Introduction to Spectral Theory on Hyperbolic Surfaces

Transcription

1 Introduction to Spectral Theory on Hyperbolic Surfaces David Borthwick Contents 1. Hyperbolic geometry 1 2. Fuchsian groups and hyperbolic surfaces 4 3. Spectrum and resolvent Spectral theory: finite-area case Spectral theory: infinite-area case Selberg trace formula Arithmetic surfaces 33 References Hyperbolic geometry In complex analysis, we learn that the upper half-plane H = {Im z > 0} has a large group of conformal automorphisms, consisting of Möbius transformations of the form (1.1) γ : z az + b cz + d, where a, b, c, d R and ad bc > 0. The Schwarz Lemma implies that all automorphisms of H are of this type. Since the transformation is unchanged by a simultaneous rescaling of a, b, c, d, the conformal automorphism group of H is identified with the matrix group PSL(2, R) := SL(2, R)/{±I}. Under the PSL(2, R) action, H has an invariant metric, ds 2 H = dx2 + dy 2 y 2, often called the Poincaré metric. To see the invariance, it is convenient to switch to the complex notation, where ds 2 H = dz 2 (Im z) Mathematics Subject Classification. Primary 58J50, 35P25; Secondary 47A40. Supported in part by NSF grant DMS

2 2 D. BORTHWICK The pullback of the metric tensor under the map (1.1) is γ (ds 2 H) = γ (z) dz 2 (Im γ(z)) 2, so the invariance of ds 2 H follows from the easily checked identities, γ 1 Im z (z) =, Im γ(z) = (cz + d) 2 cz + d Geometry of the hyperbolic plane. Because PSL(2, R) acts transitively on H, by isometries, it is immediately clear that the Gaussian curvature K must be constant for ds 2 H. Using the formula provided by Gauss s Theorema Egregium, we can easily check that this value is K = 1. This is the defining feature of a 2-dimensional hyperbolic metric, and for this reason the Poincaré upper half space H is also called the hyberbolic plane. In this section we will introduce just a few of the most basic geometric concepts in H. First, the Riemanian measure associated to ds 2 H is da(z) = dx dy y 2. Obviously, this metric inherits PSL(2, R)-invariance from the metric. The metric also determines angles, and conveniently the angles measured with respect to ds 2 H are the same as Euclidean angles, since the metrics are conformally related (ds 2 H = y 2 ds R 2). A geodesic is a curve which is locally length minimizing within the class of piecewise smooth curves. It is not hard to see, by direct computation, that vertical lines 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 ẋ2 1 + ẏ l(η [0,1] ) = 2 dt. y 0 Clearly the minimum is achieved by setting ẋ 0 and restricting to ẏ > 0, which shows that the y-axis is a geodesic and gives the hyperbolic distance (1.2) d H (ia, ib) = log b 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 sphere C { }. Vertical lines then correspond to circles on the Riemann sphere that intersect the boundary H := R { } at right angles and pass through. Since Möbius 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 circles intersecting H = R { } orthogonally, as illustrated in Figure 1. By applying PSL(2, R) transformations to (1.2), it is not hard to work out the general formula for the distance function in H explicitly, d H (z, w) = log z w + z w z w z w. Now consider a geodesic triangle ABC with vertex angles α, β, and γ. as in Figure 2. Vertices are allowed to lie on H, in which case the angle is zero.

3 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 3 Figure 1. Geodesics in H α γ β α β γ Figure 2. Geodesic triangles The Gauss-Bonnet theorem (for any two-dimensional metric) says that the total curvature of a triangle T is equal to the angle deficit, K da = α + β + γ π. T Thus the fact that the metric is hyperbolic is equivalent to the area formula: for any geodesic triangle. Area(T ) = π α β γ, 1.2. Classification of isometries. Elements of PSL(2, R) are classified according to their fixed points. For γ PSL(2, R), the fixed point equation z = γz is quadratic: cz 2 + (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, as shown in Figure 3. The conjugacy class is determined by the rotation angle. (2) parabolic: a single degenerate fixed point, which must lie in H. The action is illustrated in Figure 4. Any parabolic transformation is conjugate to the map z z + 1 (for which is the degenerate fixed point). (3) hyperbolic: two distinct fixed points in H. The transformation maps points away from one fixed point and toward the other, as shown in Figure 5. A hyperbolic transformation is conjugate to the dilation z e l z for some l R.

4 4 D. BORTHWICK Figure 3. An elliptic transformation rotates hyperbolic circles around a fixed center. Figure 4. A parabolic transformation fixes a point on H. Figure 5. A hyperbolic transformation translates between two fixed points on H. The axis (unique fixed geodesic) is shown in gray. Notes. For background material on two-dimensional Riemannian geometry, see do Carmo [10] or Pressley [34]. Ratcliffe [35] gives a thorough intoduction of hyperbolic geometry. 2. Fuchsian groups and hyperbolic surfaces A Fuchsian group is a subgroup Γ PSL(2, R) that is discrete with respect to the matrix topology (which is equivalent to Euclidean R 4 ). It follows from the discreteness that Γ must act properly discontinuously on H, meaning that each orbit Γz is locally finite. The converse is also fairly easy to argue, as follows. If Γ acts properly discontinuously on H, then the triangle inequality implies that only finitely many points in any compact neighborhood could be fixed by Γ {I}. But if there is a sequence T k I in Γ, and w H is not fixed by Γ {I}, then the sequence T k w would contradict the proper discontinuity of the action of Γ. Thus Fuchsian groups are precisely the subgroups of PSL(2, R) acting properly discontinuously on H.

5 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 5 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 in H. Figure 6 shows an example of an orbit. Note the accumulation on H; this does not contradict the properly discontinuity of the action because the accumulation occurs only in the Riemann sphere topology, not in the topology of H. The quotient is a smooth surface if and only if Γ has no elliptic elements. The term surface is frequently used for Γ\H, even in the non-smooth case; this usage presumably comes from the interpretation of Γ\H as an algebraic variety. Since Γ acts by hyperbolic isometries, the quotient inherits a hyperbolic metric from H. A fundamental domain for Γ is a closed region F such that the translates of F under Γ tesselate H. An example is shown in Figure 7. The fact that Γ is discrete implies that this tesselation will be locally finite (a compact set meets only finitely many translates of F) Automorphic forms and functions. A function f on the quotient Γ\H is 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 applications one generally considers vector valued-functions H V along with a unitary representation ρ : Γ GL(V ). The condition for an automorphic function is then f(γz) = ρ(γ)f(z). For simplicity, we will consider only the trivial representation in these notes. Another possible generalization is an automorphic form of weight k, for which the transformation rule is f(γz) = (cz + d) k f(z). Here c and d are matrix elements of γ as in (1.1). Since γ (z) = (cz + d) 2, the automorphic of even weight are sections of L k/2, where L is the holomorphic tangent bundle over Γ\H. In some contexts, the term automorphic form carries with it a restriction to meromorphic or holomorphic functions. Indeed, this is the classical usage of the term.

6 6 D. BORTHWICK 2.2. Uniformization. The Uniformization Theorem for hyperbolic surfaces essentially has two parts. The first says that any metric on a surface is conformally related to a hyperbolic metric. Theorem 2.1 (Koebe, Poincaré). For any smooth complete Riemannian metric on a surface, there is a conformally related metric of constant curvature. Given a general Riemannian surface (M, g), the equation for the curvature of g := e 2u g, with u C (M) is K g = e 2u (K g + u), where is the (positive) Laplacian operator, := div grad. The proof of Theorem 2.1 amounts to finding a solution u for which K g 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 met solutions always exist. The second part of uniformization is the characterization of surfaces of constant curvature by their universal covering spaces. We can use a global rescaling to restrict our attention to K = 1, 0, or 1, and then there is only one possibility for each case. Theorem 2.2 (Hopf). Up to isometry and global rescaling, the only smooth, complete, simply connected surfaces of constant curvature are the sphere S 2, the Euclidean plane R 2, and the hyperbolic plane H 2. The local part of the proof is fairly straightforward: in geodesic polar coordinates any metric takes the form dr 2 + φ 2 dθ 2, with φ r as r 0. The curvature is given by K = r 2 φ/φ in these coordinates. Setting K = 1, 0, 1 results in a 2nd order ODE with unique solutions, φ = sin r, r or sinh r, respectively. The global result requires the fact that a local isometry of complete Riemannian manifolds must 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 < 0 is conformally related to a hyperbolic quotient Γ\H. It is thus fair to say that hyperbolic surfaces provide the uniformizing models for most surfaces. Riemannian orbifolds are considered good if they arise as quotients of a smooth Riemannian manifold under a properly discontinuous group action, and bad if not. There is a uniformization theorem for good 2-dimensional orbifolds without boundary which says that they are isomorphic as orbifolds to the quotient of S 2, R 2, or H 2 by some discrete group of isometries (see Thurston [47]). As we noted above, the term hyperbolic refers specifically to curvature. However, because the hyperbolic isometries of H are precisely the conformal automorphisms, any quotient Γ\H inherits a natural complex structure from H. In other words, a hyperbolic surface is also a Riemann surface, which means a onedimensional complex manifold. The terms hyperbolic surface and Riemann surface are sometimes used interchangeably, especially for smooth compact surfaces, where the only non-hyperbolic Riemann surfaces are the sphere and tori. We have described the geometric Uniformization Theorem, but there is also a complex version in the context of Riemann surfaces This says that any simply connected Riemann surface is holomorphically equivalent to the either the Riemann sphere C { }, the complex plane C, or the upper half-plane H.

