The variance of arithmetic measures associated to closed geodesics on the modular surface

We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface.

1. Introduction 1.1. Equidistribution theorems for closed geodesics. Let X be a compact surface with a metric of constant negative curvature κ = −1, SX be the unit tangent bundle of X, and Φ t : SX → SX the geodesic flow. We think of SX as the set of initial conditions (z, ζ) with z ∈ X the position and ζ the direction vector. The geodesic flow is ergodic with respect to Liouville measure dx, the smooth invariant measure for the flow: Generic geodesics become equidistributed, in the sense that for Lebesgue-almost all initial conditions x 0 ∈ SX, for integrable observables on SX.
As is well known , there are infinitely many closed geodesics, in fact the number π(T ) of closed geodesics of length at most T grows exponentially with T : π(T ) ∼ e T /T as T → ∞ [40], [8]. For a closed geodesic C, let ℓ(C) be its length and µ C be the arc-length measure along C, i.e. for F ∈ C(X), This is a measure on SX, invariant under the geodesic flow and of total mass ℓ(C). Closed geodesics become, on average, uniformly distributed with respect to dx: 1 For any observable F ∈ C(SX) we have Lalley [20] determined the fluctuations of the numbers µ C (F )/ ℓ(C) for F as above of zero mean, as C varies over closed geodesics ordered by length. He showed that they are Gaussian with mean zero and variance V (F, F ) where V is the hermitian bi-linear form on functions of zero mean given by The negative curvature guarantees that the correlations in the inner integral decay exponentially as t → ±∞, so that V is convergent [33]. The bilinear form V is positive semi-definite, and is degenerate, in fact if F 0 is smooth then V (F 0 , F ) = 0 for all F if and only if F 0 is a derivative in the flow direction: F 0 = d dt | t=0 f •Φ t for some other observable f ∈ C ∞ (SX). An important involution of SX is time reversal symmetry w : (z, ζ) → (z, −ζ) which reverses the direction vector of the initial condition, and satisfies w • Φ t = Φ −t • w. It induces an involution on the set of geodesics, taking a geodesic C = {Φ t x 0 : t ∈ R} to its time reversalC = wC = {Φ s wx 0 : s ∈ R}.
Time reversal symmetry can also be incorporated in Lalley's theorem: To do so, note that for a closed geodesic C, its time-reversed partnerC is also closed and both have the same length: ℓ(C) = ℓ(C). Grouping these together yields the measure µ even C := µ C + µC of mass 2ℓ(C). By Lalley's theorem, the fluctuations of µ even C / 2ℓ(C) are again Gaussian with mean zero but with variance given by the hermitian form where F even = (F + F • w)/2 is the even part of F under w. Note that µ even C is invariant and V even is bi-invariant under the geodesic flow as well as under time-reversal symmetry w. Both hermitian forms V and V even on can be diagonalized and computed explicitly by decomposing the regular representation of PSL 2 (R) on this space, see § 3.
1.2. The modular surface. In this paper we investigate fluctuations of measures on the modular surface associated with grouping together geodesics of equal discriminant. As is well known, any of our compact surfaces X may be uniformized as a quotient of the upper half-plane H, equipped with the hyperbolic metric, by a Fuchsian group Γ. Furthermore, the group G = PSL 2 (R) of orientation preserving isometries of H acts transitively on the unit tangent bundle SX, giving an identification SX ≃ Γ\G, reviewed in § 2. The modular surface is obtained by taking Γ = PSL 2 (Z); the resulting surface is non-compact (but of finite volume) and has elliptic fixed points, but these issues will not be important for us. Closed geodesics correspond to (hyperbolic) conjugacy classes in Γ, with the length of a closed geodesic C given in terms of the trace t of the corresponding conjugacy class by ℓ(C) = 2 log(t + √ t 2 − 4)/2. In the case of the modular surface, the hyperbolic conjugacy classes correspond to (strict) equivalence classes of integer binary quadratic forms ax 2 + bxy + cy 2 (also denoted by [a, b, c]), of positive discriminant d := b 2 − 4ac with the modular group acting by linear substitutions (we need to exclude discriminants which are perfect squares). The discriminant disc(C) of a closed geodesic C is defined as the discriminant of the corresponding binary quadratic form.
For d > 0, d ≡ 0, 1 mod 4 and d not a perfect square, letf 1 , . . .f H(d) be the classes of binary quadratic forms of discriminant d. We do not assume that f j = [a j , b j , c j ] is primitive and so H(d) is the Hurwitz class number [21]. Let be the fundamental solution of the Pellian equation t 2 − du 2 = 4. Then as in [37,39] associate to eachf j the Γ-conjugacy class (it is well-defined) of the matrix This gives H(d) closed geodesics for each discriminant d, all of length 2 log ǫ d . Let µ d be the corresponding measure on SX: These measures are the arithmetic measures in the title of the paper. They have been studied extensively and the primary result about them is that they become equidistributed as d → ∞. That is, if F is bounded and continuous on SX and has mean zero, then Linnik [24] developed an ergodic theoretic approach to this equidistribution problem and recently [3] have shown that this method leads to a proof of this specific result. The first proof of equidistribution is due to Iwaniec [11] and Duke [1]. Iwaniec established the requisite estimate for Fourier coefficients of holomorphic half-integral weight forms (of weight > 5/2) and Duke obtained the estimates for weight 3/2 and weight zero Maass forms. In view of our reductions in Sections § 3 and § 6, these together imply 2 the full equidistribution on SX.
The measures µ d enjoy some symmetries (see [39]). Firstly they are invariant under time reversal symmetry: wµ d = µ d . Secondly, let r be the involution of Γ\G given by g → δ −1 gδ, where (it is well defined since δ −1 Γδ = Γ). In terms of the coordinates (z, ζ) on SX, r is the orientation-reversing symmetry The measure µ d is also invariant under r. The involutions w, r commute and their product rw is also an involution. Thus µ d is invariant under the Klein four-group H = {I, r, w, rw}. These involutions induce linear actions on L 2 (Γ\G) by f (x) → f (h(x)) with h ∈ H and x ∈ Γ\G and we denote these transformations by the same symbols. The fluctuations of the measures µ d inherit these symmetries and since we are particularly interested in comparing their variance with the classical variance V we define the symmetrized classical variance V sym on functions of mean zero on Γ\G by where F sym := 1 4 h∈H hF .
1.3. Results. We can now state our main results about the fluctuations of µ d . We normalize these measures as . This is essentially equivalent to normalizing by the square root of the total mass, H(d)2 log ǫ d , see Remark 1.4.2. The space of natural observables for which one might compute these quantities is L 2 0 (Γ\G), or at least a dense subspace thereof. This space decomposes as an orthogonal direct sum of the cuspidal subspace : u ∈ R}, and the unitary Eisenstein series [5]. The former is the major and difficult part of the space L 2 0 (Γ\G) so we will concentrate exclusively on it. One can easily extend our analysis of the variance to the unitary Eisenstein series.
