Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms

We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$-norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in $\Gamma \backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindelof hypothesis for Maass eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Maass eigenforms need not hold at or below the Planck scale. Finally, we prove similar equidistribution results in shrinking sets for Heegner points and closed geodesics associated to ideal classes of quadratic fields.

1.1.1. Random Wave Conjecture. Let B 0 (Γ) denote the set of Hecke-Maaß eigenforms of weight zero and level 1 on the modular surface Γ\H, where Γ = SL 2 (Z) and H denotes the upper half-plane; we normalise g ∈ B 0 (Γ) to be such that g, g · · = Γ\H |g(z)| 2 dµ(z) = 1, where dµ(z) = y −2 dx dy. A well-known conjecture of Berry [Ber77] and Hejhal and Rackner [HejRa92] states that a Hecke-Maaß eigenform g ∈ B 0 (Γ) of large Laplacian eigenvalue λ g = 1/4 + t 2 g ought to behave like a random wave. Here by a random wave, we mean a function of the form where η(λ) → ∞ as λ → ∞ and η(λ) = o(λ), each f is a normalised Hecke-Maaß eigenform, and the coefficients c f are independent Gaussian random variables of mean 0 and variance 1. These are a randomised model of eigenfunctions of the Laplacian in the large eigenvalue limit λ → ∞, and it is easier to prove (almost surely) results for random waves than for true eigenfunctions.
For Γ\H, there are situations in which random waves do not behave precisely like Laplacian eigenfunctions: random waves satisfy sup z∈K |g λ (z)| ≍ K √ log λ almost surely for every compact subset K, whereas Milićević [Mil10, Theorem 1] proved the existence of a dense subset of points z ∈ Γ\H for which a subsequence of Hecke-Maaß eigenforms g ∈ B 0 (Γ) may be much larger. Nonetheless, it is conjectured that Laplacian eigenfunctions should, on the whole, be well-modelled by random waves. This (admittedly loosely defined) conjecture is known as the random wave conjecture.
In this paper, we study two aspects of this conjecture: bounds for the L 4 -norm of an automorphic form, and quantum unique ergodicity in shrinking balls. The former is a special case of the Gaussian moments conjecture, while the latter is a refinement of quantum unique ergodicity.
1.1.2. Gaussian Moments Conjecture. A particular manifestation of the random wave conjecture states that the moments of a Hecke-Maaß eigenform g ∈ B 0 (Γ) should be identical to those of a Gaussian random variable in the large eigenvalue limit.
Conjecture 1.1 (Gaussian Moments Conjecture). Let K be any fixed compact continuity set of Γ\H, so that the boundary of K has µ-measure zero, and let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. Then for every nonnegative integer n, (1.2) 1 Var K (g) n/2 vol(K) K g(z) n dµ(z) converges to as t g tends to infinity. Here Var K (g) · · = 1 vol(K) K |g(z)| 2 dµ(z).
When K is replaced by a noncompact set, the Gaussian moments conjecture ought not necessarily to hold for high moments. As explained in [HeSt01, Section 4], using a heuristic appearing in [Hej99,Section 7], the transition range of the Whittaker function leads to a "tidal pulse" phenomenon near the cusp of Γ\H; when K is replaced by Γ\H, so that Var Γ\H (g) = vol (Γ\H) −1 , one can thereby show that there exists a subsequence of Hecke-Maaß eigenforms g ∈ B 0 (Γ) for which (1.2) grows like a power of t g whenever n ≥ 12 is even. This is closely related to the fact that there exists a subsequence of Hecke-Maaß eigenforms for which Nonetheless, it is not unreasonable to conjecture that the Gaussian moments conjecture holds for smaller moments when K is replaced by Γ\H. Indeed, the conjecture holds by definition for n ∈ {0, 2} and is easily shown to also be true when n = 1, as both sides vanish, while for n = 3, this can be shown to hold via the work of Watson [Wat08].
1.1.3. Quantum Unique Ergodicity. Another manifestation of the randomness of Hecke-Maaß eigenforms is quantum unique ergodicity. Conjecture 1.3 (Quantum Unique Ergodicity in Configuration Space). Let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. Then the probability measure |g(z)| 2 dµ(z) converges in distribution to the uniform probability measure on Γ\H as t g tends to infinity, so that for every continuity set B ⊂ Γ\H, as t g tends to infinity.
By the Portmanteau theorem, this conjecture is equivalent to for every bounded continuous function on Γ\H. It behoves us to mention that there is a stronger formulation of quantum unique ergodicity, namely quantum unique ergodicity in phase space, which is the cosphere bundle S * (Γ\H) ∼ = Γ\SL 2 (R): not only should the sequence of probability measures |g(z)| 2 dµ(z) equidistribute on the configuration space Γ\H, but that a microlocal lift of these measures to Wigner distributions on phase space should equidistribute with respect to the Liouville measure.
Quantum unique ergodicity in phase space, and hence also in configuration space, is known to be true via the work of Lindenstrauss [Lin06] and Soundararajan [Sou10]. However, this proof does not quantify the rate of equidistribution; in particular, it does not give explicit rates of decay for the terms (1.5) Γ\H f (z)|g(z)| 2 dµ(z) for fixed f ∈ C b (Γ\H) as t g tends to infinity. Watson [Wat08, Corollary 1] has shown that optimal decay rates for these integrals follow directly from the generalised Lindelöf hypothesis.
The n = 2 case of the Gaussian moments conjecture for the set K = Γ\Hnamely the L 4 -norm of g -shares many similarities with quantum unique ergodicity in configuration space. In fact, it is extremely closely related to a more refined version of quantum unique ergodicity, namely equidistribution on shrinking sets.
1.1.4. Randomness of Eisenstein Series. The Gaussian moments conjecture and quantum unique ergodicity ought to be true, once suitably modified, when g(z) = E(z, 1/2 + it g ) is an Eisenstein series. Eisenstein series are not square-integrable, so one must use some sort of regularisation. One method is to use Zagier's regularisation of divergent integrals [Zag82]; another is to replace E(z, 1/2 + it g ) with the truncated Eisenstein series Λ T E(z, 1/2 + it g ) for some T ≥ 1; this is defined for ℜ(s) > 1 by Λ T E(z, s) · · = E(z, s) − γ∈Γ∞\Γ ℑ(γz)>T ℑ(γz) s + Λ(2 − 2s) Λ(2s) ℑ(γz) 1−s and extended by meromorphic continuation to the complex plane; here Λ(s) denotes the completed Riemann zeta function. For quantum unique ergodicity, we need not deal with the truncated version of the Eisenstein series provided that we take into account the growth of the L 2 -norm of an Eisenstein series on compact sets. Theorem 1.6 (Luo-Sarnak [LS95, Theorem 1.1]). For any compact continuity set K ⊂ Γ\H and for g(z) = E (z, 1/2 + it g ), as t g tends to infinity.