7 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 7 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 limit points of orbits of Γ, in the Riemann sphere topology. Since Γ acts properly discontinuously, orbit points can only accumulate in H. One can see this accumulation in Figure 6. As long as w H is not an elliptic fixed point, it suffices to take the set of 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 to the following: Theorem 2.3 (Poincaré, Fricke-Klein). Any Fuchsian group is of one of the following 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 fixed point of the generator of the group, as shown on the left in Figure 8. For the particular case of Γ = z z + 1, we have Λ(Γ) = { }. Similarly, the orbit of a hyperbolic cyclic group accumulates at both of the fixed points. These groups are the only possibilities where Λ(Γ) has 1 or 2 points. All other elementary groups are finite 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 are precisely the Fuchsian groups of the first kind. All arithmetic surfaces are in this class Geometric finiteness. A Fuchsian group is geometrically finite if Γ admits a fundamental domain that is a finite-sided convex polygon. For such a group, the Dirichlet domain, D w := {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 domain with finitely many sides. The fundamental domain shown in Figure 7 is actually a Dirichlet 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 compact surface with finitely many punctures).

8 8 D. BORTHWICK z l(γ) γz π(z) Figure 9. The axis of a hyperbolic transformation descends to a closed 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 hyperbolic surfaces is only tractable in general for geometrically finite Γ. The reason is that we lose control over the geometry at infinity for geometrically infinite surfaces. A theorem of Siegel says that all groups of the first kind are geometrically finite (see Katok [23, Thm ]). But for groups of the second kind (i.e. infinite area surfaces) we must insist on this condition Geometric features. One of the most appealing aspects of the theory of hyperbolic surfaces is the fact that we can associate distinct geometric features to each class of elements of the group Γ. For example, if γ Γ is hyperbolic, then there is a unique geodesic connecting the two fixed points of γ, called the axis of γ. (In Figure 5, the axes were highlighted in 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 e l z for some l > 0. Then the vertical geodesic from 0 is the axis, and on this geodesic g acts as translation by l. This value l(γ) is called the translation length of γ. For general γ PSL(2, R) we have (2.1) l(γ) = 2 cosh 1 ( tr γ /2) In the quotient Γ\H, the axis of a hyperbolic element γ descends to a closed geodesic of length l(γ), as illustrated in Figure 9. This gives a 1-1 correspondence: closed geodesics of Γ\H conjugacy classes of hyperbolic elements of Γ. The parabolic elements of Γ create cusps in Γ\H. Any parabolic element T fixes some point p H. This element will also fix the interior O of a horocycle, a circle tangent to H at p (shown on the left in Figure 10). A cusp is defined as a quotient of the form T \O. A portion of the cusp is shown embedded in R 3 on the right in Figure 10; the full cusp is infinitely long, although of finite area. Because any parabolic element is conjugate to the translation z z + 1, all cusps are isometric to 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 parabolic subgroups of Γ. An elliptic element of Γ fixes a point in Γ\H and so gives rise to a conical singularity in Γ\H. This is just as in the definition of an orbifold, except that in our case the conical point is always generated by a global symmetry, not just a

9 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 9 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 is shown embedded in R 3 in Figure 11. There is a 1-1 correspondence: conical points orbits of elliptic fixed points of Γ Hyperbolic ends. If a geometrically finite hyperbolic surface Γ\H is not compact, then it will have ends. Under the homeomorphism that identifies Γ\H with a compact surface with punctures, the ends are defined as neighborhoods of the punctures. Since Γ\H carries a complete Riemannian metric, the punctures are necessarily moved out to infinity. The main reason that spectral theory is so tractable for geometrically finite hyperbolic surfaces is that the ends are easily classified geometrically. There are essentially only two types. The first possibility is the cusp end, as seen above in Figure 10. In normal coordinates the metric for a cusp end is (2.2) ds 2 = dr 2 2r dθ2 + e (2π) 2, where r 0, θ R/2πZ. There is no canonical location for the horocycle boundary of the cusp, but it is always possible to take the boundary length equal to at least 1 (see [3, Lemma 2.12]). So in (2.2) we can make {r 0} the standard choice of domain. The second type of hyperbolic end is the funnel. Consider a hyperbolic element γ PSL(2, R). The quotient γ \H is called a hyperbolic cylinder. The axis of γ

10 10 D. BORTHWICK Figure 12. A portion of the fundamental domain intersecting H in an interval descends to a funnel. Nielsen region = convex core cusps funnels K Figure 13. Decomposition into core plus cusps and funnels. corresponds to a single closed geodesic at the neck of the cylinder. A funnel is half of a hyperbolic cylinder, bounded by this closed geodesic. The canonical funnel metric is (2.3) ds 2 = dr 2 + l 2 cosh 2 r dθ2 (2π) 2, where r 0, θ R/2πZ and the parameter l gives the length of the bounding geodesic at r = 0. Figure 12 shows a portion of a funnel embedded in R 3. The full funnel 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 C 1 C nc F 1 F nf, where K is a compact orbifold with boundary, the C i s are cusps with boundary length 1, and the F j s are funnels with boundary lengths l 1,..., l nf. The subset K would be called the compact core of X, while K C 1 C nc is called the Nielsen region, as shown in Figure 13. The Nielsen region is also the convex core, meaning the smallest geodesically convex subset of X. If X is a noncompact 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 piecing together those portions of the domain that meet H. The details are essentially given in Fenchel-Nielsen [13], or see the proof in [3, Thm. 2.13].

11 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 11 The elementary cases not covered by Theorem 2.5 are easily understood. There is H itself, of course, whose end carries the metric (2.4) dr 2 + sinh 2 r dθ 2. We could also have quotients of H by finite elliptic groups, whose metrics would be given 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 hyperbolic or parabolic cyclic groups. The former case gives a hyperbolic cylinder, which is the union of two funnels and so fits the framework of Theorem 2.5 with K given by a circle. The parabolic cylinder is a special case. Its small end is a cusp, while the horn end carries the metric dr 2 + e 2r dθ2 (2π) 2, for r > 0. The hyperbolic plane, funnel, and horn all have the same exponential asymptotic behavior, but they are distinct as isometry types. The planar and horn ends 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 and concise 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 spectral theory of hyperbolic surfaces in the 1950 s. The essential idea was to bring techniques from harmonic analysis into the study of automorphic forms. Spectral theory was of course already a well-established topic in physics at that point, but it was studied mostly in the Euclidean context (i.e. domains, obstacles, or potentials in R n ). It turns out that the relation between hyperbolic surfaces and Fuchsian groups makes their spectral theory much easier to analyze than some of these more standard physical situations. Moreover, hyperbolic surfaces proved to be of great physical interest as relatively simple models for which the underlying classical dynamics is chaotic. The Selberg trace formula became something of a beacon to quantum physicists, who could see in it an exact version of the correspondence principle of quantum mechanics The Laplacian. The Laplacian operator associated to the hyperbolic metric on H (also commonly called the Laplace-Beltrami operator in a geometric context) is ( ) (3.1) := y 2 2 x y 2. The Laplacian s essential property is invariance under local isometry, so for H we have γ = γ, for γ PSL(2, R). This means that descends to an operator on the quotient Γ\H, even when the quotient is not smooth. At least, the definition of acting on smooth functions on Γ\H is unambigious. In order to do spectral theory, however, we need to interpret