and there is a limiting variance We call this variance B the "arithmetic variance". The structure of the bilinear form B is revealed by choosing a special basis of observables, compatible with the symmetries of the problem. Recall that the unit tangent bundle is a homogeneous space for G = PSL 2 (R), and thus it is natural to decompose the space L 2 (Γ\G) into the irreducible components under the G-action. In addition, there is an algebra of Hecke operators acting on this space, commuting with the G-action, hence also acting on each isotypic Gcomponent. We take observables lying in irreducible spaces for the joint action of G and the Hecke operators -the automorphic subrepresentations of L 2 cusp (Γ\G). Denote the decomposition of the regular representation on L 2 cusp (Γ\G) into Gand Hecke-irreducible subspaces by so π j is a cuspidal automorphic representation. In order to describe the arithmetic variance explicitly we need a more detailed description of the W π j 's. To each π j is associated an even integer k, its weight (see § 3) which we indicate by π k j . For k = 0 there are infinitely many π 0 j 's corresponding to Hecke-Maass cusp forms on X, while for k > 0 there are d k such π k j (where d k is either [ k 12 ] or [ k 12 ] + 1, depending if k/2 = 1 mod 6 or not), corresponding to holomorphic Hecke cusp forms of weight k.
for j = 1, 2, . . . , d −k and these correspond to the anti-holomorphic Hecke cusp forms. With these we have the orthogonal decompositions To each π j as above one associates an L-function L(s, π j ) given by where λ π j (n) is the eigenvalue of the Hecke operator T n acting on W π j . It is well known (Hecke-Maass) that L(s, π j ) extends to an entire function and satisfies a functional equation relating its value at s to 1 − s. In particular the arithmetical central value L( 1 2 , π j ) is well defined (and real).
1.4.1. The hermitian forms V sym and B can be computed explicitly on each U π k j (see § 3). Time-reversal symmetry w forces V sym to vanish on U π k j for k = 2 mod 4. Also orientation-reversal symmetry r fixes the weight zero spaces U π 0 j and hence takes the generating vector (see § 3) φ 0 j ∈ π 0 j into ±φ 0 j . Corresponding to this sign we call U π 0 j even or odd. According to § 3, V sym is completely determined on U π k j by its value on the generating vector; hence it follows that V sym | Uπ j ≡ 0 for the odd π 0 j 's. In the above cases where V sym | U π k j vanishes, the sign ǫ π j of the functional equation of L(s, π j ) is −1 and hence the central L-value L( 1 2 , π j ) = 0 for reasons of symmetry. In the other cases (k = 0 mod 4 and π 0 j even), ǫ π j = 1 and V sym | Uπ j = 0. One expects that in these cases L( 1 2 , π j ) = 0 as well. However if we pass from Γ = PSL 2 (Z) to a congruence subgroup, where our analysis can be carried over with similar results, then there will be π's corresponding to holomorphic forms for which L( 1 2 , π) = 0 for number-theoretic reasons, specifically the conjecture of Birch and Swinnerton-Dyer [43]. In this case the restriction of the arithmetic variance to such a subspace will vanish for reasons far deeper than just symmetry.
The fluctuations of L(1, χ d ) are mild and well-understood [4] and hence the normalizations are essentially the same. In any case one could use methods as in [12,Chapter 26] to remove the weights L(1, χ d ) and deduce Theorem 1.1 with this other normalization.
1.4.3. In § 3 we show, in a more abstract context, that the space of linear forms on an irreducible unitary representation of G which are invariant by both the geodesic flow and time reversal symmetry is at most onedimensional, and how to incorporate orientation-reversal symmetry. This shows that the form that the arithmetic and "classical" variance take is universal. That is for any family of such invariant measures, the variance B ′ , if it exists, is determined completely in each irreducible representation of G by B ′ (v 0 , v 0 ), where v 0 is either a spherical vector, or a lowest (or highest) weight vector in the representation.
1.4.4. The geometric problem is to order the µ d by the length of any of the geodesic components of the measure. We do not know how to do this. What we can do is to compute the variance of the µ d 's when ordered by the discriminant d. From the arithmetic point of view this ordering is anyway the most natural one. For many considerations these two orderings of µ d yield quite different answers (see [39]). However for the fluctuations we believe they are similar. The difficulty in proving the same result of the µ d 's ordered by t d (or ǫ d ) is apparent already for F 1 = F 2 = f a holomorphic cusp form of weight m ≡ 0 mod 4. In this case according to the formula of Kohnen and Zagier [17], we have for d a fundamental discriminant say (1.14) |µ (with * explicit and under control). Thus we would need to understand the averages The first, but big, step in this direction would be to understand (see [34] for an execution of such an analysis on a simpler problem). This appears to be beyond the well developed techniques for averaging special values of L-functions in families. We leave it as an interesting open problem.
1.4.5. The recent work [36] giving lower bounds for moments of special values of L-functions in families, together with (1.14), shows that the fluctuations of µ d (F )/d 1/4 are not Gaussian, at least not in the sense of convergence of moments.
1.4.6. The arithmetic variance B in Theorem 1.1 is the same as the quantum variance for the fluctuations of high energy eigenstates on the modular surface that were calculated in [25] and [44]. We expect that the variance for the µ d 's when ordered by length will be the same as B. This would give a semi-classical periodic orbit explanation for the singular finding [25] that the quantum variance is B rather than V even . It points yet again, just as for the local spectral statistics (see the survey [38]), to the source of the singular behaviour of the quantum fluctuations in arithmetic surfaces being the high multiplicity of the length spectrum. Similar phenomena are found for the quantized cat map [18,19].
1.5. Plan of the paper. We end with an outline of the paper and the proof of Theorem 1.1. In § 2 we give recall some background connecting the dynamics on the modular surface with the group structure on SL 2 (R). In § 3 we show that up to a scalar multiple, there is at most one linear form on the smooth vectors of an irreducible unitary representation of SL 2 (R) which is invariant under the action of the diagonal subgroup (corresponding to the geodesic flow) and the element 0 −1 1 0 corresponding to time-reversal symmetry. We show that such a linear form is determined by its value on a "minimal" vector -a spherical vector in the case of a principal series representation and a lowest/highest weight vector for holomorphic/antiholomorphic discrete series representations. We then bring in invariance under orientation reversal and apply the results to show that the bilinear forms V sym and B are determined by their values on Maass forms and holomorphic modular forms. In §4 we give present some background on half-integral weight forms, and in § 5 we discuss Rankin-Selberg theory for these, giving a mean-square result for Fourier coefficients along positive integers by modifying work of Matthes [28] for weight zero forms.
In § 6 we review the results of Maass [26], Shintani [41], Kohnen [15,16] and Katok-Sarnak [14], relating periods along closed geodesics to Fourier coefficients of theta-lifts. This allows us to express µ d (F ) in terms of Fourier coefficients of half-integral weight forms on Γ 0 (4); the precise normalizations in terms of the inner products of the forms and their θ-lifts are crucial here. This is where the factor L( 1 2 , π) appears. These results put us in a position to use the Rankin-Selberg theory of § 5 to determine the variance B, which we do in § 7.
1.6. Acknowledgments. We would like thank Akshay Venkatesh for insightful discussions of the material related to this paper. Supported by NSF FRG Grant DMS-0554373 (Sarnak and Luo) and by the Grant No 2006254 (Sarnak and Rudnick) from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