Since K is compact, one can replace g(z) with Λ T E (z, 1/2 + it g ) for some T sufficiently large dependent on K. The presence of log(1/4 + t 2 g ) essentially stems from the Maaß-Selberg relation; see Corollary 2.3.
Quantum unique ergodicity in phase space is also known for Eisenstein series; this is a result of Jakobson [Jak94, Theorem 1].
1.2. The L 4 -Norm Problem. The L 4 -norm problem for a Hecke-Maaß eigenform g is the second nontrivial case of the Gaussian moments conjecture.
Conjecture 1.7 (L 4 -Norm Problem). Let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. As t g tends to infinity, A similar statement can be formulated when g is an Eisenstein series, though some care must be taken, since Eisenstein series are not square-integrable; see [DK18].
In general, an unconditional proof of the L 4 -norm problem seems quite difficult. A weaker conjecture (see, for example, [Sar03, Conjecture 4]) is that In certain special cases, this has been shown: when g is a dihedral Maaß eigenform, this is a result of Luo [Luo14], while when g is a truncated Eisenstein series, this is a result of Spinu [Spi03] (with the implicit constant of course dependent on the truncation parameter T ).
Buttcane and Khan [BK17b, Theorem 1.1] have recently given a proof, conditional on the generalised Lindelöf hypothesis, of the L 4 -norm problem for a Hecke-Maaß eigenform g ∈ B 0 (Γ). Our first main result is to give an unconditional upper bound for the L 4 -norm of a truncated Eisenstein series that is sharper than (1.8).
Theorem 1.9. Let g(z) = Λ T E (z, 1/2 + it g ). We have that Up to the implicit constant, Theorem 1.9 should be sharp, for the Maaß-Selberg relation implies that Remark 1.10. Theorem 1.9 was previously claimed by Spinu [Spi03, Theorem 1.2], as was a proof of (1.8) for Hecke-Maaß cusp forms by Sarnak and Watson [Sar03, Theorem 3]; in both cases, however, the proofs are incomplete, as we shall discuss further in Remark 3.3.
Remark 1.11. Djanković and Khan [DK18] have recently reformulated the L 4 -norm problem for Eisenstein series by studying a regularised fourth moment of an Eisenstein series in the sense of Zagier [Zag82]; cf. Section 2.2. This has the advantage that one ought to be able to prove an asymptotic for this regularised fourth moment, whereas Theorem 1.9 only provides an upper bound for the fourth moment of a truncated Eisenstein series.
1.3. Quantum Unique Ergodicity in Shrinking Sets. A natural strengthening of quantum unique ergodicity is to determine whether equidistribution still occurs if we vary the set B with t g ; in particular, if the size of B shrinks as t g increases. This small scale equidistribution should be thought of as a reinterpretation of determining the rate of equidistribution, as opposed to determining explicit rates of decay for the terms in (1.5). Proving equidistribution in shrinking sets has applications towards bounds for the L p -norms and size of nodal domains of eigenfunctions of the Laplacian; see [HezRi16]. We denote by B = B R (w) the hyperbolic ball of radius R centred at w ∈ Γ\H: its hyperbolic volume is which is independent of the centre w.
Question 1.12. Let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. For what conditions on R, with regards to t g , is it still true that as t g tends to infinity?
In the general setting of negatively curved manifolds, this question has independently been answered by Han [Han15, Theorem 1.5] and Hezari and Rivière [HezRi16, Proposition 2.1] for a full density subsequence of Laplacian eigenfunctions with the radius R shrinking at a rate (log λ g ) −β for a particular range of β > 0 dependent on the manifold.
We should not expect equidistribution to hold when R ≪ t −1 g ; indeed, Hejhal and Rackner [HejRa92, Section 5], writing Ψ n in place of g, λ n in place of λ g = 1/4 + t 2 g , and A in place of R, state that . . . in the physics literature, c/ √ λ n is commonly referred to as the de Broglie wavelength. At length scales below c/ √ λ n , one expects the topography of Ψ n to look "essentially sinusoidal", that is, regular. It is only when A is substantially bigger than the de Broglie wavelength that one stands any chance of seeing any type of Gaussian distribution. We confirm this statement by showing that if R ≪ A t −1 g (log t g ) A for any A > 0, then there exist infinitely many points w ∈ Γ\H for which (1.13) does not hold, so that the sequence of probability measures |g(z)| 2 dµ(z) does not equidistribute on the shrinking balls of radius t −1 g (log t g ) A centred at these points. We think of R ≍ t −1 g as being the Planck scale, so that equidistribution need not occur within a logarithmic window of the Planck scale.
Theorem 1.14. Let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. For every fixed Heegner point w ∈ Γ\H, we have that |g(z)| 2 dµ(z) = Ω exp 2 log t g log log t g 1 + O log log log t g log log t g for R ≪ A t −1 g (log t g ) A for any A > 0 as t g tends to infinity. Nevertheless, we should expect equidistribution to occur at every scale larger than the Planck scale, namely R ≫ t −δ g for any δ < 1. Towards this, Young [You16] has proved the following.
In fact, with little work, we can improve the range in Young's result for Eisenstein series.
A simpler version of Question One can also reformulate Question 1.12 probabilistically by asking for which scales equidistribution holds almost surely with respect to a random eigenbasis of Laplacian eigenfunctions; positive results towards this question appear in the work of Han [Han17] and Han and Tacy [HT16].
We study a related question: instead of demanding that equidistribution hold in shrinking balls of radius R > 0 centred at w for every point w ∈ Γ\H, we relax this requirement by instead asking whether equidistribution holds in shrinking balls B R (w) for almost every w ∈ Γ\H.
1.3.1. Conditional Results. We are able to give a conditional proof of equidistribution in almost every shrinking ball when g ∈ B 0 (Γ) and R ≫ t −δ g for any 0 < δ < 1, that is, at all scales above the Planck scale.
Theorem 1.17. Let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. Assume the generalised Lindelöf hypothesis, and suppose that R ≍ t −δ g for some 0 < δ < 1. Then for any c ≫ ε t > c converges to zero as t g tends to infinity.
1.3.2. Unconditional Results. Proving unconditional results seems to be much more difficult. Nevertheless, we are able to do so when g(z) = E (z, 1/2 + it g ) is an Eisenstein series.
Theorem 1.18. Let g(z) = E (z, 1/2 + it g ). Suppose that R ≍ t −δ g for some converges to zero as t g tends to infinity, where D(g; w) is given by (5.7).
This result is consistent with Theorem 1.6 due to the following.
Lemma 1.19. In any compact subset K of Γ\H, we have that for all w ∈ K, In particular, we may rephrase Theorem 1.18 in the following way.
Corollary 1.20. Let g(z) = E (z, 1/2 + it g ), and let K be a fixed compact subset of Γ\H. Suppose that R ≫ ε t −1+ε g . Then for any fixed c > 0, vol > c converges to zero as t g tends to infinity.

