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Quantum ergodicity of Eigenfunctions on PSL2(Z)/H 2

Quantum ergodicity of Eigenfunctions on PSL2(Z)/H 2 QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\I-I 2 W~.NZm LUO and PF.TER SARNAK 1 To Wolfgang Schmidt on the Occasion of His 60th Birthday 1. Introduction Schnirelman [SH], Colin de Verdi~re [CD] and Zelditch [Z1] have proven the quantum analogue of the geodesic flow on a compact Riemannian manifold Y being ergodic. Let A denote the Laplacian on Y and % an orthonormal basis of L*-eigenfunctions of A. The corresponding eigenvalues are denoted by ),j. One forms the probability measures d~tj(z) := I%(z)I" dV(z), dV being the volume element (actually they consider a microlocalization of these measures to S~(Y), the unit cotangent bundle). If the geo- desic flow on S~(Y) is ergodic they show the existence of a full density subsequence )'Jk (i.e. one satisfying Y~ik<~X 1 "~Y~xj<X 1) for which ~jk(A)~Vol(A)/Vol(Y) for all nice sets A (e.g. geodesic balls). Zelditch [Z2] has extended this result to noncompact surfaces such as the modular surface X = PSI_~(Z)\H 2. He shows that if h e C~~ the space of smooth functions on X with compact supports, and j~ h(x) dV(x) = 0, then (1) X I<h,~>l ~ xj.~< x log 7." Here A ~, B means I AI~< CB where C depends only on -r. Selberg [SE] has shown that in this case, ~xs~< x 1 ~ ),/12 and from this one can easily deduce the quantum ergodicity using (1). Recent works of Hejhal-Rackner [H-R] and Rudnick-Sarnak JR-S] suggest that at least for this X much more is true. Namely, that there are no exceptional subsequences, that is ~t~ ~ dV as j -+ oo. This phenomenon might be called Quantum Unique Ergo- dicity. Hejhal-Rackner confirm this numerically while Rudnick-Sarnak show that certain natural candidates for such singular limits (that is, measures concentrated on closed geodesics) do not occur. 1. The research of the first author was supported by NSF Grant DMS 9304580, while he was a member at the Institute for Advanced Study during 1993-1994. The second author was partially supported by NSF Grant DMS 9102082. 208 WENZHI LUO AND PETER SARNAK This paper is concerned with this individual equidistribution conjecture for X. While we fall short of proving it we obtain a number of results in that direction. Firstly we prove the conjecture for the continuous part of the spectrum of X--that is we show that the Eisenstein series become individually equidistributed. Secondly for the discrete spectrum (cusp forms) we show that if exceptional subsequences occur they must be very sparse. We also introduce the discrepancy--a well-known measure of equidistri- bution for sequences--to quantify the measure of equidistribution of the ~Ts. This enables us to show that except for a sparse set ofj's the ~j's become equidistributed at a certain rate. For more background on this problem see the Lectures [SA]. Along the way we establish a conjecture of Iwaniec concerning the average size of Rankin-Selberg L-func- tions on their critical lines. The latter may be used to obtain new bounds for the remainder term in the Prime Geodesic Theorem (see below). We turn to a precise description of our results. The spectrum of A on L2(X) consists of three types, see Hejhal [H2]: (A) %(z) = ~/3/~, the constant function; (B) q~l(z), ?2(z), ..., an orthonormal basis of cusp forms, Aq~ + ),j % = 0; (C) E z,~ + it , t >1 0, the unitary Eisenstein series which furnish the continuous spectrum, AE -t--( 1 + t2) E = 0. We will assume that the basis % is chosen to be simultaneously eigenfunctions of the Hecke algebra. This choice is possible and in fact determines the %'s up to a scalar and hence determines ~j uniquely. It is quite likely that there is only one o.n.b. of %'s anyway, since the numerical evidence points to the spectrum being simple [H3, ST]. We define ~t ---- E z, -~ + it Note that ~x~(X) = o% so that there b( ) is no canonical normalization of ~x t . Our first result is that ~t become individually equidistributed. Theorem 1.1. -- Let A, B be compact Jordan measurable subsets of X, then lim Ez,(A) _ VoI(A) ,-~oo tzt (B) Vol(B) " The renormalizafion is actually needed since we in fact show that as t -+ o% ~,(A) ~ 48 Vol(A) log t. (2) 7~ Jakobson [J] has recently extended Theorem 1.1 to the microlocalizations ~t of ~, to S~(X). QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H ~ 209 To describe our main result concerning ~j we introduce some norms on functions on X. For H e C~(X) let II H It~,~ be defined by: 0~1+~, (3) II H [[k,. = max sup l y k ~+~.