Background on periods
2.1. The upper half-plane and its unit tangent bundle. We recall the hyperbolic metric on the tangent bundle of the upper half-plane H = {z = x + iy : y > 0}. We identify the tangent space at z ∈ H with the complex numbers: T z H ≃ C. The hyperbolic metric on T z H is then given by ξ, η z := ℜ(ξη) y 2 and the unit tangent bundle SH is then identified with The induced map on the tangent bundle T H is then Note that this is an action: if g, h ∈ SL 2 (R) then g(h(z, ζ)) = (gh)(z, ζ). A computation shows that we get an isometry of H: and thus we get the same element in PSL 2 (R).
Setting g x,y,φ = n(x)a(y)κ(φ) we find so that using the basepoint (i, i) ∈ SH of the upward pointing unit vector at i = √ −1 ∈ H, we get a bijection We may then identify functions on PSL 2 (R) and on SH: If F (z, ζ) is a function on SH we may defineF on SL 2 (R) bỹ being the endpoint of the (unit speed) geodesic starting at z in direction ζ = iye iφ . It turns out that on PSL 2 (R) the geodesic flow is multiplication on the right by e t/2 e −t/2 , that is Indeed, for an initial position (z, ζ) ∈ SH, we write (z, ζ) = g(i, i) and the the geodesic γ(t) = Φ t (z, ζ) starting at (z, ζ) will be the translate by g x,y,φ of the geodesic γ 0 (t) starting at the initial condition (i, i): γ(t) = g x,y,φ γ 0 (t). A computation shows that and therefore . Using it, one has a symmetry of the set of geodesics, corresponding to reversing the orientation. In PSL 2 (R) it Then the corresponding functionF on PSL 2 (R) satisfies F (gκ(α)) = e ikαF (g) that is transforms under under the right action of the maximal compact K = SO(2)/{±I} via the character κ(α) → e ikα . As an example we start with a function f on H and define F f (z, ζ) = ζ k f (z).