12 12 D. BORTHWICK as an (unbounded) self-adjoint operator on L 2 (Γ\H, da), where da is the hyperbolic measure introduced above. This means that we need to choose a appropriate domain within L 2 (Γ\H, da). There is a natural procedure for this, called the Friedrichs extension. We start from the domain { } D := f C0 (Γ\H) : f and f L 2 (Γ\H, da), and then the Friedrichs extension produces a larger domain in L 2 (Γ\H, da) on which 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 the H 2 Sobolev space. If Γ\H has conical points, then the Friedrichs extension is just one of a range of possible self-adjoint extensions. The choice of extension has a strong effect on the spectral theory in general, so it is important to specify in these cases. In the arithmetic 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 L 2. Physicists generally don t include the minus, but instead they write in spectral formulas. Thus, wherever one puts the minus sign, the common convention is to study a positive spectrum Eigenvalues. We say that λ is an eigenvalue of on Γ\H, with eigenvector φ, if φ L 2 (Γ\H, da) and (3.2) φ = λφ. The L 2 restriction is irrelevant in the compact case, because any solutions of the eigenvalue equation will be smooth by elliptic regularity. But in the non-compact case we can easily have smooth solutions of (3.2) that are not eigenfunctions because they grow too rapidly at infinity. Note that the action of could be extended to automorphic forms as well as functions. In the context of automorphic forms, eigenvectors of the Laplacian on Γ\H are called Maass forms. This terminology is used primarily in the arithmetic context. If Γ\H is compact, then the eigenvalues fill out the spectrum, by the following general result. Theorem 3.1 (Spectral theorem for compact manifolds). If (M, g) is a compact smooth Riemannian manifold, with Laplacian g defined by the Friedrichs extension, then there exists a complete orthonormal basis for L 2 (M, dv g ) such that with g φ j = λ j φ j, 0 = λ 0 < λ 1 λ 2. The proof is not exactly elementary, but is straightforward once a little functional analysis background is established. The key facts are that the exact domain of g is the Sobolev space H 2 (M, dv g ), and that H 2 (M, dv g ) is a compact subspace of L 2 (M, dv g ). This implies that ( g + 1) 1 is a compact self-adjoint operator on L 2 (X, dv g ). The theorem then follows from the spectral theorem for compact selfadjoint operators, which is covered in any basic functional analysis course. One could also use the heat kernel in place of the resolvent in this argument, see Buser [7]

13 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 13 λ + iε 0 λ iε Figure 14. Limits of the resolvent in Stone s formula For compact Riemannian manifolds we have some general theorems about the eigenvalue spectrum. For example, we have the Weyl asymptotic formula d/2 vol(m, g) λ k (4π) Γ( n 2 + 1) k2/d, as k, where d = dim M. The standard proof is by analyzing the small-time behavior of the trace of heat kernel of Resolvent. If Γ\H is non-compact, then the spectral theory of is necessarily more complicated. By definition, the spectrum σ( ) consists of those values of λ for which the resolvent ( λ) 1 fails to exist as a bounded operator on L 2 (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 separable Hilbert space H, there exists a measure space (Ω, µ), where Ω is a union of copies of R, a map W : L 2 (Ω, µ) H, and a real-valued measurable function a, such that W conjugates A to multiplication by a, i.e. for f L 2 (Ω, dµ). W 1 AW f = af The Spectral Theorem defines a functional calculus for the operator A, meaning we 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 finitedimensional 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 linear algebra that it is useful to introduce spectral projectors onto the eigenspaces of a matrix. For self-adjoint operators on a Hilbert space, we can use the spectral theorem to define such projectors, essentially by taking the limit of h(a) as h approaches the characteristic function of an interval [α, β] R. The result is the following: Theorem 3.3 (Stone s formula). For a self-adjoint operator A, the spectral projector onto [α, β] is given by 1 β [ P α,β = lim (A λ iε) 1 (A λ + iε) 1] dλ. ε 0 + 2πi α Figure 14 shows that limits of the resolvent that appear in the integrand of Stone s formula. The projector P α,β is actually defined as the average of the spectral projectors onto [α, β] and (α, β). This makes a difference only if there is point spectrum at one of the endpoints. From the abstract statement, it is a little hard to see how Stone s formula could be useful. And indeed, for the general complete Riemannian manifold, we can t get

14 14 D. BORTHWICK much from this result without some fairly strong extra assumptions. However, if we impose some asymptotic structure on the metric near infinity, then in many interesting 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 y s = s(1 s)y s gives a strong hint that λ = s(1 s) will be a natural substitution for the spectral parameter. Accordingly, we start from the definition R H (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 [ 1 2, 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))r H (s; z, w) = δ(z w). After switching to polar coordinates we can separate variables, and the radial equation is of hypergeometric type. The resulting formula for the Green s function is (3.3) R H (s; z, z ) = 1 Γ(s) 2 4π 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 R H (s; z, z ) extends to meromorphic of s C, confirmation that the spectral parameter was chosen wisely. Note that this continuation does not contradict the fact that ( λ) 1 should fail to exist when we hit σ( ), because our formula for R H (s) gives an unbounded operator for Re s 1 2. The same picture holds for Γ\H for any geometrically finite Γ, although meromorphic continuation is not so easily proven. Theorem 3.4. The resolvent R(s) = ( s(1 s)) 1 admits a meromorphic continuation to s C as an operator on C 0 (X). The idea of approaching the spectral theory of hyperbolic surfaces through meromorphic continuation of the resolvent was pioneered by Faddeev [12], who proved the result for finite-area hyperbolic surfaces. Selberg s approach to the spectrum was based on the continuation of Eisenstein series, which we ll introduce below. That method is restriced to the case of hyperbolic quotients, whereas the resolvent approach exemplifies the more general methods of spectral theory. In the full case where Γ\H possibly has cusps and also infinite area, Theorem 3.4 was proven by Guillopé-Zworski [15], using a parametrix construction inspired by Mazzeo-Melrose [28]. Here is an outline of the proof: (1) In the interior, we can use R(s 0 ) for some fixed large Re s 0 to approximate R(s). (2) In cusps and funnels, use the resolvents for cylindrical hyperbolic and parabolic quotients, which can be constructed explicitly in terms of hypergeometric functions, to construct a model resolvent R 0 (s).

15 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 15 λ s Figure 15. The transformation from λ to s. The arrows show the limits taken in Stone s formula. (3) Paste the model resolvents together using nested cutoffs χ j C 0 (Γ\H), with χ j+1 = 1 on supp χ j, to get the parametrix (4) Compute the error M(s) := χ 2 R(s 0 )χ 1 + (1 χ 0 )R 0 (s)(1 χ 1 ). ( s(1 s))m(s) = I K(s), and show that K(s) is compact on a weighted L 2 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 of hyperbolic ends. Indeed, the result of [15] allows the metric to be arbitrary within a compact set Spectrum of. As already noted in Theorem 3.1, in the compact case the eigenvalue spectrum is the whole spectrum. In this case the meromorphic continuation of the resolvent has poles only in the set {Re s = 1 2 } [ 1 2, 1], at the points where s(1 s) σ( ). Away from these poles, R(s) is bounded on L 2 (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 the resolvent, corresponds to s [ 1 2, 1] {Re s = 1 2 }, as shown in Figure 15. By analyzing the behavior of R(s) as s approaches the critical line Re s = 1 2, use the structure of the resolvent obtained through the parametrix construction, we can deduce the properties of the spectral projectors. For (α, β) [ 1 4, ), the substitution λ = s(1 s) transforms Stone s formula into (3.4) P α,β = 1 β 1/4[ 2πi R( 1 2 iξ) R( 1 2 ]2ξ + iξ) dξ. α 1/4 In the non-compact case, we can deduce from this formula that the projectors P α,β will have infinite rank for (α, β) [ 1 4, ). This is the defining condition for continuous spectrum. Moreover, we can see easily that the spectrum is absolutely continuous, meaning that the spectral projection onto a set of Lebesgue measure zero that contains no eigenvalue will vanish. As in the compact case, the continued resolvent R(s) will have poles in {Re s = 1 2 } [ 1 2, 1] corresponding to points where s(1 s) lies in the discrete spectrum. If these poles lie on the critical line Re s = 1 2,