<~ ,~ (0~) ~ (Oy)~ (~) , where F is the usual fundamental domain for X in H. For H an integrable function on X we denote by H the mean value Vol(X~---) H(z) dV(z). Theorem 1.9,. -- For ~,> 0 and H e C~(X), X(1/2) + ~ ' f H(z) dtLj(z ) -- H 2 '~ [[ H 1[~,8 x/<x the implied constant depending on ~ only. The upper bound here is essentially the square root of that in (1) and in fact is sharp (i.e. it cannot be replaced by any exponent less than 1/2). Theorem 1.2 asserts that on average H(z) d~,~(z) -- H is of size X71/4. We expect that this is true indi- vidually (see [SA]). As a corollary to Theorem 1.2 we can address a question of Zel- ditch [Z], as to the size of an exceptional subsequence. He showed in general that such a subsequence must be of zero density. The following asserts that an exceptional subse- quence must be very thin. Corollary 1.3. -- Let Jk be a subsequence of j's corresponding to a subset S C N and for which ~Jk -+ ~ 4= 3dV/rc. Then for any ~> 1/2, IS c~ [1, N]] = O~(N~). We also establish Theorem 1.2 for H an individual Eisenstein series, that is xj<x "'2 +it'~s) <~([ t l + 1)" X"/2'+*" By Cauchy's inequality and in view of the integral representation 4rd P(s) fF E(z, s) I q~j(z)l ~ dV(z), L(u s | us, s) = p2(s/2O P(s/2 + itj) P(s/2 -- its) the last implies the following "mean Lindel6f" conjecture of Iwaniec [I1]: L UsNU~, + it (5) 2: 2 <~(Itl-4- 1)4X 1+~, xi< x cosh(=ts) where L(u~| uj, s) is the Ranldn-Selberg L-function (see w 2 and w 3 below) and ),j = (1/4) + t~. We apply (5) to counting prime geodesics on X. Let ,~r(x) -- [{P I N(P) < x}l, 27 210 WENZHI LUO AND PETER SARNAK where P runs over the primitive conjugacy classes of F = PSL2(Z) and N(P) is the corresponding norm [H1]; ~r(x) counts the number of prime geodesics on X of length at most log x. Theorem 1.4. ,~r(X) = li(x) + O(x (7'~~ +'), for ~ > 0. Here as usual, li(x)= dr/log t. For a general discrete cofinite F C PSL,(R) the best-known bound for the remainder term is O(xS/4/log x) [RAN, SE], while for F = PSLz(Z), Iwaniec in the paper quoted above established the bound O,(x135/4sl+*). In w 6 we will give an outline of a proof of Theorem 1.4. It is based on (5) and Iwaniec's method in [I1]. It should be pointed out that the expected remainder term here is O~(xllm+*). In the analogy with the Riemann zeta function and primes, the analogue of the Riemann Hypothesis for Selberg zeta function is true. Even so the abundance of eigenvalues puts the O,(x lira+ ~) bound completely out of reach. Indeed the O(x s/*) bound is reasonably straightforward but anything beyond that invoIves capturing cancellation in the sums over the eigenvalues. To quantify equidistribution of the ~j's it seems best to avoid issues of subsequences (as has been traditional in this problem) and to investigate directly the discrepancy. Various notions of discrepancy have been introduced in connection wich equidistribution of sequences [K-N]. The spherical cap discrepancy D(~) is defined by (6) D(~) = sup ~(B) 3Vol,B_(~ BCX 7~ where the supremum is over all injective geodesic balls in X. It is clear that if D(~) -+ 0 then Vtj become equidistributed and the size of D(Vtj) gives the rate. The choice of balls in this geometry seems natural enough. A somewhat bold conjecture that emerges from Theorem 1.2 (see also [SA] and the question of Colin de Verdi~re [CD]) is that for ~> 0, D(Vtj) ~X7 11/41+~. As pointed out in [SA] this equidistribution rate if true would be optimal. Theorem 1.5. Z I D(~j)12 x '2~ + xj~<x The main achievement in Theorem 1.5 is that the exponent of X is less than 1. We have made no effort to reduce it further which is certainly possible by these methods. On the other hand the optimal exponent 112 seems out of reach by these methods. From Theorem 1.5 it follows that with exception of a very sparse set, the [zr become equidistributed at a rate which is a negative power of k s. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSLz(Z)\H ~ 211 We end the introduction with some comments about the proofs. As pointed out in [SA] these questions of equidistribution are in part related to obtaining non-trivial bounds for Ranking-Selberg L-functions on their critical lines. Concer- ning Theorem 1 an involved but pleasing computation which exploits each factor of ( 1 ) ( 1 ) E z, ~ + it E z, -~ + it differently, leads to two features: (1) The entire problem in this case reduces to estimating L-functions. (2) The Rankin-Selberg L-functions in question conveniently factor into Euler products of degree at most two. We can therefore appeal to known estimates on the Riemann zeta function and L-functions of cusp forms (Meurman [MEU]) to prove the Theorem. For the case of cusp forms, viz Theorem 2 we no longer have any direct relation to L-functions. Our method is to first establish Theorem 2 for certain families of h's called incomplete Poincar~ series. This allows us to exploit that ~j is a Hecke eigenform and to represent [ q~(z)13 with expressions involving the Fourier coefficients in quadratic polynomials. Eventually this allows us to use the Fourier coefficient--Kloosterman sum connection, that is the Petersson--Kuznetsov trace formula [KU], to convert the problem to estimating exponential