Equidistribution of Geometric Invariants of Quadratic Fields in Shrink-
ing Sets. Finally, in Section 6, we study a similar equidistribution problem in shrinking sets. Associated to each narrow ideal class A of the narrow class group Cl + K of a quadratic number field K = Q( √ D) is a geometric invariant. For D < 0, this is a Heegner point z A , while for D > 0, this is a closed geodesic C A or a hyperbolic orbifold Γ A \N A having this closed geodesic as its boundary; we explain these geometric invariants in more detail in Section 6.1.
For each fundamental discriminant D, we choose a genus G K ⊂ Cl + K in the group of genera Gen K · · = # Cl + K denotes the narrow class number of K. Duke, Imamoḡlu, and Tóth have proved the following equidistribution theorem.
as D → −∞ through fundamental discriminants, and If we sum over all genera, so that we are studying equidistribution associated to the full narrow class group, then this result is due to Duke [Duk88, Theorem 1] for Heegner points and closed geodesics, while this result becomes trivial for hyperbolic orbifolds, for there is no error term whatsoever in this case. Moreover, the equidistribution of closed geodesics has a stronger realisation: instead of merely asking for the equidistribution of closed geodesics on Γ\H, we may lift these geodesics to phase space S * (Γ\H) ∼ = Γ\SL 2 (R) and demand equidistribution with respect to the Liouville measure. This has been proved by Chelluri [Che04].
It is natural to ask whether equidistribution still occurs if B shrinks as |D| grows. Towards this, Young [You17a] has proved the following.
for fixed δ < 1/24, where Cl K denotes the class group of K and h K · · = # Cl K denotes the class number. Assuming the generalised Lindelöf hypothesis, (1.23) holds as D → −∞ through fundamental discriminants for fixed δ < 1/8.
In fact, from the method of proof, it is clear that Young's theorem applies to genera mutatis mutandis, and proves equidistribution not only of Heegner points, but also of closed geodesics and hyperbolic orbifolds.
Once again, we may weaken the demand that equidistribution hold in shrinking balls of radius R > 0 centred at w for every point w ∈ Γ\H and instead study whether equidistribution holds in shrinking balls B R (w) for almost every w ∈ Γ\H.
We prove the following conditional result.
Theorem 1.25. Suppose that R ≍ |D| −δ . Assuming the generalised Lindelöf hypothesis, we have that for converges to zero as D → ∞ along fundamental discriminants.
Unconditionally, we obtain the following weaker results.
> c converges to zero as D → −∞ along odd fundamental discriminants, while for > c converges to zero as D → ∞ along odd fundamental discriminants, and for all δ > 0 and c > c converges to zero as D → ∞ along odd fundamental discriminants.
The fact that these geometric invariants equidistribute on almost every ball of different scales should not come as a surprise, and essentially boils down to the fact that a Heegner point has dimension 0, a closed geodesic has dimension 1, and a hyperbolic orbifold has dimension 2. For Heegner points, we need roughly R 2 balls to cover Γ\H, so we require the number of Heegner points #G K corresponding to the genus G K to be at least R 2 in order to expect equidistribution; this is the scale R ≍ (−D) −1/4 . For closed geodesics, on the other hand, R balls will cover roughly 1/R of Γ\H, but a closed geodesic may intersect more than one ball, so we only require the total length A∈GK ℓ (C A ) of closed geodesics corresponding to the genus G K to be at least R; this is the scale R ≍ D −1/2 . Finally, we should expect equidistribution at all scales for hyperbolic orbifolds, since these are just (possibly uneven) coverings of Γ\H.
1.5. Idea of Proof. The chief idea behind the proof of the aforementioned small scall equidistribution theorems is to use Chebyshev's inequality to reduce the problem to bounding a variance. For example, The method of bounding the variance in order to show equidistribution in almost every shrinking ball is also used in [GW17, Theorem 1.6] for eigenfunctions of the Laplacian on T 2 , as well as in both [EMV13, Theorem 1.3] and [BRS16, Theorem 1.8], where the problem investigated is not quantum unique ergodicity, but rather the equidistribution of lattice points on the sphere.
The variance is an inner product of functions in L 2 (Γ\H), as is the fourth moment of a truncated Eisenstein series; both are thereby amenable to being spectrally expanded via Parseval's identity. The resulting spectral sum over Hecke-Maaß forms f occurring in the spectral expansion Var(g; R) when g is an Eisenstein series is essentially the same as the spectral sum for fourth moment of a truncated Eisenstein series in the range 0 < t f ≪ ε R −1+ε , whereas for t f ≫ 1/R, it is much smaller.
Finally, we use the Watson-Ichino formula to write | |g| 2 , f | 2 as a product of Lfunctions. This reduces the problem to bounding certain moments of L-functions, with the length of these moments corresponding inversely to the radius of the shrinking ball.
Though not a manifestation of the random wave conjecture, the equidistribution problems in Section 1.4 nonetheless involve equidistribution on Γ\H, and the proofs of Theorems 1.25 and 1.26 contain many of the same ingredients as the proofs of Theorems 1.17 and 1.18. The chief difference is that in place of | |g| 2 , f | 2 , we have Weyl sums; akin to the Watson-Ichino formula, these can be expressed as a product of L-functions via the work of Duke, Imamoḡlu, and Tóth [DIT16].
1.6. Connections to Subconvexity. The rate of equidistribution for quantum unique ergodicity for Hecke-Maaß eigenforms g ∈ B 0 (Γ) can be quantified via explicit rates of decay for for fixed f ∈ B 0 (Γ) and ψ ∈ C ∞ c (R + ) as t g tends to infinity. Via the Watson-Ichino formula, this is equivalent to obtaining subconvex bounds of the form for some absolute constant δ > 0. Similarly, quantifying the rate of equidistribution for quantum unique ergodicity for g(z) = E(z, 1/2 + it g ) is equivalent to obtaining subconvex bounds of the form For quantum unique ergodicity in almost every shrinking ball of radius R for Hecke-Maaß eigenforms g ∈ B 0 (Γ), on the other hand, we will show that we require bounds of the form That is, we require subconvex moment bounds for L-functions uniformly in two parameters: t f and t g . Thus this is a problem of hybrid subconvexity. Proving such bounds unconditionally seems to be currently out of reach for moments involving GL 3 ×GL 2 Rankin-Selberg L-functions. For g(z) = E(z, 1/2 + it g ), on the other hand, the required subconvex moment bounds are and the fact that these moments only involve GL 2 L-functions makes this problem tractable. It is for this reason that we are able to prove Theorem 1.18 unconditionally, whereas Theorem 1.17 is conditional.