16 16 D. BORTHWICK Figure 16. A portion of the resonance set for a hyperbolic surface of genus zero with 3 funnel ends. then the corresponding eigenvalues lie inside the continuous spectrum and are called embedded eigenvalues. It turns out that embedded eigenvalues can be ruled out if Γ\H has at least one funnel, so they occur only for Fuchsian groups of the first kind. 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, ) and discrete spectrum contained in [0, ). (3) If Γ\H has infinite-area, then there are no embedded eigenvalues, i.e. the discrete spectrum is finite and contained in (0, 1 4 ). We had already noted (1) as a general result for compact manifolds in Theorem 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 continuation. In the non-compact case the resolvent may have poles for Re s < 1 2 that do not correspond to eigenvalues. The full collection of poles of R(s) (including those coming from eigenvalues) is the set R Γ of resonances of Γ\H. These are counted with a multiplicity defined as the rank of the residue operator at the pole. Figure 16 shows a numerical calculation of the resonance set for a simple hyperbolic surface. Notes. Reed-Simon [36] is a standard reference for the basic spectral theory used 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-area case, where Γ\H has at least one cusp but no funnels. Arithmetic surfaces, which we will discuss later, fall into this category, so this is the case of greatest importance in number theory. For simplicity, let us start by assume that Γ\H has a single cusp. By conjugating Γ if necessary, we can assume that this cusp corresponds to a parabolic fixed point

17 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES Figure 17. Standard fundamental domain position for a surface with one cusp at. 1 2 at H, with stabilizer Γ := z z + 1 Γ. The fundamental domain F is assumed to be bounded by Re s = 1 2, as illustrated in Figure 17. We can identify L 2 (Γ\H, da) with L 2 (F, da). The cusp forms then constitute the subspace { 1 } H cusp := f L 2 (F, da) : f(x, y) dx = 0 for a.e. y > 0. 0 Note that we re still talking about functions here; the terminology cusp form is standard even for this special case. (The definition does of course extend to automorphic forms, but we ll focus on the function case.) If we expand f L 2 (F, da) as a Fourier series in the x-variable, (4.1) f(z) = n Z c n (y)e 2πinx, then the defining condition for a cusp form is equivalent to the vanishing of the zero mode f H cusp c 0 (y) = 0. For surfaces with multiple cusps, cusp forms are required to satisfy the vanishing condition separately with respect to each cusp. Consider the Fourier decomposition of an f L 2 (F, da) that satisfies the eigenvalue equation ( s(1 s))f = 0. The coefficients from (4.1) then must satisfy [ ] (4.2) y 2 y 2 + 4π 2 n 2 y 2 s(1 s) c n (y) = 0. For n = 0 the two independent solutions are obvious: (4.3) c 0 (y) = A 0 y 1 s + B 0 y s. The non-zero mode equation is of Bessel type. The solutions for n 0 are c n (y) = A n yis 1/2 (2π n y) + B n yks 1/2 (2π n y). These Bessel I and K modes either grow or decay exponentially as y, respectively. Since only the latter behavior is allowed for f L 2, the coefficients A n all vanish for n 0. We conclude that cusp forms satisfying the eigenvalue equation must decay exponentially as y. This enforced exponential decay can be used to show that

18 18 D. BORTHWICK the restriction of to H cusp has purely discrete spectrum. The eigenfunctions of the restriction of to H cusp are called Maass cusp forms. Embedded eigenvalues (for which λ 1 4 ) must come from Maass cusp forms, because a zero mode of the form (4.3) with Re s = 1 2 could not be L2. Below the continuous spectrum (λ < 1 4 ), eigenvalues may or may not come from cusp forms. Selberg s trace formula shows that certain arithmetic surfaces have an abundance of Maass cusp forms. But Phillips and Sarnak showed that these disappear when the arithmetic surface is deformed to an ordinary hyperbolic surface [33]. They conjectured that H cusp is small or empty for a generic cofinite Fuchsian group. Wolpert [49] was able to prove this conjecture under the hypothesis that the cuspidal spectrum is simple. The general belief now seems to be that H cusp is non-trivial only in the arithmetic case, but this issue is as yet unresolved Eisenstein series. Let us continue our discussion under the assumption of a single cusp at with stabilizer Γ Γ. From (4.3) we can see that y s is a zero-mode solution of the eigenvalue equation in Γ \H, but of course it is not invariant under the full group Γ. We can try to remedy this by averaging over the rest of the group, setting E(s; z) := (Im γz) s (4.4) γ Γ \Γ = γ Γ \Γ y s cz + d 2s. This construction is called an Eisenstein series. It is not too hard to argue that the series converges for Re s > 1 (this follows from (6.2)), but continuation beyond that is not at all obvious. Meromorphic continuation can be established by various routes; see, for example, the elegant proof given by Colin de Verdiére [9]. From a spectral theory viewpoint, the natural method is to connect the Eisenstein series to the resolvent, whose meromorphic continuation is already understood. (As we noted above, this approach 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 R H (s; γz, w) = γ Γ \Γ R Γ \H(s; γz, w). The resolvent R Γ \H(s) can be computed explicitly as a Fourier series in the x- variable [3, Prop. 5.8], and this calculation shows that R Γ \H(s; z, z ) = ys y 1 s 2s 1 + O(y ), as y with y held fixed. Thus we have the relation (4.5) lim y y s 1 R(s; z, z ) = 1 2s 1 E(s; z). The parametrix construction can be used to insure that the limit on left-hand side is well-defined, and the meromorphic continuation of E(s; z) then follows immediately from that of R(s).

19 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES Spectral decomposition: finite-area case. For finite area Γ\H, we must distinguish two forms of discrete spectrum. The cusp form spectrum, which includes all embedded eigenvalues, has already been introduced above. The rest of the discrete spectrum, consisting of eigenvalues below 1 4 which do not come from cusp forms, is called the residual spectrum. The span of the residual eigenfunctions is denoted H res. We are now prepared to parametrize the full spectrum. On H cusp H res we have an complete eigenbasis {φ j, λ j } for. To complete our decomposition, let us denote the remaining part of the Hilbert space as the continuous Hilbert space, so that H cont := (H cusp H res ), L 2 (Γ\H, da) = H cusp H res H cont. The spectrum of on H cont is purely continuous, and we can use Eisenstein series as continuous analog of the eigenbasis on this subspace. The full spectral decomposition result is the following: where Theorem 4.1 (Roelcke-Selberg). A function f L 2 (Γ\H, da) can be writen f(z) = a(r) := F b j := E( ir; z)a(r) dr + b j φ j (z), E( 1 2 F + ir; z)f(z) da(z), φ j (z)f(z) da(z). See [46, Thm. III.7.1] for a proof. This spectral decomposition theorem is analogous to the Fourier transform in R n, in that the function f is expressed as a superposition of oscillatory states of various frequencies. The Eisenstein series play the role of Euclidean plane waves in this analogy Scattering matrix: finite-area case. We continue to work with the example of a finite-area quotient Γ\H with a single cusp. We ve noted that E(s; z) is the analog of a plane wave, but it is more accurate to think of it as the superposition of incoming and outgoing waves. As y, (4.6) E(s; z) = y s + ϕ(s)y 1 s + O(y ), for a meromorphic function ϕ(s). The y s term corresponds to an incoming solution of the wave equation and y 1 s to an outgoing term. That is, if we use E(s; z) to build a solution of the wave equation, 2 t u = u, via separation of variables, we ll find u(t, z) = e i λt E(s; z), where λ = s(1 s). For s = iν with ν > 0, e i λt y s = [ y exp i ] λt + iν ln y, exhibits a wavefront moving inward. Likewise, the y 1 s term yields an wavefront moving outward. If there are multiple cusps, then for each cusp j we define a coordinate y j as the y coordinate when the corresponding parabolic fixed point is moved to and

20 20 D. BORTHWICK stablized by Γ. In this case a different Eisenstein series E i (s; z) is assigned to each cusp i. As above, E i (s; z) is defined as an average of y s i over Γ \Γ. From the asymptotic expansion of E i (s; z) in the j-th cusp, E i (s; z) δ ij y s j + ϕ ij (s)y 1 s j, for a set of meromorphic functions ϕ ij (s). The physical interpretation is that E i (s) contains a single incoming waveform, in cusp i, and ϕ ij (s) gives the coefficients of the resulting outgoing solution in cusp j. These coefficients collectively define the scattering matrix S(s) := [ϕ ij (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) := det S(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 that the resonances will give rise to scattering poles provided the coefficient of y s in the residue of R(s; z, z ) at the pole does not vanish. It turns out that this will always be the case unless the resonance is caused by a cusp form. Cusp forms decay exponentially in the cusps and thus do not give rise to scattering resonances. We thus essentially have the following relationship between resonances and scattering poles: resonances = scattering poles + cusp resonances, but this simple statement is not quite accurate. Residual eigenvalues generally give rise to scattering poles, but there is a possible exception in the case of residual spectrum. Because of the symmetry ϕ(1 s) = 1/ϕ(s), a resonance at ζ corresponds also to a zero of ϕ(s) at 1 ζ. Since the only resonances with Re s > 1 2 lie in the interval ( 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 eigenvalue Λ = 0, which appears by default for any finite-area surface. Notes. There are many references for the spectral theory of cofinite Fuchsian groups, 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 L 2 (Γ\H, da) satisfying ( s(1 s))u = 0. Using the structure of the resolvent, we can argue that in any funnel end, such a function must an asymptotic expansion u k ρ s+k f k,