sums. To do so effectively Well's bound on Kloosterman sums is used as a key arithmetical ingredient. Also crucial in our analysis are the recent bounds of Iwaniec [I2] and Hoffstein-Lockhart [HN-L] for Fourier coefficients of cusp forms in the j aspect. Theorem 1.5 is derived from Theorem 1.2 using only Fourier analysis and geometry. The key point here is the structure of the spectral development of the characteristic function of a geodesic ball. All of the above results may be proven for congruence subgroups of PSL~(Z) (save for the possibility of small eigenvalues Xj < 1/4 intervening in Theorem 1.5). However inasmuch as we use heavily Poincar~ and Eisenstein series (and related Fourier coefficients) we do not know how to establish these results for compact arithmetic surfaces. 2. Eisenstein Series This section is devoted to the proof of Theorem 1.1. The Eisenstein series E(z, s) for X are defined by 1 y" (7) E(z, s) = ~] y(yz)" = ~ ,~o ~r~\r ,~ ~lcz +dl ~"' where ~R(s) > 1, z = x + ~y, F = PSL.(Z) and Foo ={z~z+n,n~Z}. The Fourier development of E(z, s) is well known [SA2]: 2Y 1/~" ~ n 8- ,/2 (8) E(z,s) =y" + ~(s)y 1-" + ~-~,=1 al_2,(n ) K,_,/2(2~ny ) cos(2rcnx), 212 WENZHI LUO AND PETER SARNAK where ~(s) = ~-,/2 P(s/2) ~(s), q~(s) -- ~(2s -- 1), ~(2s) ~(~) = y~ d ~ a[. and K is the Bessel function. In order to prove the equidistribution of ~t we consider its inner products with various functions spanning Lz(X). We begin with inner products with Maass cusp forms %.. Set (1)(1 %(z) E z,~+it E z,~--it yZ . (9) J,(t)-=fx~d~=fx To investigate this we first consider (10) L(s) = ~s(~) E ~, ~ + it E(~, ~) y--r Note that all of the above integrals converge rapidly since % is a cusp form. Now for ~R(s) > 1 we can "unfold" the integral in (10) using the definition (7) to get (11) Is(s ) = q~s(z) E z, ~ § it y~ --o y2 Since E(z, s) = E(-- ~, s) it follows that I~(s) =- 0 if q~ is an odd cusp form (the cusp forms are of two types %(-- 5) -~r ~= 1,-- 1) so we may assume that % is even. In this case it has a Fourier development ao (12) %(z) =yl/2 ,~ p~(1) Xs(n ) K,9(2T:ny ) cos(2~nx). Here (1/4) + t~ = X s and the coefficients Xs(n ) satisfy the multiplicative relations which are a consequence of %(z) being a Hecke eigenform. (We will ignore the normalization pj(1) since in the discussion of this section j is fixed.) These amount to oo Xs(n ) _ II(1 -- xj(p)p -~ +p-~")-~ (13) L(%,s):= Y~ n ~ So o~ (yl/2 ,~=1 Xs(n ) K,j(2rcny) cos(2~nx)) 2112 co ) 4(1 -[- 2it) ~ nu ~-2"(n) K~,(2rcny) cos(2rrnx) y" dxdy n ~ 1 y2 oo ( ~=lX (n) K~tj(2rcny ) n~ t a-~t(n) K,t(27:ny))Y" dy__ if; 4(1 § 2it) K~tj(27:y ) K~t(2rcy) f dy__ ~(1 + 2it) .~1 ,e ] Y QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H 2 which on evaluation of the integral ([G-R]) yields it) R(,) r(s+i j+it) 2 2 2 2--S (14) 4(1 4- 2it) P(s) /Z ~t X~(n) ~- 2it (/z ) where R(s) = n= 1 /Z s Now R(s) may be expressed in terms of the L-function L(%, s) in (13): (15) R(s) ----- L(%, s -- it) L(%, s 4- it) ~(2s) To prove (15) first we factor R(s) (we write X(n) for X~(n) in short), R(s) = H R~(s) io where oo (16) R~(s) = Z x(p j) p*~t ._2.(pa) p-~, J=0 Now ~_~.(p~) = Z p-~"~ = ~=o 1 -- p-2. SO R~(s) = ~ x(p j) p~jt 1 -- p-~.(J+l~ 5=0 1--p-~" P-J~ co oo --p-~" ]~ x(p~)p -~l'+"') 1 -- p-~" ( ).= ZoX(p~ p-Ol,-u, i=0 p-- 2it ) 1( 1 1 -- x(p) p-~,+-I + p-21~+m 1 -- p-2U 1 -- x(p) p-I,-.) + p-2~.-m -- 1 -- p--28 (1 -- x(p)p -'*-"' + p-2,.-.,) (1 -- x(p)p -''+"' +p-Z("+"')" This proves (15). Using this in combination with (14), (11) and (9) we get P + P 4- it~ _ it) = 271:-- 2it 214 WENZHI LUO AND PETER SARNAK One can check using the functional equation for L(%.,s) that the RHS of (17) is real as it has to be in view of (9). An interesting point here is that J~(t) = 0 if L(q%,l/2) =0. We are now ready to investigate the behavior of Jj(t) as t -+ oo. Using Stirling's formula (I F(a -+- it)] ~e-=ltl/21t ]o-(~m), we see that the r factors in (17) will yield a factor ,--- c [ t I- a/2 as t ~ oo (here t~ is fixed). This together with the arithmetic estimates: (A) (logt) -a ~ 1~(1 + it) l ~ logt, (1 )[ (B) L ?~,-~+it <j,,It ~>0, yields Proposition 2.1. IJ~(t)l <~,~ It[ -'~/~'+*, for all r O. The estimate (A) above is well-known in the theory of the Riemann zeta function (IT]). Actually later we will need an improvement of (A) due to Weyl. Indeed this method of Weyl of estimating exponential sums is used crucially by Meurman [MEU] (1) in his proof of the estimate (B) above. The standard convexity bound for L % ~ -t- it l ( 1 )] i~lm+~ is L q%,~ + it <j,~ It . Clearly this would not suffice to show J~(t)-+0 as t -+ oo. However any improvement (in the exponent) of the standard bound would suffice for our purpose. This completes the analysis of the inner products of ~xt with cusp forms. We turn now to inner products of ~t with incomplete Eisenstein series. Let h(y) ~ C~~ +) be a rapidly decreasing function