Integrals of Automorphic Forms and L-Functions
However, this is no longer the case when we replace the Eisenstein series with the truncated Eisenstein series The following explicit formula for the inner product of two truncated Eisenstein series is known as the Maaß-Selberg relation.
Using the Taylor expansions together with the fact that |ϕ(1/2 + it g )| = 1 and that It remains to use Stirling's formula to find that and [IK04, Theorem 8.29] to give the bound

The Watson-Ichino Formula.
To deal with spectral sums involving terms of the form | |g| 2 , f | 2 , one can use the Watson-Ichino formula, which essentially states that the square of the integral over Γ\H of the product of three automorphic forms is equal to a product of completed L-functions involving those automorphic forms. In particular, if f, g ∈ B 0 (Γ), then from [Ich08, Theorem 1.1] and [Wat08, Here Λ(s, π) denotes the completed L-function of an automorphic representation π of GL n (A Q ): this is of the form (2.7) Λ(s, π) = q s/2 π L ∞ (s, π)L(s, π), where q π denotes the conductor of π, L ∞ (s, π) is the archimedean part of Λ(s, π), which is of the form π −ns/2 n j=1 Γ( s+κπ,j 2 ) for some κ π,j ∈ C, and L(s, π) is the usual nonarchimedean part of Λ(s, π). Note that the numerator in the Watson-Ichino formula factorises: Similar results also hold when either f or g is replaced with an Eisenstein series.
A similar result also holds when g is an Eisenstein series.
Finally, when f is also an Eisenstein series, the integral is no longer convergent. One can work around this issue by replacing this integral with a regularised integral. This is defined by Zagier [Zag82] in the following way. Let F : Γ\H → C be a continuous function of moderate growth, so that there exists c j , α j ∈ C and nonnegative integers n j such that for all N ≥ 0 at the cusp at infinity, with no α j equal to 0 or 1. Then there exists a function E(z) that is a linear combination of Eisenstein series and derivatives of Eisenstein series E(z, α), each satisfying ℜ(α) > 1/2, such that for some δ > 0, at the cusp at infinity. The regularised inner product of two functions f, g such that f g = F is continuous and of moderate growth is defined to be Moreover, if f and g depend on complex parameters, then we may extend both sides via analytic continuation where possible.
Proposition 2.10 ([Zag82, Equation (44)]). We have that In practice, it is the nonarchimedean part L(s, π) of a completed L-function Λ(s, π) that is difficult to deal with; this is because the asymptotic behaviour of the archimedean part of a completed L-function can be inferred via Stirling's approximation.
Lemma 2.12. The product of the archimedean parts of the completed L-functions in Propositions 2.8, 2.9 (with t = t f ), and 2.10 (with s 1 = s 2 = 1/2 + it g and s = 1/2 + it f ) is equal to where Proof. The product of the archimedean parts of the completed L-functions is The result then follows directly from Stirling's approximation.
On occasion, we also need to deal with lower bounds for L(1, sym 2 f ). This is less complex than values of L-functions within the critical strip 0 < ℜ(s) < 1; indeed, the following is known.

Sharp Bounds for the L 4 -Norm of a Truncated Eisenstein Series
3.1. The Spectral Expansion of the L 4 -Norm. We wish to determine sharp bounds for g 4 L 4 (Γ\H) = Γ\H |g(z)| 4 dµ(z) with g(z) = Λ T E(z, 1/2 + it g ) in terms of t g . Our first step is to express this quantity as a spectral sum, which requires the spectral decomposition of L 2 (Γ\H).
In particular, the following spectral expansion of the L 4 -norm of g is simply Parseval's identity with g 1 = g 2 = |g| 2 .
Corollary 3.2. Let g ∈ L 2 (Γ\H) be of rapid decay. Then This is reduced to understanding bounds for the inner product of |g| 2 with eigenfunctions of the Laplacian. The first term in this expansion is the inner product of |g| 2 with the constant function and Corollary 2.3 shows that It remains to treat the cuspidal and continuous spectra.