2.2.
Quotients. Let Γ ⊂ PSL 2 (R) be a Fuchsian group , M = Γ\H and SM the unit tangent bundle to M . The identification SH ≃ PSL 2 (R) descends to an identification SM ≃ Γ\SH ≃ Γ\ PSL 2 (R) 2.2.1. Automorphy conditions. Let k ≥ 0 be an integer, and f : H → C is a function on the upper half-plane satisfying the (weak) automorphy condition that is F f is Γ-invariant so descends to a function on SM = Γ\SH, and via the identification F →F gives a Γ-invariant functionF f on PSL 2 (R): Moreover, F f has K-type k since from the definition we find and therefore the functionF f on the group PSL 2 (R) transforms under the right action of K = SO(2)/{±I} by the character κ(α) → e ikα .

2.2.2.
Closed geodesics on M . We consider closed geodesics on M , that is an initial condition (z 0 , ζ 0 ) ∈ SH so that there is some and hence that (the equality is in PSL 2 (R), that is the matrices agree up to a sign). Changing the initial condition (z 0 , ζ 0 ) to a Γ-equivalent one (z 1 , ζ 1 ) = δ(z 0 , γ 0 ), δ ∈ Γ (so that we get the same point in SM = Γ\SH) replaces γ by its conjugate δγδ −1 . Thus we get a well-defined conjugacy class γ C corresponding to the geodesic C. The conjugacy class is hyperbolic as its trace satisfies | tr γ C | = 2 cosh(T /2) > 2.

2.3.
A correspondence with binary quadratic forms. An binary quadratic form f (x, y) = ax 2 + bxy + cy 2 (also denoted by [a, b, c]) is called integral if a, b, c are integers, and is primitive if gcd(a, b, c) = 1. The discriminant of f is b 2 − 4ac. The modular group SL 2 (Z) acts on the set of integral binary quadratic forms by substitutions, and preserves the discriminant.
There is a bijection between SL 2 (Z)-equivalence classes of primitive binary quadratic forms of positive (non-square) discriminant and primitive hyperbolic conjugacy classes in PSL 2 (Z) defined as follows: Given a primitive hyperbolic element which is primitive by definition, and has discriminant and is primitive. Then B(γ(f )) = f and gives a bijection between primitive hyperbolic conjugacy classes in PSL 2 (Z) and equivalence classes of primitive binary quadratic forms of non-square positive discriminant.

2.4.
Periods. Consider a (primitive, oriented) closed geodesic on M ; it is determined by a primitive hyperbolic conjugacy class γ ∈ Γ, Let C be the lift of the closed geodesic to to the unit tangent bundle SM . For any function F on SM , we define the period of F along C by choosing a point on the lifted geodesic (z 0 , ζ 0 ) (that is an initial condition) and setting where T > 0 is the length of the geodesic, that is the first time that Φ T (z 0 , ζ 0 ) = γ(z 0 , ζ 0 ).