at 0 and 0% that is, for any positive integer N, h(y) = ON(y ~) when 0 <y ~< 1, and h(y) = ON(y -~) when y > 1. Let H(s) be its Mellin transform (18) H(s) = h(y)y_S dy --o Clearly H(s) is entire in s and is of Schwartz class in t for each vertical line a + it. The inversion formula gives = H(s)y' ds (19) h(y) 1 I ~+~w for any a e R. For such an h we form the convergent series y, h(Y(TZ)) = 1 f H(s) E(z,s) ds; (2O) F~(z) vGr~\F ~ J~(s)=2 QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H * 215 F h belongs to C~176 and is rapidly decreasing in the cusp (i.e. as y -+ oo). Hence we may form 1 H(s) E(z,s)ds E z,~+it y~ I( 2~i (,) = 2 _ 1 H(s) y*ds E z,-~+it y, fo'~ Ii ( 1 ) 2dxdy 2hi (*) = 1 H(s)Sd s yl/2+. + ~ + it yi/~-*t 2hi (~) = 2 f0i, ( + 14(1 -& 2it)I' = I K"(2nnY)[2 " Now q~(i[1 +it)~=\ 1, so that the first term above contributes (as t-+m) (21) 2 fo o h(y) d_y + a rapidly decreasing function of t depending on h. The second term which we denote by I~(t) is Oo t/v I K. (2ny)]~y-:- ds. I2(t) = nil ~(1 + 2it)12 (~,=~ ' :: ].L 3' The series can be evaluated as was first done by Ramanujan [RA] l a_2,(n) 1~ _ ~(s) ~(s -- 2it) ~(s + 2it) .=1 n ~ ~(2s) The y-integral is evaluated in terms of F-functions as before. We obtain (note that t is fixed while s is the variable for the integral) In(t) = in 14(1 ~- 2it)t ~(2s) F(s) ds (s) = g _ _ 2 ~ B(s) ds, say. in ]4(1 + 2it)] ~ ~(,)=~ 216 WENZHI LUO AND PETER SARNAK Now shift the integral to ~R(s) = 112, 4hi + 2 [ B(s) ds + O(t-~~ In(t) in 1~(1 + 2it)[ 2Res"=lB(s) in 1~(1 § 2it)[ 2 3~(s)=112 The O-term comes from the contribution of poles at s = 1 i 2it. To justify shifting the contour we use Stifling formula to estimate the F-factors and the fact that H(a + it) is rapidly decreasing in t. In fact using this and Weyl's bound (22) ~(2 + it)<~t a/6,+", we find that B(s) ds 4~ t u/8)+' t -~/~ = t-(116)+~ (23) in 1~(1 + 2it)] 2 (s)= 1/2 This corresponds to the bound in Proposition 2.1. The residue term is more complicated. Write B(s) as ~(s) G(s) where G(s) is holomorphic at s = 1. Then ( G' ) Ress= 1B(s)= G(1) 2,(+~-(1) , where ,( is Euler's constant. Now a simple cal- culation gives G(1) = --24 H(1) 7~ G' H' ~' ~' ~-(1) =~ (1) -t-C+~(1--2it) +-~(1 + 2it) and r'(s_ i,) r' +r,2 + + C being independent of t. According to the Weyl-Hadamard-de la Vall6e Poussin bound [T] ~' log t (24) -~ (1 -+- it) ~ log log~ and Stirling's approximation -~ + it = log t -}- O(I), we have =-- (logt ] Ress=i B(s) 48~ H(1) log t + O k~] (25) QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL,(Z)\I'I ~ 217 as t-+ oo. Note that ye 9 H(1) = fo ~ This leads to Proposition 2.2. -- Let F E C=(X) be of the form F~ as in (20). Then -~- y2 ] log t as t ---~ oo. With Proposition 2.1 and 2.2 we are ready to establish the following Proposition which by standard approximation arguments implies Theorem 1.1: Proposition 8.3. -- Let F e C00(X) (i.e. F is a continuous function of compact support in X); then -~ yZ ] log t as t ----~ oo. Proof. -- It is easy to see that the functions of the form F h as above together with the cusp forms ~0~ are dense in C0(X ) (the space of continuous functions vanishing in the cusp). Let F E Co0(X ) and ~ ~ 0; then we can find G = G 1 + G2 with G 1 a finite sum of cusp forms and G 2 in the space of incomplete Eisenstein series with corresponding h E C~0(R+), such that [[ G -- F ][~ < ~. If H = G -- F then H is rapidly decreasing in the cusp and so we can find an h I/> 0 which is rapidly decreasing and for which Hl(z ) = E h~(y(yz) /> ] H(z)[, "rE Poo\P and fx H~(z) dV(z) < 5~. Hence by positivity of d~t we have ~<-- lim 1 f H(z) d~t(z ) Jx From this the Proposition follows easily. To end this section we remark that while ] < F, ~z t > [ is expected to be of size t-1/2 for F in the space of cusp forms (and smooth), the above shows for F an incomplete Eisenstein series and of mean zero, that ( F, ~z t >]log t ~ 0, but this convergence is very slow. 28 218 WENZHI LUO AND PETER SARNAK 8. Incomplete Eisensteln Series and Poincar6 Series In this section we will establish Theorem 1.2 in a very special but important case, i.e., H(z) is either an incomplete Eisenstein series or an incomplete Poincar6 series. For these functions, we have the advantage of being able to "unfold" the integral d~j(z) appeal automorphic theory fx H(z)and then to L-function and the Petersson- Kuznetsov formula. In the next section we will see that Theorem 1.2 holds in general by an approximation argument. Proposition 8.1. -- Let h(x) be a smooth function on (0, oo), supported in (xo, oo), x o > O, such that for some C~,k >>. 