ = βΓββ−1 and that β gives the same Mobius transforma-tion as β−1, we have

〈Tpf, g〉 =1√p

∫βΓββ−1\H

f(β−1z)g(z) dA(z)

=1√p

∫Γβ\H

f(z)g(βz) dA(z)

= 〈f, Tpg〉

In the proof of Lemma 7.4, we can see a hint of the proper abstract definitionof Hecke operator. The essential ingredient is the fact that the groups ΓZ andβ−1ΓZβ are commensurable, meaning that Γβ and Γβ (as defined above) have finitecodimension in ΓZ. This was the fact that allowed us to decompose ΓZβΓZ as a(finite) collection left cosets and then define a Hecke operator as a sum over thesecosets. A distinguishing feature of general arithmetic groups Γ is the existence of alarge class of groups β−1Γβ commensurable with Γ; we have treated only the caseΓ = ΓZ for simplicity.

Since the Tn commute with each other and with the Laplacian, self-adjointnessimmediately gives the following:

Corollary 7.5. The full ring of Hecke operators Tn together with ∆ can besimultaneously diagonalized.

In particular, we can choose a basis of Maass cusp forms such that

∆φj = λjφj , Tnφj = τj(n)φj .

Indeed, it is conjectured that the cusp spectrum of X is simple, which would implythat Maass cusp forms are automatically Hecke eigenfunctions, with no furtherdiagonalization needed.

In any case, all known Maass cusp forms of the modular surface are simpleeigenvalues, so these eigenfunctions—a few are shown in Figure 25—must have the


Figure 25. Density plots of Maass cusp forms (image by AlexBarnett/Holger Then)

Figure 26. The effect of T3 is to average copies of the functiondrawn from the four shaded regions.

Tn symmetry for each n. The effect of Tn will be to take set of snapshots from theoriginal image and superimpose them after rescaling and translation back to theoriginal fundamental domain. Figure 26 illustrates this for T3; each shaded regionis rescaled to the size of F and then these four pictures are combined. For a Heckeeigenfunction, this procedure must result in a copy of the original image (with thedensity mutiplied by |τj(n)|). It is amusing to stare at the density plots in Figure 25and imagine taking these recombinations for arbitrary n.

7.4. L-function connection. One primary reason that Maass cusp forms areimportant in number theory is the connection to Dirichlet L-functions. These areseries which generalize the Riemann zeta function.

Suppose that φ is a Maass cusp form, with λ = ζ(1− ζ) and Hecke eigenvaluesgiven by τφ(n). The standard L-function associated to φ is

Lφ(s) :=∑n

τφ(n)n−s,

42 D. BORTHWICK

which converges for Re s sufficiently large. By Lemma 7.3, the Hecke eigenvaluessatisfy

τφ(m)τφ(n) =∑

d | gcd(m,n)

τφ(mn/d2).

From this product relation, one can deduce an Euler product formula of degree 2for Lφ(s),

Lφ(s) =∏p

11− τ(p)p−s − p−2s

.

Consider the Fourier decomposition of φ(z) in the (periodic) x-variable,

φ(z) =∑k 6=0

ck√yKζ−1/2(2π|k|y)e2πikx,

where ζ is the location of the resonance associated to the cusp form. The form ofthe coefficients is dictated by the equation (4.2). We drop the n = 0 terms anduse only the K-Bessel solution because φ is a cusp form and must therefore decayrapidly as y →∞. The action of the Hecke operator is then

Tnφ(z) :=∑k 6=0

1√n

∑ad=n

d−1∑b=0

ck

(ayd2

)1/2

Kζ−1/2(2π|k|ay/d)e2πik(ax+b)/d.

Note that b appears only in the final exponent and that this term will average tozero unless d divides k. If d|k then the sum over b simply contributes a factor of d,so that

Tnφ(z) :=∑k 6=0

∑ad=nd|k

ck√yKζ−1/2(2π|k|ay/d)e2πikax/d.

Now we can rewrite the sum in terms of the integer m = ka/d,

Tnφ(z) :=∑m 6=0

∑ad=na|m

cdm/a√yKζ−1/2(2π|m|y)e2πimx.

Using the fact that φ was a Hecke eigenfunction, we then have the relation

τφ(n)cm =∑

a | gcd(m,n)

cnm/a2 .

In particular, by setting m = ±1 we deduce

c±n = τφ(n)c±1.