21 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES scattering cusp form Figure 18. A portion of the resonance plot for PSL(2, Z)\H for f k 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 cylinder with circular boundary.) If Re s = 1 2, then u can be in L2 only if f 0 = 0. But then, by applying to the expansion of u at infinity, one can see by induction that all the f k must vanish. In the compactified picture, this means that u must vanish to infinite order at the boundary ρ = 0. Then a unique continuation argument can be applied to show that u = 0. (See Mazzeo [27] for the original version of this argument, 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 in the 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 in the 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 analog of 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 ) := lim ρ 0 ρ s R(s; z, z ), where z = (r, θ ) are the standard funnel coordinates and ρ = 2e r. The justification for term Poisson operator becomes clear if we examine the corresponding situation for H. As long as we stay away from, the coordinate y makes a suitable boundary-defining coordinate ρ. With this choice, we see from (3.3) that E H (s; z, y) := lim y s R H (s; z, z ) y 0 = 1 Γ(s) 2 [ ] s y 4π Γ(2s) (x x ) 2 + y 2.

22 22 D. BORTHWICK For s = 1 this is indeed the classical Poisson kernel associated to the boundary value problem for harmonic functions in the upper half-plane. For s 1 the operator E H (s) maps compactly support functions on R to solutions of the eigenvalue equation ( s(1 s))u = 0 in H. This follows from ( s(1 s))r(s) = I, but we could also check it explicitly by differentiating E H (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 in X. For s 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 ρ 1 s f + ρ s f, with f the original boundary function and some other f C ( X). In the context of a funnel end, it is the map (5.2) S(s) : f f that is called the scattering matrix. This time we re really not talking about a matrix 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 the expression (4.5) for an Eisenstein series. And, just as in the finite-area case, for Re s > 1 we can express the kernel E(s) as an average over translates of E H (s; z, y) by Γ. Hence the terms Poisson kernel and Eisenstein series are often used interchangeably in this context. The only possible source of confusion is the fact that the 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 the structure 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 projections in terms of the Poisson operators: P α,β = 1 2πi β 1/4 E( iξ)e( 1 2 iξ)t 4iξ 2 dξ. α 1/4 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) and points (one for each cusp). The Poisson operator is defined just as above, by an integral kernel given as a boundary limit of the Green s function. It just becomes a bit more notationally complicated, with one component of the operator for each boundary element. Likewise, the scattering matrix is obtained as in (5.2), but it has a separate component for each pair of boundary elements. The diagonal funnel-funnel terms are pseudodifferential operators, while the off-diagonal terms are smoothing operators. See Borthwick [3, Ch. 7] for a complete description of this structure. Notes. For more details on the material in this section, see Borthwick [3].

23 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES Selberg trace formula The central result in the spectral theory of hyperbolic surfaces is the Selberg trace formula. This connects the spectral data associated to the Laplacian to the geometric structure of the quotient. In physical terms, we could view this as a connection between quantum and classical mechanics, with the quantum side represented by the spectral theory and classical mechanics by the geometry of geodesics on the surface. Indeed, the trace formula can be given a heuristic formulation in terms of Feynman path integrals (see Gutzwiller [18]). There exist more general trace formulas connecting quantum physics with classical mechanics, but these are typically asymptotic expansions as Planck s constant goes to zero. The theory of such expansions is called semiclassical analysis. The Selberg trace formula is not only a prototype for these expansions, but also a special case where the expansion is exact rather than asymptotic. The crucial extra ingredient in the hyperbolic case is the algebraic structure that serves as an intermediary between the analytic and geometric sides of the formula Smooth compact case. For ease of exposition, we start by considering the case where Γ\H is smooth and compact. The Laplacian thus has spectrum {λ j } corresponding to a complete orthonormal basis of eigenvectors {φ j }. To any f C [0, ) we can try to define an operator K f with integral kernel (6.1) K f (z, w) := γ Γ f(d(z, γw)). For u L 2 (Γ\H), the action of this operator is K f u(w) = K f (d(z, γw))u(w) da(w) F = f(d(z, w))u(w) da(w). H Of course, we need to insist on some decay of f at for these expressions to converge. It is not too hard to prove for any geometrically finite Γ that (6.2) # { γ Γ : d(z, γw) t } = O(e t ). Note that the left-hand side counts the number of orbit points in Γw inside the disk B(z; t). We can assume z, w F and choose a compact subset F 0 F containing both points. Each orbit point of w sits inside a separate translate of F 0 by the group. Whenever d(z, γw) t, the image γf 0 is contained in a disk of radius t + d centered at z, where d = diam(f 0 ). Since the images of F 0 must be disjoint, we have an estimate # { T Γ : d(z, T w) t } Area(B(z; t + d)) C cosh(t + d), Area(F 0 ) which implies (6.2). It follows that the sum (6.1) defining K f will converge uniformly 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, this makes K f a smoothing operator L 2 (Γ\H, da) C (Γ\H).

24 24 D. BORTHWICK The Selberg trace formula computes the trace of K f in two different ways. On the one hand, since K f is trace class and has a continuous kernel, the trace could be written as (6.3) tr K f = K f (z, z) da(z). Γ\H On the other hand, if we let {κ j } be the eigenvalue spectrum of K f, then the trace is simply (6.4) tr K f = j These two expressions for the trace would be valid for any operator on a compact surface defined by a smooth kernel. The novelty in our case is that both expressions are explicitly computable in terms of f. For the spectral side (6.4), we start with the observation that K f is self-adjoint and commutes with the Laplacian, which follows directly from the symmetry κ j. z d(z, w) = w d(z, w). Then, since K f and are commuting self-adjoint operators, they can be simultaneously diagonalized by a basic result of functional analysis. In particular, we can assume that the eigenfunctions {φ j } of also satisfy K f φ j = κ j φ j, for some κ j. To compute κ j, we write this eigenvalue equation in integral form, (6.5) κ j φ j (w) = f(d(w, z)) φ j (z) da(z). H Now specialize to w = i and let s j = λ j 1/4. Note that y sj satisfies the same eigenvalue equation as φ j, y sj = λ j y sj. If C(u) denotes the circular average of u with respect to elliptic rotations centered at i, then both C(φ j ) and C(y sj ) satisfy the radial version of the eigenvalue equation ( 2 r + coth r r )u = λ j u, where r = d(i, ) here. Matching the boundary conditions at r = 0 gives C(φ j ) = φ j (i)c(y sj ) Inserting this into the integral (6.5), and noting the invariance of d(i, z) under rotations about i, gives κ j φ j (i) = φ j (i) f(d(w, z)) y sj da(z). This gives the crucial formula expressing κ j in terms of f, κ j = f(d(i, z)) y sj da(z) H Using this result we can write the spectral side of the trace formula as ( ) (6.6) tr K f = h λ j 1 4, H j=0

25 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 25 where (6.7) h(r) := H f(d(i, z)) y r da(z). The restriction on the growth of f translates to the assumption that h extends to an analytic function on { Im z < 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 K f heat operator e t and (6.4) becomes is the tr e t = j e tλj. Now let s turn to the geometric side (6.3) of the trace formula. Our assumption that Γ\H is smooth and compact implies that Γ contains only hyperbolic elements. The length trace computation starts from tr K f = K f (z, z) da(z) F = f(d(z, γz)) da(z). γ Γ F The trick is to organize the sum over Γ as a sum over conjugacy classes, and then express these in terms of lengths of closed geodesics. Let Π be a complete list of representatives of conjugacy classes of primitive elements of Γ. Then we can write (6.8) Γ {I} = {σγ k σ 1 }. γ Π k N σ Γ/ g The innermost union over σ gives the conjugacy class of γ k in Γ, and by the definition of Π, each non-trivial conjugacy class in Γ corresponds to exactly one γ Π and 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 corresponding set of lengths (as defined by (2.1)) forms the primitive length spectrum of Γ\H: L(Γ) := {l(γ) : γ Π}. Note this is the oriented length spectrum; since γ and γ 1 are listed separately in Π, each length will appear twice in L(Γ). Using the conjugacy class decomposition (6.8) of Γ, we write the trace as tr K f = f(0) Area(Γ\H) + f(d(z, ηγ k η 1 z)) da(z) γ Π F k N η Γ/ g A change of variables in the integral gives f(d(z, ηγ k η 1 z)) da(z) = F ηf f(d(z, γ k z)) da(z) We then observe that the union of ηf over η Γ/ γ forms a fundamental domain for the cyclic group γ. We could replace this by any other fundamental domain