1, I h"'(x) l -< c,,~ x -~, i, k ~> o. Let Ph, o(Z) = Y~ h(y(vz)). Then for any a > O, T >1 1, we have X I<Pho, iUj(Z)]=>-P~o] 2 ~,ClllC T 1+*, , , , 8,8 tj<T 1 I~ P~,o(~) uv(~). where Ph, o -- Vol(F\H) \H Proof. --By unfolding the integral and using (12) we have, with pj(n) = 0j(1) Xj(n), fo o luj(z)12( ~ h(y(yz)))dV(z)= ~ ]0j(k)[ 2 K (r) Set hl(x ) = h(x-1), and hi(x) x 8-1 dx = h(x) x -8- ~ dx. (26) G(s) = [~ 3o Then G(s) is entire and by Mellin inversion 1 [ G(s) x_Sds, a>0. (27) hi(x) = ~ Jco~ It is clear from the definition of G(s), by partial integration, that C1,1 (28) G(s) ~~ l) ... (s-t+ 1)[ QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H 2 219 for l/> O, sr and 1 >> ~=9t(s)>/ %>0. By Mellin transform and a well-known formula (see (35)), it follows that K~i(r ) h r _dr _ ! [" G(s) 2=i ~,(2= I k I)' J, where ~ > 1. Then we have [ us(z ) { 2 Ph, o(Z) dV(z) (29) fr\a _ if8rd__ o, G(s) L(us| co oo where L(u s| = ~] [ps(n) l 2n -8=los( 1)12 ~lX]( n) n-*" We move the line of integration to ~(s) = 1/2 and pass the simple pole of the inte- Ph, o(z) dV(z), in view of grand at s= 1 with residue Ph,0--Vol(r\H) 1 Ir \H dy = 3=_ 1 G(1), which Res,= 1 L(u s | us, s) = 12= .2 cosh(=tj) and Ph,0 = 3r:-1 1] ~ h(y)y follows from the unfolding method. In order to finish the proof, we need to understand the behavior of L(u s | uj, s) uniformly in j and s, which is of independent interest. Let L(2~(u s) stand for the second symmetric power L-function [SHI] attached to the Maass-Hecke form us(z), and Rs(s ) = ~(2s) ~ Z](n) n-8, the Rankin-Selberg convolution L-function. We have oo ~o Rj(s) = ~(s) L(e'(us, s), ~: X](n) n -8 = ~(s) Z ),/n 2) n-8. n=l n=l Hence co oo L(2)(us, s) = ~(2s) Z Xs(n ~) n -8 = ~= .=1 los(n) n-8, can) = Z xs(k2), 12k = n for 9t(s) > 1. It is well known from the work of Shimura [SHI] that L(2)(u~ ~., s) is entire and 220 WENZI-II LUO AND PETER SARNAK is invariant under the change of variable s -+ 1 -- s. Define and let w = (1/2) + ito. Consider the integral (x > O) X--S LI2~(uj, s + w) F(s + l) ds, (3o) --1 f 2~i o~ where l is a positive integer, and ~ = (1/2) + I/log tj. Clearly (30) equals co y~ cj(r,) F(nx), r = 1 .t0 1 U(s+l) t- ds= e -~*-~d~. where F(t) = ~ ol s Moving the line of integration in (30) to --., we pass the simple pole of the integrand at s = 0 with residue F(l)L~21(u~, w), and get 1 I L'~'(u~, s + w) r(s + l) x-~ ds. r(/) L~~ w) + f~ ,-o~ s Now the integral equals -- 1 ~ L~21(u x ~ , j,--s+w) r(--s+l)--ds 2~i Jlo~ s 11 ~ o ~i~ s + ~ 0~/s + ~/~ (s ~+ ~) ~Is + ,/, ~ Moreover 0j(s + ~) - t~'~+~,-~(~'+;-~) -~ (l + o(I s + ~ I ~+~) t71) 0A- s + w) ~1 (1 + oil s + t;1). In [13] itis shown that 2~,<~ X~(n) < tj N, hence L~2~(uj, s + ~) = Rj(s + ~)/~(s + ~) < t~, where ~ is an arbitrarily small positive number. We conclude that the above integral equals ( =3 1,0,~,+~ xo t~ (31) _ ~.0 t;~.o ~ ~(~) F w, .=~ ~---~ xC + o(I w ,), QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H 2 221 where (32) V(w, t) = P(-- s + l) -- ds. al 2 Hence, we have derived the "approximate functional equation" r(l) L(2)(us, w) = ~ q(n) .=1 n w F(nx) + ~0 q~"" Z cj(n) F w, ( ~] + o(I w I '~'~'+~ x ~ t]). .=17 xt U Now we integrate both side of the above expression from ~81~ i to e~al2/t~ with respect to the measure dx/x, which yields, for ts ~ T (we denote T < tj ~< 2T by t~ ,~ T), 5, F -- n= 1 7/, w ~Tlti Y fTIt~ ~lc~(n ) ( ~_y) dy [is/z,+aT_am+a). @ ~3ito l~2ito ---=- F w, -- + 0 (I w = n w Tier Y We observe that: 1. F(w, t) ~t -~ I w [~, t > 0. This is easily seen by moving the line (a) in (32) to (a). 2. F(w,t)~z(t/]w[) -l+(lm, t>0. This is also easily seen by moving the line (~) in (32), but this time to (l- 1/2). Thus, if t/Iw I > T ~ then for any N, F(w, t) ~ T-N(] w lit) ~" by choosing l = IN/a] § 4. Hence, by Cauchy's inequality, we have ~--1 F Z I L(~>(u~,w)l ~ ~ I] ;/ = y T<tj~2T T<tj~<2T /2 Z =F w, -- +O( Tl+alw 12e T<ti~2T. n=l n w y Now ffwe choose l = [10/a] -{- 4, and note that [cj(n)/n~l  n (1/2,+8 since I ),~(k)l < k 1/~ [SA2], we have s = F w, ~lw[ 2. n~(mlwl)X+e n w Also ~>~TI+~ n w 222 WENZHI LUO AND PETER SARNAK Using [D-Ill X ] 52 a. vj(n)[ 2< (T 2+N) T ~ X [a,~l 2 , ty~<T n~<N n~<N the lower bound [I2], vj(1) >> t~ -~ (where o~(n) = cosh(~t/2) v~(n), vj(n) = vj(1) Xj(n)), the definition of ej(n), the above remarks on F, and Cauchy's inequality, we deduce that (33) Z I L'2'(uj, w)l 2 < TZ+~lw [ ~+~. T< t]~2T Finally, by the upper bound v~(1) ~ t~. [HN-L], and the crude bound ~(w) ---- O(I w W2), we have proven Theorem 3.2. (34) I L(u~| w)[ ~ < T~+~ I w l '11/~'+~ tj<T cosh 2~tj This establishes (4) and the conjecture of Iwaniec (5). We return to (29). By Cauchy's inequality, (fr~Hlu~(z) [2 ph, o(Z ) d~(z)--Ph, of o(1/2) But according to Theorem 3.2 and Stirling's formula, r Idsl 52 I 'G(s)"L(uj| tj~T ~sl~<T/10 Tl+e Cs,8 Also, we have trivially, again using Theorem 3.2 and (28), I [ tj ~ T ~s I/> T/10 Cs,8 T 1+~. This completes the proof of Proposition 3.1. Proposition 3.3. -- Let h(x) be a smooth function on (0, ~), supported in (Xo, oo), x o > O, such that for some C~, k >1 1, ]h~)(x)[~< C~,kx -k, i,k>/ 0. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H s 223 Let Pj,.