3.2.
Ranges of the Spectral Decomposition for the L 4 -Norm. We divide the spectral expansion of the L 4 -norm of g(z) = Λ T E(z, 1/2+it g ) given in Corollary 3.2 into different parts, then analyse each part individually. There are two main ranges of the continuous spectrum to consider, which depend on a small fixed parameter δ > 0: • the initial range 0 ≤ |t| ≤ 2t g + t 1−δ g , and • the tail range |t| > 2t g + t 1−δ g . Both of these ranges will be shown to contribute a negligible amount via subconvexity estimates for the L-functions appearing in the integral.
For the contribution from the cuspidal spectrum, the summation over B 0 (Γ) may be broken up into different ranges depending on t f . There are four main ranges of the cuspidal spectrum left to consider, which depend on a fixed small parameter δ > 0: • the short initial range 0 ≤ t f ≤ t 1−δ g , • the bulk range t 1−δ We divide the spectral sum into these particular ranges due to the size of the product of analytic conductors of L-functions. The analytic conductor of which is large when t f lies in the bulk range, but is small in the short initial range, and drops in the short transition range. For this reason, the main contribution will be shown to arise from the bulk range, while the contribution from the two short ranges will be shown to be negligible. Assuming the generalised Lindelöf hypothesis, this can be proven directly; see [BK17b, Section 5]. Finally, the exponential decay in (2.13) arising from the archimedean components of the completed L-functions indicates that the tail range contributes a negligible amount.
Remark 3.3. In [Spi03, Chapter 6], Spinu sketches an unconditional proof of Theorem 1.9. The proof, however, only treats the spectral sum in the range αt g < t f < 2(1 − α)t g for any fixed α > 0 (essentially the bulk range), in which the contribution of the spectral sum ought to be nonnegligible. The remaining ranges, which all ought to contribute a negligible amount, are left unaddressed. This same issue is present in a claim of Sarnak and Watson [Sar03, Theorem 3(a)] of the bound g L 4 (Γ\H) ≪ ε t ε g for Hecke-Maaß cusp forms, under the assumption of the Selberg eigenvalue and Ramanujan conjectures (but not the generalised Lindelöf hypothesis, as in [BK17b, Theorem 1.1]). Sarnak (personal communication) subsequently has retracted this claim, and instead only claims this bound for the contribution of the spectral sum in the bulk range, as the method he uses is unable to treat the short initial range.
We are able to treat the short initial and transition ranges, left untreated by Spinu, by applying the work of Jutila [Jut04], Ivić [Ivi01], and Jutila and Motohashi [JM05] on certain hybrid moments of L-functions. We do not know how to treat these ranges when g is a Hecke-Maaß cusp form.
3.3. Spectral Methods to Bound the Continuous Spectrum. From Corollary 3.2, we must bound Here c is any constant less than 1/2 − 2θ, where θ is a positive constant such that |g| 2 , f 2 .
Lemma 3.6 ([Spi03, Theorem 4.2]). We have that This allows us to use Proposition 2.9 and Lemma 2.12. We divide the cuspidal spectrum into four ranges, as discussed in Section 3.2. The convexity bound for the associated L-functions together with the Weyl law shows that the tail range is negligible. So it remains to bound the first three ranges.

Lemma 3.8 ([Spi03, Proposition 5.5]). We have that
Remark 3.9. Spinu uses the large sieve only to prove Lemma 3.7 and employs a more complex method in proving Lemma 3.8; nonetheless, one can in fact use the local large sieve, as stated in [Luo14,Lemma] ≪ (log t) 2/3 (log log t) 1/3 .

It therefore suffices to show that
for some δ ′ > 0. We divide the short transition range 0 < t f < t 1−δ g into dyadic intervals H ≤ t f < 2H, of which there are roughly log t g intervals, on which

It then suffices to show that for
This bound follows from the work of Jutila [Jut04], Ivić [Ivi01], and Jutila and Motohashi [JM05]. It is worth noting that the purpose of these works is to obtain Weyl-type subconvexity bounds for Hecke-Maaß eigenforms f ∈ B 0 (Γ), so long as |t| is not too close to t f ; here q(f, s) denotes the analytic conductor of L(s, f ). Conveniently, their methods to obtain such bounds involve obtaining bounds for the exact type of spectral sum that we are studying.
Lemma 3.10. For t ≥ 0 and H ≫ 1, we have that Proof. For H ≥ t 1/2 , this follows from [JM05, Theorem 2], which states that for t ≥ 0 and H ≫ 1, For H ≤ t 1/2 , this follows from the subconvexity bound Corollary 2], and from [Jut04,Theorem], which states that for t ≥ 0 and 1 ≪ G ≪ H, Corollary 3.11. For any δ > 0, we have that and Ivić's [Ivi01] bounds for moments of L(1/2, f ) in short intervals of t f close to 2t g . A similar idea works when g is a truncated Eisenstein series. We must show that for some δ ′ > 0. We use the Cauchy-Schwarz inequality to see that this spectral sum is bounded by t −3/2 g times the square root of the product of The first sum is bounded by and a similar expression holds for the second sum. We then apply the following lemma to show that each sum is bounded by a constant multiple dependent on ε of t 3−δ 2 +ε g , from which the result follows. Similarly, for H ≫ 1, 0 ≤ t ≪ H 3/2−ε , and 0 ≤ G ≤ (H + t) 4/3 H −1+ε , we have that Corollary 3.13. For any 0 < δ < 2/3, we have that 3.8. Spectral Methods to Bound the Bulk Range. In [Spi03, Chapter 6], Spinu proves the bound where ρ(z, w) · · = log |z − w| + |z − w| |z − w| − |z − w| denotes the hyperbolic distance on H. The function u : H × H → [0, ∞) is a point-pair invariant. From this, a function k : [0, ∞) → C gives rise to a point-pair invariant k(z, w) · · = k(u(z, w)) on H. The Selberg-Harish-Chandra transform maps sufficiently well-behaved functions k : [0, ∞) → C to functions h : R → C. This transform is given in three steps as follows: Note that h(t) is real whenever t is real. We shall take k(z, w) = k R (z, w) equal to the indicator function of a small ball of radius R centred at a point w, normalised by the volume of this ball. So and consequently We require the following asymptotics for h R (t), which are extremely similar to the analogous result for T 2 ; see [GW17, Lemma 2.1].