26 26 D. BORTHWICK F γ, so that η Γ/ γ F f(d(z, ηγ k η 1 z)) da(z) = f(d(z, γ k z)) da(z). F γ In particular, if we conjugate γ to z e l z, where l = l(γ) then the convenient choice is F γ = {1 y e l }. We then compute f(d(z, ηγ k η 1 z)) da(z) = f(d(z, e kl z)) da(z) η Γ/ g F = {1 y e l } l sinh(kl/2) kl f(cosh t) 2 cosh t 2 cosh kl sinh t dt. This essentially completes the evaluation of the length side of the trace formula: tr K f = f(0) Area(Γ\H) + l sinh(kl/2) l L(Γ) k N kl f(cosh t) 2 cosh t 2 cosh kl 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 direct but not so simple calculations (see Buser [7, 7.3]), we find and kl f(0) = rh(r) tanh πr dr, f(cosh t) 2 cosh t 2 cosh kl sinh t dt = ĥ(kl) Theorem 6.1 (Selberg Trace Formula - smooth compact case). For h satisfying the conditions given above, ( ) h λ j 1 4 = Area(Γ\H) rh(r) tanh πr dr 4π j=0 + l L(Γ) k N l sinh(kl/2)ĥ(kl). 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 topological term 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 e t/ (4πt) 1/2 0 l L(Γ) k=1 re r2 /4t sinh(r/2) dr l 2 sinh(kl/2) e l /4t.

27 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 27 By studying the divergence of both sides as t 0 we can use a Tauberian argument to prove the Weyl asymptotic formula, # { λ j r } Area(Γ\H) 4π The behavior of the heat trace as t 0 is actually quite well understood in much greater generality, and this type of Weyl asymptotic for the spectrum holds for any compact manifold, with the power of r given by the dimension. By considering a full asymptotic expansion of the heat trace as t 0, we can recover the full length spectrum Λ(Γ) from that side of the heat trace formula. On the other hand, the eigenvalue spectrum is easily recovered from the asymptotic expansion of the heat trace as t. These facts yield a direct proof of the following: Corollary 6.2 (Huber s Theorem). For Γ\H smooth and compact the eigenvalue 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 λ 0 = 0, we can deduce from this limit that l L(Γ) k=1 l 2 sinh(kl/2) e l /4t = 2(4πt) 1/2 e t/4 (1 + o(1)), as t. Applying a Tauberian argument to this asymptotic yields another famous corollary of the Selberg trace formula, also due to Huber: Corollary 6.3 (Prime geodesic theorem). As x, # { l Λ(Γ) : e l x } = Li x + r. m Li(x sj ) + O(x 7/8+ε / log x). where the values s j ( 1 2, 1] correspond to the eigenvalues {λ 1,..., λ m } of in [0, 1 4 ) according to the relation λ j = s j (1 s j ). j=1 Here Li x denotes the log integral function, Li x := 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) := ( 1 e (s+k)l ). l L(Γ) k=1 The product converges for Re s > 1 by the estimate (6.2). And in the compact case it extends to an entire function of s C. We cannot compute Z(s) directly using the trace formula. However, with the choice 1 h(z) = z 2 + (s 1/2) 2 1 z 2 + (a 1/2) 2,

28 28 D. BORTHWICK the trace formula connects the zeta function to a regularized trace of the resolvent on the spectral side, tr[r(s) R(a)] = 1 Z 2s 1 Z (s) 1 Z 2a 1 Z (a) χ [ 1 s + k 1 ]. a + k k=0 The resolvent itself is not trace class, but the difference R(s) R(a) will be trace class in the compact case. We defined resonances to be the poles of the resolvent, so this version relates the resonance set to the divisor of the zeta function. Indeed, it reveals the beautiful fact that the zeros of Z(s) occur at the resonances, with some extra zeros at N 0 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 with multiplicities proportional to χ General finite area case. The argument for the trace formula doesn t change much in the compact orbifold case, where we allow elliptic elements in the group. The only new feature is that in the decomposition (6.8) of Γ into conjugacy classes, we must include a sum over representatives of primitive elliptic conjugacy classes. For each such representative η we ll have a contribution to the trace formula of the form m 1 k=1 σ Γ/ T F f(d(z, ση k σ 1 z)) da(z) = m 1 k=1 F η f(d(z, η k z)) 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 compute these integrals and then express them in terms of h, F η f(d(z, η k z)) da(z) = 1 m sin(πk/m) e 2πkr/m h(r) dr 1 e 2πr Thus in the compact orbifold case the only change to the trace formula is a some over elliptic conjugacy classes, n c j=1 1 m j sin(πk/m j ) e 2πkr/mj h(r) dr, 1 e 2πr that appears on the length side. The trace formula is substantially harder to prove for non-compact Γ\H, because the operator K f is not trace class. Moreover, the integral K f (z, z) da(z) Γ\H 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 y j N within each cusp. On both sides of the trace formula terms appear that will diverge like log N as N. The formula is obtained by carefully canceling these terms on both sides before taking taking N.

29 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 29 The details of this argument are a little too technical for us to get into, but we 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 1 2π ϕ ϕ ( ir)h(r) dr h(0) tr[ϕ ij( 1 2 )], where [ϕ ij (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, we add a term 1 Γ π Γ (1 + ir)h(r) dr h(0) 1 ĥ(0) log 2 π j Theorem 6.5 (Selberg trace formula - finite area version). Let X be a noncompact finite-area hyperbolic surface. If that g C0 (R) is even then ( ) h λ j 1/4 1 ϕ 2π ϕ ( ir)h(r) dr = Area(Γ\H) 4π n c π r tanh(πr)h(r) dr + l L(Γ) k N l sinh(kl/2)ĥ(kl) Ψ(1 + ir)h(r) dr (n c tr[ϕ ij ( 1 2 )])h(0) n c π log 2 ĥ(0), where {λ j } are the eigenvalues of X, n c is the number of cusps, and Ψ(z) is the digamma 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 spectrum determine each other, and also χ and number of cusps. (2) Prime Geodesic Theorem: for finite-area hyperbolic surfaces # { e l x } Li x + Li(x sj ), where {λ j = s j (1 s j )} are the eigenvalues in (0, 1 4 ). (3) Weyl-Selberg asymptotic formula for finite-area hyperbolic surfaces: # { λ j r } 1 r 1/4 4π ϕ r 1/4 ϕ ( 1 Area(Γ\H) 2 + it) dt r. 4π 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 conjugacy classes). 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 proportional to the number of cusps Infinite-area surfaces. In the infinite-area case, convergence of the trace is even more of an issue, and we cannot develop a trace formula for the full class of operators K f. Analytically, it proves most convenient to focus on the

30 30 D. BORTHWICK trace of the wave operator, cos(t 1/4), whose kernel is one of the fundamental solutions of the wave equation t 2 u + ( 1 4 )u = 0. Even in the compact case, the wave operator is not trace class and its trace must be understood as a distribution in t rather than a function. But additional regularization is required for an infinite-area surface. Using the boundary defining function ρ introduced above, we can define a formal trace called the 0-trace. For a smooth kernel K(z, z ) with a polyhomogeneous asymptotic expansion in powers of ρ, ρ at infinity, we define 0-tr K := FP K(z, z) da(z), ε 0 ρ ε 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 integrate over 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 with the classical summation formula. Theorem 6.6 (Guillopé-Zworski). For a smooth geometrically finite hyperbolic surface, Θ(t) = 1 e (ζ 1 2 )t + n c 2 4, as a distribution on R +. ζ R The theorem proven in [16] is actually more general, applying to surfaces with hyperbolic ends but with an arbitrary metric inside some compact set. See [3] for a detailed exposition of the proof of the version stated here. The Poisson formula gives a realization of the spectral side of the trace formula, with resonances taking up the role played by eigenvalues in the compact case. It s worth noting that if we had used the heat operator, the putative spectral trace would be a sum of e tζ(1 ζ) over ζ R. In the infinite-area case, the resonances 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 on the distribution of resonances than we currently have, there is no way to regularize this spectral heat trace. For the length side of the wave trace formula, we can apply the same method used in the compact case, breaking up the group into a sum over primitive conjugacy classes and then computing the 0-traces explicitly for each class. The result [17] is: Theorem 6.7 (Guillopé-Zworski). For a smooth, geometrically finite, nonelementary hyperbolic surface Γ\H, Θ(t) = l 4 sinh(kl/2) δ( t kl) + χ cosh(t/2) 4 sinh 2 (t/2) l L(Γ) k=1 + n c 4 coth( t /2) + n cγδ(t),