~(z) = ~2 h(y(yz)) e(mx(yz)). ve roo\r Then for any ~ > O, m 4: O, T >>. 1, we have x I < p~,~, I ~(~)I ~ > I ~ <~ m2(C~,2 + Co~,o) T'+*" tj<~ Proof. -- Without loss of generali W we may assume that the first Fourier coef- ficient p~(1) of uj(z) is real. It follows from the unfolding method that I uj(z)12 ( E h(y(yz)) e(mx(yz))) dV(z) Y E F~\ r flail ;o o ( ---- Y, pj(k) pj(k + m) K.i(r) K~q l +-~ r h --r k*0, --m = pj(1) Y~ N p~(k 2 -I- km]d) K.j(r) Kui 1 + ~ r h dl,~k*O,--,~/e 7~dlk[ r Here we use, for k> 0, pj(k)= pj(1)Xj(k), O~(-- k) = % pj(k), ~5 =  1, and the multiplicafivity of Hecke eigenvalues dl (n, r -~ " By a standard dyadic partition it suffices to cstimatc thc sums with t~ ~ T, k ~ K, K > 0. The cases where dK >i AT (A is sufficiently large) can be ignored, since from [G-R] Kit (x) = io o e-X cooh~ cos t-~ d-r, we have K.(x) ~x -le-~% x/>0. So the contribution from these terms is exponentially small, in view of 05(1) ~ cosh(r:t/2) t~., X~(k) ~ k 1/2, and h(y) is supported in y >~Y0 > 0. Henceforth, we assume dK ~< AT. Similarly, we can assume m ~ T, because otherwise r I 1 + re~d* [ > AT. Recall [G-R] x -x K~(ax) K~(bx) dx (35) Io ~ 1--X--~+v l_~,l_b~) 2 ' 2 ' ' 9t(a + b) > 0, ~lX< 1 -- {9t~[ -- 191v [, 224 WENZHI LUO AND PETER SARNAK where oo F(,, ~,y, z) = ~] ~(~ + 1) ... (~ + n-- 1) ~(~ + 1) ... (~ + n-- 1) z" ,=0 "((7 § 1) ... (V + n -- 1) nl is the hypergeometric series. Taking 9~= l--a, ~t =v----its, a=b = 1 or a = b = I 1 + m/dk [, and using ab ~< (a e + b e)/2 and Stirling's formula we obtain ti~TY~ 11 k~x ~ 95(1) p~(k~ + km/d) Kitj(r ) Kit j 1 + ~ r h 2~ d[ k 1 , ~o 9 C 2 K e T'. 0,0 Here we have applied the following result due to Kuznetsov [KU, D-I2]: (36) Y~ I v~(/)I e e-9/T = 27:-2 T 2 -~- O((T -k l l/e) d(l)). ti~<T Hence we can assume K >> T 1/2, and m ~ K 2/3. With the above reduction, we proceed to prove Proposition 3.3. As before, set hl(x ) : h(x-1), and define G(s) as in (26). We use the Mellin transform to obtain (with a : e) fo (r) r K~@r) K~ti( [ 1 + m/dklr) h ~ ~- _ 1 f G(s) fo"rS_lK~q(r) K~ti([1 +m/dklr) drds. 2~i o) (2~ dk) 8 To deal with the inner integral we apply the formula (35) and [G-R p. 1040] r '-1 K~ti(r ) K,ti((1 + mldk) r) dr : 2 -s+~ r l'rs/'~-x(1 -- "r) "/2-~ 1 + -~- -4- x dr. If we can show (37) Y~ I IF_, a k v~(k 2 -4- km/d) f(k, t~)]2 ~ mT2+, K, tj~T k~K QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H I 225 where a k=a~(m,d,'r,s) = k 2 1 +~+'~ m ]"i 1+~ f(k, t~) = f(ra, ~, k, "~, q) = 1 + -~- +, k ak// then from Cauchy's inequality, (28), Stirling formula, and by considering [ ~s[~< T/10 and [~s[> T/10 separately, we obtain Proposition 3.3. In the following we will prove (37). It suffices to treat the ease where [t;--T[~<T 1-'. We infer that E e -C"i-*'m-'' [kExa~ v~(t? + kmld)f(k, t,)l ~ = Y, ak ~z Z v~(k ~ + kin/d) ~(P + lm/d) h(t~) + O(1), k,l~K j where k(t) = e-(" - ~,/~a-,,, f(k, t) f(l, -- t) + e- '" + T,l~t-.,t f(k, -- t) f(l, t). Applying Kuznetsov's formula [KU] to the inner sum, we deduce that ~ vj(k 2 + km/d) ~j(l ~ + lm/d) h(t~) " I/ ~ l,I = -~- t tanh(rct) h(t) dt -- - ~ i~;( 1 + 2it) r_ ~,(k ~ + kmld) d.(/~ + trald) dt + - Y~ c -1 S(k ~ + km/d, l ~ + lm/d; c) J2,, cosh(~zt---~ dr. 713 r ~o Here S(m, n; c) is the Kloosterman sum and d I d 2 ~ n Since '_' t tanh(~t) h(t) dt ~ T s+', d,(k' + km/d) d,(l ~ + lm/d) dt ~ m" T 1+', Io h(t) ]~(1 + 2it)i" 29 226 WENZHI LUO AND PETER SARNAK their contribution is at most m ~ T ~ +" K. It remains to estimate the sum of Kloosterman sums which we shall do for each modulus e separately. If c > K 2 T -z +', we estimate the integral transform of Bessel's function 2i L,, 4~/(k~ + krn/d ) (l 2 + lm/d) t cosh(~t) dt 7r j_Q ~ C by moving the line of integration to ~t = -- B, where B is a sufficiently large integer, depending upon r From Poisson integral formula [G-R] (z/2)~ f[ J~(z) = cos(z cos 0) sin 2~ OdO and Stirring formula for F(s), it is easy to see that the resulting integral is very small. The residues of the integrand at the relevant poles are also very small. Therefore it remains to consider terms with e ~< K ~ T -1 + ~. We have h(t) j~,, ~/(k~ + km[d) (/a-k- lm/d) t dt cosh(~t) ao C f T -[- TI-z 1~ T (4=~r = Y. u J2, cosh(~t~ dt -4- O(T-1). ~=+I jT_TI_$ log2 T C We need the following Van der Corput's 1emma [T]: Lemma 3.4. -- (1) If f '(x)>l ~ > 0, or if(x)<<. -- ~ < 0, x e [a, b], then e ul') dx ~ -. ff 1 tz (2) Iff"(x) >>. r> O, orf"(x)<~ -- r< O, x e [a, b], then e u(*) dx ~ ~. ff 1 Now (see [ER]) 1 1 e -~ ' (1 + O(r-1)), J,,(x) - (r + TM + x -r to,(x) = r~-U-+ x ~ + r log X QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSI.~(Z)\H z and 0 V~r~+x2+r Or %(x) = -- log 02 1 -- ~,(~) --- Or* ~ + x 2 If T/(K2/r > A -1, using part (1) of Lemma 3.4 and Weil's bound for Kloosterman sum [W] 1 1 [ S(m, n; c)[ <~ a(c) (m, n, c)~d, we deduce that the corresponding contribution is 1 K 9 T :+" max %/7 T 1 +' ~ -- ~ KT'. ~K~-~+~ ~ T K If T/(K2/c) ~< A-1, and c >> m, we use (1) of the Lemma to deduce that the corresponding contribution is max ~/c 1 TX +, ~ KT ". IfT/(K*/r ~< A -1, and r ~ m, we use (2) of the Lemma to deduce that the corresponding contribution is + g max ~/~ 1 T 1 +~/-~2 ~ %/~T1 ~ c Hence, summing over k, l, we obtain (37). 4. Appro~dmatlon The estimates for (P, ~ ) which were established in the previous section for incomplete Poincar6 series may be used to obtain similar bounds for a general smooth function F. To do this we need to approximate such functions by P's. Let F e C~176 Let Cz, C2,..., GT. be a decomposition of X into neighbourhoods, C z of the cusp, (3 2 of the point i, C 8 of the point p = e ~/s and C j, j = 4, ..., L containing no elliptic fixed points. If we choose a partition of unity subordinate to this decomposition, we can write F = ~'=1 Fj where Fj has the same smoothness properties as F and each Fj is supported in (3~. So for our purposes here we can assume to begin with that F is supported in such a neighbourhood Cli0" Let Ci be the lift to H of C~ into a fundamental K2/____fc 228 WENZHI LUO AND PETER SARNAK domain. Let ~ be the Fco periodic function on H which is equal to F in C~.,. Clearly F(x,y) is smooth and is supported iny/> 1/2. Also (38) F(z) -- Y, F(yz), Wio y~ rco\F where W~o = 1 ifjo + 2, 3 and otherwise is the order of the stabilizer. Expanding F(z) in a Fourier series in x gives cO (39) ~(z) = Z h.(y) e(mx), in=-- co hence co (40) F(z) = W~--~ ,~=-. ~: V~.,.(z). This will serve as our means of approximating F by the P's. From partial integration we see that, for ~t 1> 0 and k >t 0, 0J+~ F(z) , (41) Y~I h~'(y) [ ~< ([ m I + 1) -~ sup 3P [ .~>1,~ (ay) ~ (ex) ~ hence, for ~t I> 0, and L /> 0, (42) max C,.~(hm) ~< (1 m + 1 ])-~ I[ F I]r,,L+~. i,k~L The norm is the one introduced in (3). Now Z ( P,...... P-J ) I. W~o [ < F, ~ ) -- ~ I = [(( P~o,0 ~J ) -- P~o,o) + m:t:o Applying Gauchy's inequality to this with weight ~,, > 0, ~o = 1, gives I<F,~>--FI ~ '~ (Z~: ~) (I ( P,.o.O ~# > --P~.ol~ + Z ~.1 < Pn,....~j > Is). m m ::t: 0 Summing this for [ t~]~< T yields I= Y, I<F,~j>-~l ~ ty~<T Z ~. X I <Ph.,.~> D. ,n,o tj~T m q~<T Using (42) with ~t-----3, We are read 7 to apply the main result of section S. ~,, = ([ m I + 1) st" and the bounds of section 3, we get I 4. II F 11~..8 T 1 +" This established Theorem 1.2. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSI.~(Z)\H * 229 Finally we establish the above quantity for F = 9xk" We can estimate ~xk(z) from its Fourier expansion. It is easily seen that ~b.(z) is exponentially small for y ~ t~ xv, [I-S]. In fact in [I-S] a stronger estimate has been esta- and also that [[ ?z i ][oo ~--~ blished. However we use the crude bound which together with the Fourier expansion yields Corollary 3.1. -- Let kk ~e 0; then ti<~T Note that this bound is only of use when T is much larger than h since the trivial bound for the above sum is T2[ t, [- This Corollary is the analogue of Theorem 3.2 which is equivalent to *i< T ., ~ + it , ~ ~, (I t I + 1) 6T~+'- 5. Discrepancy This section is devoted to proving the upper bounds for the discrepancy D(~t~) as claimed in Theorem 1.5. The noncompactness of X leads to some technical compli- cations. To deal with these, choose 20 > 1 and set Vol(B) I D~o(~j) = sup ~t~(B) Vol(X) B(~, r) C X, ~ 6F, ~(~)~ Yo Vol(B) , and D,o(~Zj) ---- sup ~.j(B) Vol(X), so that D(~) = max(DU~ D,o(~)), and in particular (44) We will estimate D ~~ and D~o differently and we begin with D,o(bS). Let k(z, ~) be a point-pair invariant on H (see [SE]) and K(z, ~) = ~ver k(Yz, ~)" If Zm~,,~(z) is the characteristic function of the ball B (which we are assuming is injective in X) and if we define k,(z, ~) = 1 when d(z, ~) < r, and = 0 otherwise, then we have (45) X~(z) = K,(z, ~). 230 WENZHI LUO AND PETER SARNAK According to the spectral expansion [SE] we have, at least in the L~-sense, oo (46) K,(z, ~) = Z h,(t~) V~(z) r k=O +~ h,(t) E z,-~+i E ~,-~+it dr. 1i Here h,(t) is the Harish-Ghandra-Selberg transform [SEL] of k,. In order to use the expansion (46) to estimate the discrepancy we must smooth K, so as to deal with absolutely convergent series. For ~ > 0 let d?. be an approximate identity, that is, t~(z, ~) /> 0 is supported in a ball of radius ~, and H+,(z, ~) aV(z) = 1. We can and will also choose d?~(z, ~) so that +,(z, ~) ~ r and its Harish-Ghandra- Selberg transform h'" satisfies [ h~'(t)[ ~ 1 for I t]~< 1/~ and is rapidly decreasing for [ t[ >> l/e. Given the ball B = B(~, r) as above, let A 1 = B(~, rl) , A, = B(~, r2) where r 1 -= r -- 2r and r 2 -- r + 2~ (if r 1 < 0 then A 1 is taken to be the empty set and ?(Ax = 0). For a function F(z) defined on F\It we set (v 9 ~) (z) := f F(~) ~,(~, z) aV(~t, where ~,(z, ~) = Z d?,(yz, ~). y~F It is easily seen that with these choices (47) k,~ 9 ~,(z) ~< ?