Using the integral identity,∫ ∞0

tα−1Kν(t) dt = 2α−2Γ(α−ν2 )Γ(α+ν2 ),

we can see that∫ ∞0

φ(iy) ys−3/2 dy = c Γ( s−ζ+1/22 )Γ( s+ζ−1/2

2 )∞∑k=1

τφ(k)(πk)−s

= cπ−sΓ( s−ζ+1/22 )Γ( s+ζ−1/2

2 )Lφ(s),


where c = (c1 + c−1)/4. Since φ is modular, we have φ(iy) = φ(iy−1) in particular,so that we can deduce∫ ∞

0

φ(iy) ys−3/2 dy =∫ ∞

0

φ(iy) y−s−1/2 dy,

by changing variables y → 1/y.The modularity of φ therefore implies the functional relation

π−sΓ( s−ζ+1/22 )Γ( s+ζ−1/2

2 )Lφ(s) = πs−1Γ( 1−s−ζ+1/22 )Γ( 1−s+ζ−1/2

2 )Lφ(1− s)

s In particular, this shows that Lφ(s) admits an analytic continuation to s ∈ C.Many spectral questions about Maass cusp forms are related to properties of

L-functions. For example, Artin’s conjecture connects two-dimensional representa-tions of Galois groups, via the associated L-functions, to Maass cusp forms witheigenvalue 1

4 .

7.5. Quantum ergodicity. One fundamental question that spectral theoristsare interested in is the asymptotic distribution of eigenfunctions, and in particularhow this distribution relates to the underlying classical geodesic flow. Althoughsuch questions are originally motivated by physics, they are also quite natural inthe context of arithmetic hyperbolic surfaces, where they can be interpreted interms of certain special values of L-functions.

A surface is said to be classically ergodic if the geodesic flow is ergodic. Thismeans that the all invariant measurable sets have either zero or full measure, or,equivalently, all functions invariant under the geodesic flow are constant a.e. Thefact that a hyperbolic surface is classically ergodic is a famous result of E. Hopffrom the 1930’s. (Anosov and Sinai later showed that all compact manifolds of neg-ative curvature have this property). Classical ergodicity is expected to correspondwith ‘quantum ergodicity’, which means roughly that eigenfunctions will becomeequidistributed as j →∞.

To interpret this notion of equidistribution precisely, consider a compact Rie-mannian manifold (M, g) with Laplacian ∆ and normalized eigenfunctions φj.We can associate probability measures directly to these eigenfunctions by setting

φj 7→ νφj = |φj |2 dV.

However, it is more customary to use a more refined phase space version of thismeasure, called the microlocal lift. This is a measure µφj on the unit cotangentbundle, defined by

µφj (a) = 〈Op(a)φj , φj〉,where a ∈ C∞(S∗M) and Op(a) is some quantization of a as a zeroth order pseu-dodifferential operator on M .

A manifold is said to be quantum ergodic if µφj approaches Liouville measuredµL on S∗X. This is related to classical ergodicity by the following result:

Theorem 7.6 (Shnirelman [42], Colin de Verdiere [8], Zelditch [50]). If acompact Riemannian manifold (M, g) is classically ergodic, then there exists a sub-sequence φjk whose indices have density one in N, such that

µφjk (a)→∫S∗M

a dµL,

for any a ∈ C∞(S∗M).

44 D. BORTHWICK

If the result is extended to include the decay of off-diagonal correlations, it iseven possible to prove a converse statement. We refer the reader to the survey ofZelditch [52] for more background on these results.

There is a version of quantum ergodicity for non-compact hyperbolic surfacesof finite volume; see Zelditch [51]. In this case, the property has two alternateforms, based either on measures associated to the discrete spectrum as above, oron the analogous measures coming from the continuous spectrum, defined by theEisenstein series E( 1

2 + it; z). (In the latter case the quantum limit is t→∞.)The quantum unique ergodicity conjecture (QUE), made by Rudnick-Sarnak

[37], postulates that for compact manifolds of negative curvature, the full sequenceµφj converges to Liouville measure, so that the limiting quantum measure is unique.For the case of continuous spectrum (Eisenstein series) on the modular surface, QUEwas established by Luo-Sarnak [26] and Jakobson [21].

More recently, there has been a great deal of development in the arithmeticcase, starting with Lindenstrauss’s proof [25] of QUE for Hecke eigenfunctions inthe compact arithmetic case. Soundararajan [43] extended this result to the non-compact case, so that QUE is now fully proven for arithmetic hyperbolic surfaces.

There is also a ‘holomorphic’ version of QUE for hyperbolic surfaces, whichconcerns the equidistribution of holomorphic Hecke eigenforms of weight k as k →∞. This has also been established quite recently, in work of Holowinsky andSoundararajan. For an account of these recent developments see the expositoryarticles of Sarnak [39] and Zelditch [52].

Notes. Much of the material in §7 was adapted from Apostol [1], Terras [46],or Sarnak [38].

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46 D. BORTHWICK

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Department of Mathematics and Computer Science, Emory University, Atlanta,Georgia, 30322, USA

E-mail address: [email protected]

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