Lemma 4.2 (Cf. [Cha96, Lemma 2.4]). As R tends to zero, we have that
if Rt tends to zero, if Rt tends to infinity.
Proof. If R and Rt both converge to zero, then the dominated convergence theorem implies that If R converges to 0 and Rt converges to some value in (0, ∞), then similarly via [GR07,8.411.10]. So it remains to prove the case that R converges to 0 and Rt tends to infinity. To do this, we let We show that is pointwise convergent as R tends to zero and is uniformly convergent to 0 as x tends to infinity, from which the Moore-Osgood theorem allows us to interchange the order of limits taken in order to obtain the desired asymptotic. Indeed, the dominated convergence theorem once again shows that h(R, x) converges to as R tends to zero. For the uniform convergence as x tends to infinity, we integrate by parts and make the substitution r = 2 R arsinh sin v sinh R 2 , yielding Using stationary phase, with the two critical points being the endpoints ±π/2, we find that there exists some R 0 > 0 such that sup R∈(0,R0) For a function k : [0, ∞) → C, we may form the automorphic kernel which is Γ-invariant in both variables. When k(u) = k R (u), we write K(z, w) = K R (z, w).
Lemma 4.3. If f : Γ\H → C is an eigenfunction of the Laplacian with eigenvalue 1/4 + t 2 f , then Proof. This follows from [Iwa02, Theorem 1.14]. Note that there it is assumed that not only is k(u) compactly supported, but that it is smooth; this, however, is not essential to the proof. Instead, we merely require that k(z, w) be twice differentiable in both variables µ-almost everywhere.
Proposition 4.4 ([Mil10, Theorem 1]). For every fixed Heegner point w ∈ H, |g(w)| = Ω exp log t g log log t g 1 + O log log log t g log log t g as t g tends to infinity.
Proof of Theorem 1.14. For g ∈ B 0 (Γ), It follows by the Cauchy-Schwarz inequality that Theorem 1.14 then follows from Lemma 4.2 and Proposition 4.4.
Remark 4.5. Theorem 1.14 also holds for Maaß newforms g ∈ B * 0 (Γ 0 (q)) for any q > 1, for Proposition 4.4 is proved in this generality (and in fact in even further generality).
Remark 4.6. Since it is conjectured that max w∈K |g(w)| ≪ K,ε t ε g for every compact subset K of Γ\H, we cannot expect any significant improvement to Theorem 1.14 via this line of reasoning.

Proof of Conditional Results.
In this section, we prove the following.
Theorem 1.17 then follows directly via Chebyshev's inequality. Our starting point towards proving Proposition 5.1 is the following spectral expansion of Var(g; R).
Proposition 5.2. Let g ∈ B 0 (Γ) be a Hecke-Maaß eigenform normalised such that g, g = 1. Then Var(g; R) is equal to Proof. Via Lemmata 3.1 (namely Parseval's identity) and 4.3, |g| 2 , K R (·, w) is equal to Upon squaring and integrating over w, we obtain the desired identity.
Proof of Proposition 5.1 for 0 < δ < 1. We use Propositions 5.2 and 2.8 and Lemmata 4.2 and 2.12. We then divide the spectral expansion in Proposition 5.2 into various ranges. Just as in Section 3.2, there are two main ranges of the continuous spectrum to consider: • the initial range 0 ≤ |t| < 2t g + t δ g , and • the tail range |t| > 2t g + t δ g . The division of the cuspidal spectrum into parts depends on δ. When R ≍ t −δ g with 0 < δ < 1, the ranges are: • the short initial range 0 < t f ≤ t δ g , • the polynomial decay range t δ g < t f < 2t g + t 1−δ g , • the tail range t f ≥ 2t g + t 1−δ g . Thus Var(g; R) is bounded by a constant multiple dependent on ε of • From [BK17b, Lemma 2.1], the initial and tail ranges of the continuous spectrum are bounded by t −1+ε g . • The convexity bounds for L(1/2, f ) and L(1/2, sym 2 g ⊗ f ) show that the tail range of the cuspidal spectrum is rapidly decaying. • For the other two ranges, the generalised Lindelöf hypothesis implies that the product of these two L-functions is bounded by a constant multiple dependent on ε of t ε g , and then the Weyl law for Γ\H and partial summation imply that the contribution of the cuspidal spectrum is bounded by t δ−1+ε g . This completes the proof.
Proof of Proposition 5.1 for δ > 1. In this case, the division of the cuspidal spectrum into parts involves an additional range, and there is a dependence on an small fixed parameter δ ′ > 0: • the short initial range 0 < t f ≤ t 1−δ ′ g , which once again is bounded by , which is asymptotic to 6/π from the proof of [BK17b, Proposition 2.2], • the short transition range 2t , which is negligible. This completes the proof.
Remark 5.3. Just as with Theorem 1.14, the bound Var(g; R) ≪ ε t −(1−δ)+ε g for R ≍ t −δ g with 0 < δ < 1 in Proposition 5.1 also holds for Maaß newforms g ∈ B * 0 (Γ 0 (q)) for any q > 1. Indeed, [IK04,Theorem 15.5] gives the spectral decomposition of L 2 (Γ 0 (q)\H), though there are Eisenstein series corresponding to each cusp and the orthonormal basis of Maaß cusp forms are no longer necessarily Hecke-Maaß eigenforms. Nonetheless, Blomer and Milićević have given an orthonormal basis of B 0 (Γ 0 (q)) involving linear combinations of oldforms and newforms [BM15, Lemma 9], and a similar basis exists for the space of Eisenstein series [You17b], and these can be coupled with the work of Hu on the Watson-Ichino formula in this generality [Hu17].
Remark 5.4. In fact, the method of proof of [BK17b, Proposition 2.2] together with Lemma 4.2 show that if R ∼ (Ct g ) −1 for some positive constant C, then [GR07,(8.473.1) and (6.552.4)], which converges to 6/π as C tends to infinity.