31 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 31 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 wave trace that one obtains from Theorem 6.7 holds more generally, but one usually must include a smooth error term. This is a famous result of Duistermaat-Guillemin [11] in the compact case, extended by Joshi-Sá Barreto [22] to asymptotically hyperbolic manifolds. The combination of the two wave trace results gives a distributional version of the Selberg trace formula for infinite-area surfaces: Corollary 6.8. For smooth, geometrically finite hyperbolic surfaces, e (ζ 1 2 )t = l δ( t kl) 2 sinh(kl/2) ζ R l L(Γ) k=1 (6.10) + χ cosh(t/2) 2 sinh 2 (t/2) + n c [coth( t /2) 1], 2 as a distribution on R +. Guillopé-Zworski [15, 16, 17] also established bounds on the distribution of resonances, including the resonance counting function, N(t) := #{ζ R : ζ 1 2 t} t2, which counts resonances in the region shown in Figure 19. They proved upper and lower bounds of the optimal order, i.e., N(t) t 2, The upper bound is proved by estimating a Fredholm determinant constructed from the resolvent and is needed in the proof of the trace formula. The lower bound then is 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), but the trace formula does lead to a Weyl law for the scattering phase [16], which could be viewed as a substitute. In fact, we can use this scattering phase asymptotic to refine the upper bound on resonances, see Borthwick [4]. The resulting bound is a 2N g (t) n f l j t 2 dt χ o(1) a 2, 0 j=1

32 32 D. BORTHWICK δ Figure 20. Location of the first resonance at the exponent of convergence δ. where the l j s are the lengths of the closed geodesics bounding the funnels. This estimate is sharp in the finite-volume case, where Parnovski [30] showed that N(t) χ t 2. (This does agree with the Weyl asymptotic in the compact case remember that we are counting resonances rather than eigenvalues, so the power is doubled). Another fairly direct consequence of Corollary 6.8 is the analog of Huber s theorem, that the resonance set and length spectrum determine each other up to finitely many possibilitity [5]. This fact was used to prove that the resonance set determines an infinite-area hyperbolic surface up to finitely many possibilities in Borthwick-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 better than in the finite-area case. The product formula for the zeta function converges for Re s > δ, where δ is the exponent of convergence, usually defined as { } δ := inf s 0 : e sd(z,t w) <, T Γ which is independent of the choice of z, w H, serves also as the abscissa of convergence 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 finite area. The exponent δ is quite interesting in its own right. On the one hand, the Patterson-Sullivan theory of measures on the limit set [31, 44] shows that δ is the Hausdorff 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 an eigenvalue if and only if δ > 1 2. For the 3-funnel surface shown in Figure 16, this value is δ , as illustrated in the resonance plot in Figure 20. The meromorphic extension of the zeta function to s C was proven before the trace formula in this context, by Guillopé [14]. But, as in the compact case, the trace formula gives additional information about its divisor. From it we can derive a factorization of Z(s) in terms of the resonance set. Define the Hadamard product, H(s) := ( 1 s ) e s/ζ+s2 /2ζ 2, ζ ζ R

33 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 33 Figure 21. The standard fundamental domain for the modular group, with the corresponding tessellation of H. and let G (s) = (2π) 1 Γ(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) = e q(s) G (s) χ Γ(s 1 2 )nc H(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 and n c. Notes. The material on the trace formula in the compact case is adapted from McKean [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 arithmetic surfaces the idea is this restriction is applied to the entries of the matrices forming the Fuchsian group. If we do this directly in PSL(2, R), the result is the modular group Γ Z := 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 within that representation. The resulting subgroup is discrete and thus defines a Fuchsian group, Γ := { γ PSL(2, R) : ρ(γ) GL(n, Z) }, The full definition of arithmetic Fuchsian group includes all the groups obtained by this construction, as well as any subgroups of finite index.

34 34 D. BORTHWICK F X = Γ Z \H i p = e πi/3 i p Figure 22. Constructing the modular surface by folding the fundamental domain and identifying edges Modular surfaces. We will focus mainly on the modular group Γ Z in this exposition. This group is generated by the elements ( ) ( ) τ =, σ =, corresponding to the maps, τ : z z + 1, σ : z 1 z. The map τ is of course the standard parabolic translation fixing, the generator of Γ. The element σ is a rotation of order 2 about the fixed point i. The modular group also has an elliptic fixed point of order 3: the point p = e iπ/3 is fixed by στ 1. The standard fundamental domain for Γ Z is F = { z H : Re z 1 2, z 1}, as shown in Figure 21. From this domain the quotient surface X := Γ Z \H, called the modular surface, can be constructed by folding along the central axis Re z = 0. This is illustrated in Figure 22. Note that since the fundamental domain is a geodesic triangle with angles 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 Γ Z. For N > 1 the principal congruence subgroup of level N is defined to be Γ Z (N) := { g Γ Z : g I mod N }. For example, the level 2 subgroup is just {( ) odd even Γ Z (2) = even odd } PSL(2, Z). Figure 23 shows the standard fundamental domain for this group, which is a geodesic triangle all of whose vertex angles are 0. The quotients of H by principal congruence subgroups are denoted by X(N) := Γ Z (N)\H. These are also called called modular surfaces, but the modular surface always refers to X = X(1). The surface X(2) has genus zero with 3 cusps and no elliptic fixed points, and so is geometrically quite simple. The geometry of X(N) gets more

35 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 35 Figure 23. The fundamental domain and tessellation for Γ Z (2). complicated as N increases, with the genus of X(N) approximately equal to N 3 for N large. There are other arithmetic Fuchsian groups and even other kinds of congruence subgroups of Γ Z. But number theorists are particularly interested in Maass cusp forms on X(N) Spectral theory of the modular surface. For the rest of this section we 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 the framework of 4.2, and the Eisenstein series is defined as in (4.2) by E(s; z) := Γ \Γ Z ys cz + d 2s. To compute the scattering matrix we need to work out the asymptotic expansion of E(s; z) as y. We first note that for ( ) a b γ = Γ c d Z, the coset of Γ takes the form {( )} a + cz b + dz (7.1) Γ γ =. c d Note that determinant condition ab cd = 1 implies gcd(c, d) = 1. To each relatively prime pair (c, d) Z 2 we can associate an element of Γ Z. If we assume that c 0, to remove the sign ambiguity, then (7.1) shows that the corresponding coset in Γ \Γ Z is fixed by the choice of (c, d). We conclude that (7.2) E(s; z) = y s + c=1 d Z gcd(c,d)=1 y s cz + d 2s. Recall that the non-zero Fourier modes of E(s; z), with respect to the periodic x variable, vanish exponentially as y. So to find the scattering matrix, we need only to consider the 0-mode, c 0 (s; y) := 1 0 E(s; z) dx.

36 36 D. BORTHWICK In the sum in (7.2), we write d = cn + r, where n Z and 1 r < c and gcd(r, c) = 1. For fixed c 1 the contribution to the zero-mode is 1 r<c n Z gcd(r,c)=1 1 0 y s dx = c(z + n) + r 2s 1 r<c gcd(r,c)=1 = 1 r<c gcd(r,c)=1 y s c 2s y 1 s [x 2 + y 2 ] s dx Γ( 1 2 )Γ(s 1 2 ) c 2s. Γ(s) Since the r dependence has been integrated out, the sum over r just yields a factor of the totient function φ(c) := #{1 r < c : gcd(r, c) = 1}. With these calculations the formula for the zero mode becomes c 0 (s; y) = y s + y 1 s Γ( 1 2 )Γ(s 1 2 ) Γ(s) c=1 φ(c) c 2s. The sum can now be evaluated in terms of the Riemann zeta function by the Dirichlet series identity, This yields n=1 φ(n) n z = ζ(z 1). ζ(z) (7.3) c 0 (s; y) = y s + y 1 s Γ( 1 2 )Γ(s 1 2 ) ζ(2s 1) Γ(s) ζ(2s) From the asymptotic (7.3) and the definition (4.6) we instantly read off the scattering matrix ϕ(s) = Γ( 1 2 )Γ(s 1 2 ) Γ(s) ζ(2s 1). ζ(2s) Of course, our computations were valid only in the realm of absolute convergence of these series, namely Re s > 1. But the result clearly extends meromorphically to s C. We conclude that the scattering poles of X correspond precisely to the nontrival zeros of ζ(2s), plus the pole at s = 1 coming from ζ(2s 1) (which corresponds to the default eigenvalue λ = 0). The resulting resonance picture is shown in Figure 24 (with the locations of cusp forms taken from numerical computations of Hejhal [19]). The Selberg trace formula easily shows that the the modular surface must have many cusp forms that do not appear in the scattering matrix. The Weyl-Selberg asymptotic 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 forms dominate the spectrum.