2.4.1.
An alternative expression for the period. For a hyperbolic matrix γ = a b c d , define a binary quadratic form (not necessarily primitive) The two zeros w ± of Q γ are the the fixed points of γ, which are the intersection with real axis of the semi-circle in the upper half-plane which determines the closed geodesic. By (2.2), the fixed points w ± of γ on the boundary are g 0 (0) and g 0 (∞): that is iff e T g −1 0 (w) = g −1 0 (w), and since T = 0 this forces g −1 0 (w) = 0, ∞.
Let f : H → C satisfy the automorphy condition (2.1) of weight 2k for Γ, where z 0 lies on the semi-circle between the fixed points of γ and the contour of integration 3 is along the geodesic arc linking z 0 and γz 0 . Let be the discriminant of the quadratic form Q γ . Then r k (f, γ) is simply related to the period of f on the geodesic defined by γ [13, proposition 4]: Note that the RHS above makes sense also for non-primitive forms, and is dilation invariant: J(tB) = J(B).

Symmetry considerations
3.1. Background on the representation theory of SL 2 (R). Let π be an irreducible infinite dimensional unitary representation of SL 2 (R) on a Hilbert space H which factors through G = PSL 2 (R). Let K = SO (2), and let H (K) be the space of K-finite vectors in H, that is vectors whose translates by K span a finite dimensional subspace. Then H (K) is dense in H and consists of smooth vectors, and the Lie algebra sl 2 acts on H (K) by dπ, the differential of the action of G. According to Bargmann's classification of such π's (we follow the exposition in Lang [22]), there are orthogonal one-dimensional subspaces H n , with n even, which are K-invariant and together span H (K) . To be more precise, we consider two cases: i) That there is no highest or lowest K-type, this being the spherical, or Maass case: If f is holomorphic, the integral is independent of the contour with H n one dimensional for n even, say H n = Cφ n , and the φ n satisfy are the standard basis of the Lie algebra sl 2 (R), are in the complexified Lie algebra sl 2 (C), and dπ(E ± ) are the weight raising/lowering operators. Here s ∈ C is a parameter which, since we assume that π is unitary, lies on the imaginary axis iR or in the interval (−1, 1). Note that since we are assuming the representation factors through G = PSL 2 (R), only even weights appear.
ii) H has a lowest or highest K-type. In the first case there is an even positive integer m 0 > 0 so that These π's correspond to holomorphic forms of even weight. In the case there is a highest K-type, there is a negative even integer m 0 < 0 so that Again H m = Cφ m for m ≤ m 0 even, so that φ m satisfy (3.2) with s = −m 0 − 1 and the highest weight vector φ m 0 is annihilated by the raising operator: In case (i) we denote by φ π the K-invariant (spherical) vector φ 0 . We normalize it so that φ 0 , φ 0 = 1 and then it is unique up to multiplication by a complex scalar of unit modulus. In case (ii) we denote by φ π the similarly normalized lowest/highest weight vector φ m 0 . We will call these φ π 's "minimal vectors" of the representation.

Linear forms.
We consider linear forms η on H (K) which are invariant by the "geodesic flow" and "time reversal symmetry", that is • η is annihilated by dπ(H), where H = 1 0 0 −1 ∈ sl 2 is the infinitesimal generator of the group of diagonal matrices A: In this case we say that η is A-invariant 4 .
• η is fixed by π( In this case we say that η is invariant under time-reversal symmetry. This is shown by giving an explicit formula for η(φ n ) in term of η(φ π ). Since the cases (i) and (ii) have slightly different features we deal with them separately.
But the LHS of (3.12) is zero since we are assuming (3.9). Hence for n even and in particular n ≡ 2 mod 4 we have It follows that for m ≥ 4, m ≡ 0 mod 4 that This together with (3.11) determines η on H (K) explicitly in terms of η(φ 0 ). Conversely, (3.11) and (3.13) with η(φ 0 ) = 1 define a unique Aand winvariant linear form on H, which we denote by ξ π,φπ . So in this case the space of such linear forms is one-dimensional and any such form η satisfies We turn to case (ii) and show that the space of A-invariant linear forms on H (K) is one-dimensional. Consider say the lowest weight case: Take the lowest weight vector φ m 0 , m 0 > 0 even. From (3.2) and (3.6) we have Thus the space of A-invariant linear forms on H (K) is one-dimensional. It is spanned by ξ Hπ,φπ where ξ Hπ,φπ (φ m 0 ) = 1 and is defined by (3.14) and (3.15). Again any A-invariant linear form η on H (K) satisfies The case of highest weight vectors and A-invariant forms is the same.
If we now impose the further condition that η be w-invariant in these case (ii) representations, then invariance under time-reversal symmetry gives as in (3.11) that η(φ m ) = 0, for m ≡ 2 mod 4 . This coupled with (3.14) means that if m 0 ≡ 2 mod 4 then η = 0. That is of m 0 ≡ 2 mod 4 then there is no non-zero linear form invariant under A and w.
If m 0 ≡ 0 mod 4 then from our discussion, every A-invariant linear form is automatically w-invariant and in this case such linear forms satisfy (3.16).
3.3. Orientation reversal symmetry. We now examine the role of an additional possible symmetry, "orientation reversal" r. It need not act on irreducible representations of PSL 2 (R). What we do is given an irreducible unitary representation π on a Hilbert space H, we consider Hilbert spaces U which in the spherical case is the original representation H, and in the case of the discrete series H m where there is a lowest weight vector of weight m > 0, we define U = H +m ⊕ H −m to be the direct sum of the irreducible representations with lowest weight m and that with highest weight −m. We write U (K) for the dense subspace of K-finite vectors in U.
An orientation-reversing symmetry of U is a unitary map r of U which is an involution, that is rπ(E + ) = π(E − )r .