5.2.
Proof of Unconditional Results. We first sketch how to prove Theorem 1.16.
Next, we cover the proof of the following, from which Theorem 1.17 will be derived.
Since the constant term of F (z) is we have that F (z) − E(z) = O(y 1/2−δ ) for some δ > 0 at the cusp at infinity, and consequently F − E ∈ L 2 (Γ\H). Lemmata 3.1 (namely Parseval's identity) and 4.3 then imply that The left-hand side is equal to F, K R (·, w) − E, K R (·, w) , and Lemma 4.3 allows us to calculate E, K R (·, w) explicitly. On the right-hand side, the inner product E, f vanishes whenever f ∈ B 0 (Γ), being the linear combination of inner products of Eisenstein series with a cusp form, and similarly F − E, 1 vanishes via [Zag82, Equation (36) and Section 2]. Finally, we claim that the inner product 2it) .
We now define (5.7) Here γ 0 is the Euler-Mascheroni constant and (1 − e(mw)) denotes the Dedekind eta function; note that ℑ(w) 6 η(w) 24 is a Maaß cusp form of weight 12 and level 1 that is nonvanishing outside the single cusp of Γ\H. That D(g; w) is, in some sense, the "true" average of |E(z, 1/2 + it g )| 2 on compact sets, rather than log Proof of Lemma 1.19. This follows from (2.4), (2.5), and (2.6), together with the fact that ℑ(w) 6 η(w) 24 is nonvanishing in K.
With this in hand, we can finally give the spectral expansion of Var(g; R).
Proposition 5.10. Let g(z) = E (z, 1/2 + it g ). Then Var(g; R) is equal to Proof. This follows directly from Lemma 5.9 after an application of Parseval's identity in Lemma 3.1.
Proof of Proposition 5.5. We use Propositions 5.10 and 2.9 and Lemmata 4.2 and 2.12. We then divide the spectral expansion in Proposition 5.10 into various ranges. The two ranges of the continuous spectrum are: • the initial range 0 ≤ |t| < 2t g + t δ g , and • the tail range |t| > 2t g + t δ g . The cuspidal spectrum can be broken into five ranges, which depend on a small fixed parameter 0 < δ ′ < 1 − δ: • the short initial range 0 < t f ≤ t δ g , • the short initial polynomial decay range t δ The continuous spectrum is readily dealt with: For the cuspidal spectrum, we have the following: • The convexity bounds for L(1/2, f ) and L(1/2 + 2it g , f ) show that the tail range is rapidly decaying. • The short initial range is bounded by a constant multiple dependent on ε of t − min{1−δ,1/6}+ε g upon dividing into dyadic intervals and applying Lemma 3.10.
• For the bulk polynomial decay range, we divide into dyadic intervals and use Lemma 3.7, which shows that this range is bounded by t • We divide the short transition polynomial decay range into intervals of length t 1/3 g , use the Cauchy-Schwarz inequality, and apply Lemma 3.12, which gives the bound t − 7 2 (1−δ)+ε g . Proposition 5.5 is proven upon taking δ ′ = 5 7 (1 − δ).
Using stationary phase as in the proof of Lemma 4.2, or alternatively using [Cha96, Lemma 2.4], we have that |h R 2t g + i 2 | 2 ≪ t −3(1−δ) g , while Stirling's approximation implies that Next, we note that , then for all sufficiently large t g , So piecing everything together, we find that if c ≫ ε t −2δ+ε Taking T = ct 3 2 (1−δ) g yields the result.

Equidistribution of Geometric Invariants of Quadratic Fields
6.1. Geometric Invariants of Quadratic Fields. Let K = Q( √ D) be a quadratic field of discriminant D. We denote by h + K · · = # Cl + K the narrow class number of K and h K · · = # Cl K the (wide) class number of K; note that Cl + K = Cl K , so that h + K = h K , except when D > 1 and O × K contains no elements of norm −1, in which case h + K = 2h K . Each narrow ideal class A of Cl + K is associated to an SL 2 (Z)-equivalence class of binary quadratic forms Q(x, y) = ax 2 + bxy + cy 2 of discriminant D.
Associated to equivalence classes of binary quadratic forms are geometric invariants: if D < 0, this is a Heegner point z A ∈ Γ\H, while if D > 0, these are a closed geodesic C A ⊂ Γ\H and a hyperbolic orbifold Γ A \N A whose boundary is C A . This last geometric invariant was introduced by Duke, Imamoḡlu, and Tóth in [DIT16].
6.1.1. Heegner Points z A . Given a binary quadratic form Q(x, y) = ax 2 + bxy + cy 2 of discriminant b 2 − 4ac = D < 0, the point lies in H. The equivalence class of binary quadratic forms containing Q(x, y), and hence the corresponding ideal class A ∈ Cl K , thereby corresponds to a point z = z A in Γ\H, which we call a Heegner point. 6.1.2. Closed Geodesics C A . Given a binary quadratic forms Q(x, y) = ax 2 + bxy + cy 2 of discriminant b 2 − 4ac = D > 0, the points determine the endpoints of a closed geodesic in H. The equivalence class of binary quadratic forms containing Q(x, y) thereby corresponds to a closed geodesic C = C A in Γ\H. The length ℓ(C A ) · · = CA ds of C A , with ds 2 = y −2 dx 2 +y −2 dy 2 , is equal to 2 log ǫ + K , where ǫ + K > 1 is the smallest unit with positive norm in the ring of integers O K of K, so that ǫ + K = (x + y √ D)/2 with (x, y) ∈ R 2 + the fundamental solution to the Pell equation x 2 − Dy 2 = 4. Note that ǫ + K is equal to ǫ K , the fundamental unit of K, if O × K contains no elements of norm −1, whereas ǫ + K = ǫ 2 K if O × K does contain elements of norm −1.