37 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES Figure 24. The resonances of the modular surface. The cusp forms lie on Re s = 1. The resonances with Re s = 1 4 come from Riemann zeros Hecke operators. The abundance of cusp forms in the modular surface noted at the end of the last section holds for other arithmetic hyperbolic surfaces as well. This contrasts sharply with the general hyperbolic case, where cusp forms are thought to be very rare and perhaps do not occur at all. The natural question to ask 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 cofinite Fuchsian groups? The answer is that the arithmetic surfaces posses extra, hidden symmetries, in the form of a family of transformations called Hecke operators. For the modular group the definition of a Hecke operator is fairly simple. For each n N we define an operator T n on L 2 (Γ Z \H, da) by (7.4) T n f(z) := 1 n d 1 ad=n b=0 ( az + b f d Since it s built as an average over isometries of H, it s clear that T n commutes with the Laplacian H. What is not immediately obvious is that T n f retains its invariance under Γ Z so that its action descends to the quotient Γ Z \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) T 3 f(z) = 1 3 [f(3z) + f ( z 3) + f ( z ). ) + f ( z )]. Note that invariance under τ : z z + 1 is immediately obvious, so it suffices to consider the other generator, σ : z 1/z. Under this symmetry, we have ( T 3 f 1 ) = 1 ( [f 3 ) ( + f 1 ) ( ) ( )] z 1 2z 1 + f + f. z 3 z 3z 3z 3z Using the invariance of f under Γ Z, we see that the first 2 terms just switched places: ( f 3 ) ( ( z = f, f z 3) 1 ) = f(3z). 3z

38 38 D. BORTHWICK For the other terms we can use the matrix identity, ( ) ( ) ( ) = (note that the first matrix on the right has determinant 1, and so lies in Γ Z ), to see that ( ) ( ) z 1 z + 2 f = f. 3z 3 Similarly, ( ) ( ) 2z 1 z + 1 f = f. 3z 3 With these identications, we have shown ( T 3 f 1 ) = T 3 f(z), z and thus T 3 f is again an automorphic function for Γ Z. 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 Hecke symmetries is very special and delicate. From the correct point of view, the general proof is much easier. The trick is to define and consider the coset space Γ Z \Υ n. Lemma 7.1. The set, A n = Υ n := {A GL(2, Z) : det A = n}, {( ) } a b : ad = n, 0 b d 1 0 d (i.e., the set of matrices appearing in (7.4)), contains a complete set of distinct representatives for Γ Z \Υ n. Proof. To prove the lemma we start from ( ) a b α = Υ c d n, and try to find a representative of Γ Z α in A n. If q = gcd(a, c), then there is an element of Γ Z with bottom row given by ( c/q, a/q) and we have ( ) ( ) ( ) a b =. c/q a/q c 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 of Γ, ( 1 k 0 1 ) ( ) a b = 0 d ( ) a b + kd. 0 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 Γ Z, then T n f( ) is also, so T n maps L 2 (Γ\H, da) to itself.

39 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 39 Proof. Suppose γ Γ Z and α A n. Then det(αγ) = n, so αγ Υ n. Lemma 7.1 then says that αγ Γ Z α, for some α A n. Then, assuming that f is automorphic with respect to Γ Z, we have f(αγz) = f(α z). The map α α induced by γ is bijective, so that T n f(γz) = 1 f(αγz) = 1 n n f(α z) = T n f(z). α A n α A n In view of Lemma 7.1, we could rewrite the definition of T n as (7.6) T n f(z) = 1 f(αz), n α Γ Z \Υ n since f(αz) clearly depends only on the coset Γ Z α generated by α. Another significant feature of the Hecke operators is that they form a commutative ring. Indeed we have a nice composition formula (which explains why the normalization 1/ n is used in the definition of T n ): Lemma 7.3. T m T n = d gcd(m,n) T mn/d 2. For gcd(m, n) = 1, this follows pretty easily from the definition. For the general case, one can first prove that T p kt p = T p k+1 + T p k 1, for p prime and k N, and then argue by induction. (See [46, III.3.6] for the details.) One final property is also rather significant from a spectral point of view: selfadjointness. As with the other properties of Hecke operators detailed above, this is not terribly difficult to prove, but not immediately obvious either. This version of the proof is adapted from Shimura [41]. Lemma 7.4. As an operator on L 2 (Γ Z \H, da), each T n is self adjoint. Proof. By Lemma 7.3 it suffices to consider T p for p prime. In that case we can write Υ p = Γ Z βγ Z, for β = ( ) p Let Γ β := Γ Z β 1 Γ Z β, and choose a set of representatives {η j } for Γ β \Γ Z, so that (7.7) Γ Z = j Γ β η j is a disjoint union. It s then easy to deduce a corresponding decomposition Γ Z βγ Z = j Γ Z βη j,

40 40 D. BORTHWICK so that {βη j } serves as a set representatives for Γ Z \Υ p. By (7.6) we then have, for f, g L 2 (Γ\H, da), f, T p g = 1 f(z)g(βη j z) da(z) p j = 1 p η jf F f(z)g(βz) da(z). where F is a fundamental domain for Γ Z. By (7.7), the set j η j F is a fundamental domain for Γ β, to the expression becomes f, T p g = 1 f(z)g(βz) da(z). p Γ β \H On the other hand, we could make exactly the same argument starting from β := pβ 1, since Υ p = Γ Z βγz also. Apply this reasoning gives T p f, g = 1 f( βz)g(z) da(z). p Γ β\h Then, if we note that Γ β = βγ β β 1 and that β gives the same Möbius transformation as β 1, we have T p f, g = 1 f(β 1 z)g(z) da(z) p = 1 p βγ β β 1 \H Γ β \H = f, T p g f(z)g(βz) da(z) In the proof of Lemma 7.4, we can see a hint of the proper abstract definition of Hecke operator. The essential ingredient is the fact that the groups Γ Z and β 1 Γ Z β are commensurable, meaning that Γ β and Γ β (as defined above) have finite codimension in Γ Z. This was the fact that allowed us to decompose Γ Z βγ Z as a (finite) collection left cosets and then define a Hecke operator as a sum over these cosets. A distinguishing feature of general arithmetic groups Γ is the existence of a large class of groups β 1 Γβ commensurable with Γ; we have treated only the case Γ = Γ Z for simplicity. Since the T n commute with each other and with the Laplacian, self-adjointness immediately gives the following: Corollary 7.5. The full ring of Hecke operators {T n } together with can be simultaneously diagonalized. In particular, we can choose a basis of Maass cusp forms such that φ j = λ j φ j, T n φ j = τ j (n)φ j. Indeed, it is conjectured that the cusp spectrum of X is simple, which would imply that Maass cusp forms are automatically Hecke eigenfunctions, with no further diagonalization needed. In any case, all known Maass cusp forms of the modular surface are simple eigenvalues, so these eigenfunctions a few are shown in Figure 25 must have the

41 INTRODUCTION TO SPECTRAL THEORY ON HYPERBOLIC SURFACES 41 Figure 25. Density plots of Maass cusp forms (image by Alex Barnett/Holger Then) Figure 26. The effect of T 3 is to average copies of the function drawn from the four shaded regions. T n symmetry for each n. The effect of T n will be to take set of snapshots from the original image and superimpose them after rescaling and translation back to the original fundamental domain. Figure 26 illustrates this for T 3 ; each shaded region is rescaled to the size of F and then these four pictures are combined. For a Hecke eigenfunction, this procedure must result in a copy of the original image (with the density mutiplied by τ j (n) ). It is amusing to stare at the density plots in Figure 25 and imagine taking these recombinations for arbitrary n L-function connection. One primary reason that Maass cusp forms are important in number theory is the connection to Dirichlet L-functions. These are series which generalize the Riemann zeta function. Suppose that φ is a Maass cusp form, with λ = ζ(1 ζ) and Hecke eigenvalues given by τ φ (n). The standard L-function associated to φ is L φ (s) := n τ φ (n)n s,