Action of r on weight vectors.
We first note that due to the commutation relation (3.18), r must reverse weights, that is with |c n | = 1 since r is unitary, and c n c −n = 1 since r 2 = I. In particular, in the spherical case when there is a vector φ 0 of weight 0, we must have We say the spherical representation U is even if if the sign is +, and odd if the sign is −.
In the case of the discrete series representations U = H +m ⊕ H −m , m > 0, we choose a lowest weight vector φ m ∈ H +m of unit length, which we call the minimal (or generating) vector, that rφ m is a unit vector of weight −m, and normalize a choice of highest weight vector of unit length by taking (3.22) φ −m := rφ m .
We claim that the choice of minimal vector φ 0 in the spherical case and φ m in the discrete series case uniquely determine r. Indeed, starting with φ m , the lowest weight vector for m > 0, we get from That is for the discrete series r exactly interchanges φ n and φ −n : (3.23) rφ n = φ −n , |n| ≥ m, n = m mod 2 In the case of the spherical representations, the same analysis shows that (3.24) rφ n = ǫφ −n , n ∈ 2Z where ǫ = ±1 is determined by (3.21).

r-invariant functionals.
If η is a linear functional on U, invariant under the action of A and time-reversal symmetry w, then the functional η r : v → η(rv) is also invariant under A and w since r commutes with A and with w. We wish to determine when η r = η. Proof. We start with the spherical case. There is a one-dimensional space of functionals invariant under A and w, and we take the unique one satisfying η(φ 0 ) = 1 .
Hence η r , being itself invariant under A and w, must be a multiple of η, and because r 2 = I we have η r = ±η .
We claim the sign is determined by the sign in (3.21), that is if rφ 0 = ǫφ 0 then η r = ǫη .
It suffices to check this on the spherical vector φ 0 , that is to show η r (φ 0 ) = ǫ.
Since η r is also A-invariant, we have We claim that this happens if and only if c + = c − , which will show that the space of A-invariant functionals which are r-invariant is exactly one-dimensional in this case. Indeed, we have 3.6. Bilinear forms. We apply the above uniqueness of linear forms to biinvariant sesqui-linear forms on U × U. Let T (v, v ′ ) be such a form, that is linear in v, conjugate-linear in v ′ and invariant under A, w and r in each variable separately. For instance, we can take where η j , η ′ k are invariant linear forms. From the prior discussion, if π is of type (ii) with m 0 ≡ 2 mod 4. Otherwise T is completely determined by value T (φ π , φ π ) at the minimal vector φ π . In fact T is the product of linear forms where ξ U ,φπ is the unique invariant linear form taking value 1 at the minimal vector φ π .
3.7. Application to the classical and arithmetic variances. We apply these remarks to the measures µ d and to the classical variance V . We consider the discrete decomposition of the regular representation of G = PSL 2 (R) on L 2 cusp (Γ\G). For an irreducible sub-representation, form the space U π as above. 3.7.1. The arithmetic measure µ d is a linear form on U π invariant under A, w and r. Hence µ d (F ) ≡ 0 if π is a discrete series with weight m 0 ≡ 2 mod 4 and otherwise (3.25) µ Hence if F 1 and F 2 are in U π 1 and U π 2 the sesqui-linear µ d sums take the form This gives a universal reduction for computation of the variance of µ d to the cases 3.7.2. The classical variance V is by its definition diagonalized by the irreducibles in the decomposition of L 2 (Γ\G). We define projections onto the set of w-invariant functions and onto the set of functions invariant under both w and r, We wish to completely determine V sym and V even . For an irreducible π, V sym vanishes on U π if π is discrete series of weight m 0 ≡ 2 mod 4 and otherwise is given by It remains to determine V sym (φ π , φ π ). Lemma 3.3. i) For π spherical with parameter s = ir, For π discrete series of weight m = 0 mod 4 (m > 0), where Beta is Euler's beta function.
Proof. In the spherical case we need to compute V sym (φ 0 , φ 0 ). For spherical representations, we saw that A-invariance and w-invariance automatically imply r-invariance, hence on such spherical U, Moreover, since φ 0 is spherical, wφ 0 = φ 0 and hence φ even By its definition, V respects the orthogonal decomposition into irreducibles; since φ m ∈ π m and rφ m = φ −m ∈ π −m lie in distinct irreducibles, we get V (φ m , rφ m ) = 0 = V (rφ m , φ m ). Moreover we have for any F 1 , F 2 . To see this, note first that r is induced by the measure preserving map x → δxδ −1 of SX = Γ\G and hence Moreover, r commutes with the geodesic flow and so Then applying the raising operator E + via the regular representation gives an operator L E + satisfying (3.28) Also by the unitarity of π, Using the coordinates k(θ 1 ) e r/2 0 0 e −r/2 k(θ 2 ) on G and the formula for L + in these coordinates, we deduce from (3.28) and (3.29) that f (k(θ 1 ) e r/2 0 0 e −r/2 k(θ 2 )) = e −im(θ 1 +θ 2 ) g(r) where g satisfies the ODE We integrate (3.30) and find that where Beta is Euler's beta function. Thus we find 4. Half-integral weight forms 4.1. Basic properties. Let Γ be a discrete subgroup of SL 2 (R) of finite covolume. Given a character χ : Γ → S 1 , an automorphic function of weight k and character χ for Γ is a function f : H → C satisfying with suitable growth conditions at the cusps of Γ. It is cuspidal if it vanishes at the cusps. The Laplacian of weight k is defined as The Laplacian ∆ k maps forms of weight k to themselves, and maps cusp forms to themselves. A Maass cusp form of weight k is a cuspidal automorphic function of weight k (for some character χ) which is an eigenfunction of ∆ k . Let W κ,µ be the standard Whittaker function, normalized so that at infinity The asymptotic behaviour of W κ,µ (y) near y = 0 is The Petersson inner product is defined for a pair of (cuspidal) functions of the same weight k and character χ, as dxdy y 2