Hyperbolic Orbifolds
be a real quadratic field of discriminant D > 1. Associated to a narrow ideal class A ∈ Cl + K is an invariant ((n 1 , . . . , n ℓA )), where ℓ A is a positive integer and n 1 , . . . , n ℓA are integers; this is the primitive cycle, unique up to cyclic permutations, occurring in the minus continued fraction expansion of each point w ∈ K for which 1 > w > σ(w) > 0 and wZ + Z ∈ A. We define the elements S · · = ± 0 1 −1 0 , T · · = ± 1 1 0 1 of PSL 2 (Z), which generate PSL 2 (Z) as the free product of S and T . For each k ∈ {1, . . . , ℓ A }, define This is an elliptic element of order 2 in PSL 2 (Z). We set Γ A · · = S 1 , · · · , S ℓA , T n1+···+n ℓ A , which is a thin subgroup of PSL 2 (Z). The Nielsen region N A of Γ A is the smallest nonempty PSL 2 (Z)-invariant open convex subset of H. Then Γ A \N A is a hyperbolic orbifold, which naturally projects onto Γ\H. The boundary of Γ A \N A is a simple closed geodesic whose image in Γ\H is C A , and the volume of Γ A \N A is πℓ A .
Remark 6.1. In fact, Γ A depends on the choice of w. The resulting hyperbolic orbifold Γ A \N A ends up being only unique up to translation; however, the projection of Γ A \N A onto Γ\H is independent of the choice of w.
6.2.1. Variances and Weyl Sums. We define The proofs of Theorems 1.25 and 1.26 follow via Chebyshev's inequality from the following two propositions.
Proposition 6.4. We have that Proof. This follows from the spectral expansion of K R and Parseval's identity.
To bound these variances, we require upper bounds for the Weyl sums as well as lower bounds for #G K , A∈GK ℓ (C A ), and A∈GK vol (Γ A \N A ).
Lemma 6.5. We have that Proof. We have that #G K = 2 1−ω(|D|) h + K and ℓ(C A ) = 2 log ǫ + K , while it is shown in [DIT16, Proposition 1] that The class number formula states that The result then follows from the Landau-Siegel theorem and the bound L(1, χ D ) ≪ log |D|.
6.2.2. Genus Characters. The character group Gen K of Gen K is the group of real characters of Cl + K . These genus characters are indexed by unordered pairs of coprime fundamental discriminants d 1 , d 2 ∈ Z satisfying d 1 d 2 = D. To each pair d 1 , d 2 , we let χ = χ d1,d2 denote the genus character corresponding to d 1 , d 2 : this is a real character of the narrow class group Cl + K that extends multiplicatively to all nonzero fractional ideals via χ(p) · · = χ d1 (N (p)) if (N (p), d 1 ) = 1, χ d2 (N (p)) if (N (p), d 2 ) = 1, for any prime ideals p ∤ d K , where χ d1 , χ d2 are the primitive real Dirichlet characters modulo d 1 , d 2 respectively. In particular, χ is a quadratic character unless either d 1 or d 2 is 1, in which case it is the trivial character.
We abuse notation and write G K for an element in the coset of Cl + K corresponding to the genus G K . This allows us to write and analogous identities for W GK (zA),∞ (t), W GK (CA),∞ (t), and W GK (ΓA\NA),∞ (t). This has the advantage that we are able to show in each case that the square of the sum over A ∈ Cl + K is essentially equal to a product of L-functions.
Proof. This follows from [DIT16, Theorem 3], akin to the proof of Lemma 6.7.
For unconditional results, we make use of the following bounds. Proof of Proposition 6.3. We bound the variance by breaking up into ranges as in the proof of Proposition 6.2. Instead of applying the generalised Lindelöf hypothesis, we use the generalised Hölder inequality with exponents (3, 3, 3). Via the bounds in Lemmata 6.11 and 6.12, together with the Weyl law, we obtain the result.

Representations of Integers by Indefinite Ternary Quadratic Forms.
We briefly describe how the results in this section can be interpreted in terms of indefinite ternary quadratic forms. For simplicity, we only discuss the case of negative discriminant and summing over all genera; for positive discriminant, a detailed presentation can be found in [ELMV12, Section 2]. Consider the indefinite ternary quadratic form Q(a, b, c) = b 2 − 4ac.
It is natural to ask whether the normalised level sets G D cover V Q,−1 (R) randomly as D tends to −∞ along fundamental discriminants. Each level set V Q,D (Z) is countably infinite, and V Q,−1 (R) is isomorphic to C\R, which is not of finite volume, so one cannot immediately rephrase this random covering as equidistribution.
On the other hand, the group SO Q (Z) · · = A ∈ SL 3 (Z) : Q(Ax) = Q(x) for all x = (a, b, c) ∈ Z 3 acts transitively on V Q,D (Z), and the quotient space SO Q (Z)\G D is finite for all fundamental discriminants D, with cardinality equal to h K . Moreover, SO Q (Z) is a discrete subgroup of SO Q (R) of finite covolume, and V Q,−1 (R) ∼ = SO Q (R)/K with K equal to the maximal compact subgroup of SO Q (R), and so the space SO Q (Z)\V Q,−1 (R) is of finite volume. Thus to ask whether the normalised level sets G D randomly cover V Q,−1 (R) can be rephrased as asking whether the finite sets SO Q (Z)\G D equidistribute in the finite volume space SO Q (Z)\V Q,−1 (R). This has a positive answer by naturally realising this result in terms of the equidistribution of Heegner points on Γ\H, as proved by Duke [Duk88, Theorem 1]. Indeed, the fact that Q is indefinite implies that SO Q is isomorphic to the split special orthogonal group SO 1,2 , and we have the accidental isomorphism SO 1,2 ∼ = PGL 2 , while K ∼ = SO 2 (R). From this, we see that SO Q (Z)\V Q,−1 (R) ∼ = PGL 2 (Z)\PGL 2 (R)/SO 2 (R) ∼ = Γ\H, while SO Q (Z)\G D is naturally identified with the set of Heegner points {z A ∈ Γ\H : A ∈ Cl K }.
With this reinterpretation in mind, we now see that Proposition 6.2 implies that under the assumption of the generalised Lindelöf hypothesis, almost every shrinking ball of radius R ≍ (−D) −δ with 0 < δ < 1/4 in SO Q (Z)\V Q,−1 (R) contains a normalised equivalence class of points (a, b, c) ∈ Z 3 that represent the integer D by the indefinite ternary quadratic form Q(a, b, c) = b 2 − 4ac. This complements [BRS16, Theorem 1.8], where the analogous result is proved for the definite ternary quadratic form Q(a, b, c) = a 2 + b 2 + c 2 .