Maass operators.
For any real k, define the raising operator and the lowering operator The raising operator K k takes Maass forms of weight k to forms of weight k + 2 and the lowering operator Λ k takes Maass forms of weight k to forms of weight k − 2. Then The effect of the Maass operators on Petersson inner products is given as: If f, g have weight k and character χ, then The action of the Maass operators on the eigenfunctions f ± k (z, s) is

Maass operators and Fourier expansions.
We want to see the Fourier expansion of a "raised" Maass form in terms of its original. So start with a Maass form F of weight 1/2 and eigenvalue λ = 1/4 + r 2 with Fourier expansion Applying the Maass raising operator K 1/2 , we get a form K 1/2 F of weight 5/2 whose Fourier expansion is obtained by (4.4) as Applying the lowering operator Λ 1/2 we get a form Λ 1/2 F of weight −3/2 with Fourier expansion is obtained by (4.5) as .
Let E(z, s) be the standard Eisenstein series for Γ 0 (4): The series is absolutely convergent for ℜ(s) > 1, with an analytic continuation to ℜ(s) > 1/2 except for a simple pole at s = 1, where the residue is Res s=1 E(z, s) = 1 vol(Γ 0 (4)\H) = 1 2π One starts with the integral which is analytic in ℜ(s) > 1/2 except for a simple pole at s = 1 with residue R(F ) = F, F 2π By the "unfolding trick", we have and hence the Dirichlet series has a simple pole at s = 1 with residue Consequently we find Similar considerations show that if we take forms F of weight k + 1/2 and G of weight ℓ + 1/2 (k and ℓ possibly different) which are orthogonal then we have and that if F is a Maass form f weight1/2 for Γ 0 (4) with Fourier expansion and G is a holomorphic form of weight k + 1/2 then These arguments will also give the asymptotics of the sum of squares −N ≤n≤N |4πnρ(n)| 2 of Fourier coefficients with both positive and negative indices. However for our application we need to be able to separately sum only coefficients indexed by positive integers, that is we require the asymptotics of the series N n=1 |4πnρ(n)| 2 To do so, we make use of the arguments in the paper by Matthes [28], which we adapt for our case, see also [27,29].
We next compute the residue at s = 1 when F ′ = F : The arguments and ideas needed to establish this have been essentially provided in [28].
As a consequence, we deduce by a standard Tauberian argument that Corollary 5.3. Let F be as above. Then while if F and F ′ are orthogonal then Proof. Applying the Wiener-Ikehara Tauberian theorem (see [23] for example) to the Dirichlet series 4πn|ρ ′ (n)| 2 n s , and ∞ n=1 4πn|ρ(n) + ρ ′ (n)| 2 n s respectively, we infer by proposition 5.1 that, as N → ∞, Similarly applying the Wiener-Ikehara theorem to the Dirichlet series and consequently, In view of the asymptotics (4.1), (4.2), (4.3), the integral is absolutely convergent for ℜ(s) > 0 and hence M k (s) is analytic in that region. The asymptotic behaviour of M k (s) is given by Lemma 5.4. Assume that |k| < 1/2. Then as |s| → ∞, Proof. This is a direct generalization of Lemma 4.1 in [28], which deals with the case r ′ = r. As in [28], we use the integral representation where the the path of integration runs from −i∞ to i∞ and is chosen so that all poles of Γ(v − k) are to the left, and all poles of Γ( 1 2 − v ± ir) are to the right of L; this is possible since we assume that |k| < 1/2. Inserting this into the formula for M k (s) gives Then one uses the formula One then shifts the contour of integration to the line ℜ(v) = k − 1/2, picking up a single residue at v = k, and estimates the remaining integral as in [28] giving The conclusion of the Lemma now follows from Stirling's formula. Proof. Holomorphy in ℜ(s) > 0 follows from that of M k (s). As in [28], one shows that there is a recurrence relation By the recurrence relation (5.8), we infer that By using Lemma 5.4 with Stirling's formula, we deduce that 6 lim n→∞ (s + n)Γ 2 (s + n) det M (s + n) Γ(s + n + µ + ν)Γ(s + n + µ − ν)Γ(s + n − µ + ν)Γ(s + n − µ − ν) = −2 .
Therefore we conclude that and thus det M (s) = 0 for ℜ(s) > 0.

5.4.
Proof of Proposition 5.1. We define the Eisenstein series of weight −2 and 2 by These series are absolutely convergent for ℜ(s) > 1, with an analytic continuation (no poles) to ℜ(s) > 1/2.
Using the Maass raising operator K 1/2 , we get a form K 1/2 F of weight 5/2 with Fourier expansion and using the lowering operator Λ 1/2 we get a form Λ 1/2 F of weight −3/2 with Fourier expansion Consider the Rankin-Selberg integrals Since E ±2 (z, s) are analytic in ℜ(s) > 1/2, B(s) is analytic in ℜ(s) > 1/2; and since E(z, s) is analytic in ℜ(s) > 1/2 except for simple pole at s = 1, A(s) is analytic for ℜ(s) > 1/2 except for possibly a simple pole at s = 1, where the residue is so that A(s) is analytic also at s = 1 if F and F ′ are orthogonal. By the standard unfolding trick, we find that for ℜ(s) > 1, Proof. Let k > 0 be any half integer. In our case k = 1/2. We have, by unfolding the integral, that

Theta lifts and periods
We summarize the results on theta lifts of Shintani [41] and Kohnen [15,16] in the holomorphic case, and of Katok and Sarnak [14] in the Maass case.

7.2.
Proof of Theorem 1.2. We wish to compute the bilinear form where f, g ∈ L 2 cusp (Γ\G) are smooth and K-finite, and the sum is over discriminants. We take f and g to lie in the irreducible subspaces U π defined in (1.11), that is subspaces of L 2 cusp (Γ\G) irreducible under the joint actions of G, the orientation reversal symmetry r and under the Hecke algebra. We wish to show that for such f , g, if f , g lie in distinct (hence orthogonal) subspaces U f , U g , and to compute B(f, f ). By the results of § 3, it suffices to consider "minimal", or generating vectors, that is to consider holomorphic forms or Maass forms. In particular we need to show that for such f , where c(f ) = 6/π, f Maass form 1/π f holomorphic Since both B and V sym vanish when f is holomorphic of weight ≡ 2 mod 4 or an odd Maass form, it suffices to treat the cases of holomorphic forms of weight divisible by 4 and of even Maass forms. To do so, we recall that in these cases we have identified µ d (f ) with simple multiples of the Fourier coefficients of theta-lifts θ(f, z). Thus we may use Rankin-Selberg theory (Corollary 5.3) to recover (7.1) and (7.2) once we have made the correct identifications. We treat separately the case of holomorphic forms and Maass forms. R + (f ) = Γ(k) √ πΓ(k + 1 2 ) Note that by (3.27) f, f and therefore the residue at s = 1 of D(s) is Consequently we find We note that by (3.26) |Γ( 1 4 + ir)| 4 2π|Γ( 1 2 + 2ir)| 2 φ, φ = V sym (φ, φ) , and hence Therefore we get