(A(z) ~< k,2 9 ~,(z). For l ----- 1, 2 the expansions of these functions take the form co (48) k,,, ~,(z) = ~__20h,,(t~) h'"(t~) ~(~) ~(~) 1;(1)(1) + ~ hn(t ) hC~'(t) E z, ~ + it E ~, -~ + it ,it. The mean value over X of the functions in (47) differ by a quantity which is O(r and we may therefore conclude that Vol(B) ] (49) ~(B) Vol(X) ~ ~ + ,=~ y~ I ~,01~ hn(tk) h'"(t,) ( ~, V, ) %(~) 1 " (.,-~+)E 1 it)dt +_~f_hn(t) h,,,(t)(~,E 1 it) (~,-~+ . QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H z 231 Now I Z h,~(t,) h("(t,) < ~, ,~ ) +~(~)1 ~ k,o <~ ( Z [ h,t(t~)I ~ ] h'"(t~)II ~(~)I ~) ( y' I < ~,, ~ > I ~ I ~'='(t~)I), k*o k*O which, according to the estimate on h ~, is ( Y~ I ~,,(t~)i~1 ~(~)i S) ( y~ I ( ~, ~ ) I~) 9 [tkl.~ 11~, k# 0 I~kl .~l/r Using (46) we see that (50) x I ~,,(t~)i ~ I r~(~)I ~ r I I K,,(~, ~)I ~ aV(~). The right hand side can easily be estimated uniformly in ~, r for ~(~) ~< Yo and one finds it to be O(1 + sSyo). We have shown that ~.o l ~kl .~ 1/r k* 0 The same considerations hold verbatim for the Eisenstein series contribution. On taking supremum over all balls with ~(~)-<<Y0 we are led to (51) [ i)~o(~j)12 ~ ~2 + (1 + a"Yo) ( ~ I ( ~-,, % ) [ ~ Itkl ~l/*,k*o Next we turn to estimating D~~ which we do in a crude fashion. First we need an upper bound for ~q.<T 1%(z)]2. Ifk~(z, ~), ~ = I/T is an approximation to the identity as before (and for which h(t) >t O, which can be arranged), then Ko(z, z) = X k~(z, vz) >> E I ~,(z)I S. v~r ~j~r On the other hand the middle term above is easily estimated as being ~ T2+ Ty, where y = ~(z). It follows that (52) Y~ 1%(z)[2 ~ T 2 + Ty. q.<<T From the Fourier expansion of %(z), it is easily seen that I%(z)12 is exponentially small for y >> t t, ]. Hence in considering balls B(~, r) when computing DV0(~z~), with lt, l.< T, we can ignore those balls with center ~ satisfying ~(~) >> T. (Note also that the volume of such a ball is O(T-2).) For B = B(~, r) withy0 < ~(~) ~< T we have Vol(B) Vol(B) i~ a~ dy (53) ( ~;, B ) Vol(X) ~< Vol(X----~ + I ( ~,, B ) I ~< + I q~(z) y~ 0 232 WENZHI LUO AND PETER SARNAK Taking supremum over all of these balls and using the fact that D(~i) ~< I we conclude that dx 4y (54) I D~~ ~ .e ! + I ,~/z)I ~ f S; Yo o y2 " Gathering the bounds, we have (55) I n(~@ [ 2 ,~ r + (1 -t- *Syo) ( ~ I < bq, % ) t ~ I tkl '~ l/t, /g ::l= 0 ~dy I ~/z)I ~ f We now sum these inequalities for t~[~ T and apply Corollary 3.1, (43) and (52), where in the latter we use y ~< T, to get E t[' + Itl~ T '+'~ + -- Y~ D(~) ~ ~. T 2 ~ + (I +Yo ~) ,~, Yo tj ~ T t [ ~ llr for ~l > 0. That is T 2 D(btj) 2 ~ T ~ z 2 + (1 +Y0 z3) T~+* ~--19 ..{_ --. ts~<'r Yo Now optimize the choices of ~ = T -a, Yo = T% This yields the bound T 2-c2/211 for the RHS. This concludes the proof of Theorem 1.5. 6. Appendi~ In this section we will give an outline of the proof of Theorem 1.4 which follows closely the method in [I1]. Indeed using the argument below together with the analysis in w 9 of [I1], one can improve the exponent 7/10 slightly. However the bound O,(x ~2/s~+ ") which is mentioned following equation (12) of [I1] appears to remain out of reach. On the other hand, the exponent 2/3 can be deduced conditionally on the Lindel6f Hypo- thesis for the usual Dirichlet L-functions (see [I4], p. 189). We first show that XUi exp(-- t/T) ~ T 5/* XVS(log T) 2. (56) Let h(~) be a smooth function supported in [N, 2N] whose derivatives satisfy ]h~V'(~)[ ~N -~, forp=0,1,2 .... and = h(~) d~ = N. I~ co QUANTUM E, RGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H z 233 Thus, in review of [I1 lemma 8] and (5), we have h(n) [ vj(n)12 = ~ N + r(tj, N), 52 [ r(t~, N)[ ~ T ~ N~/~(log T) ~. tj~<T We deduce that ~h(n) (~ l vj(n)l ~ X ~*i exp(-- tilT)) = ~ ~ (~[ v~(n)[~'k(n)) X"i exp(-- tJT) ti _ _ 12 E X *ti exp(-- tilT) + O(T 2 N-Ira(log T)2). 7~ ~ q Therefore, we only need to treat (57) ~ ] vj(n)l 2 X"i exp(-- tilT) for n ~ [N, 2N]. Let q~(x) be a smooth function on [0, oo] such that [~(x)[ <x, x-+0, I #"(x)t < x -", x -~ oo, for p = 0, 1, 2, 3. Define ~o = ~ 1 E Jo(y) ~(y) , ~%(x) = ~XJo(~X) J0(~y) q~(y) dy d~, f fo ~.(x) = ~XJo(~X) Jo(~y) ~(y) dy ar~, /fo~ --o ~(t) - 2i sinh ,~t [L.(x) -- J_~.(x)] ~(x) ax '~ Io '~ With these definitions we have ~(~) = ~(~) + ~(~) 30 234 WENZHI LUO AND PETER SARNAK and the Kuznetsov formula [KU] reads. For ll, l~ t> 1, Y, c}(b) ~(t,5 + _2 I * ~(t) V j(/1) 4,(h) 4,(t,) at ' '~ 30 1~(1 + 2it)[ = = a,,,,, Oo + ~ c-' S(/1, 12; c) ~,~ (4~ ~i-~l, lc). For ~ we choose -- sinh ~(.) = x exp(ix cosh ~), 29 = log X + ~. Then [D-I2 lemma 7] sinh(~ -t- 2i~) t ~(t) = sinh ~t -- cosh [~ % = 2= ~ sinh ~ ~' -- sinh 29 txJo(tx) (cosh ~ ~ -- t~) -8/2 at, -- sinh 29 I 0~ t~)_8/~ ~H(x) - ~ j~ tXJo(tX) (cosh ~ ~ -- at. It is easy to see that ~(t$) = XUi e-ti/T + O(e-'~ti), and q~o ~ X- x[~, - 2 f~ ~(t) J0 1 ~(1 + 2it)I n (d~'(n))2 dt,~ T(log T) 2 d2(n). Furthermore, Jo(y) ~ min(1,y -~/2) and [ S(n, n; c) l <~ (n, v) 112 c v~ d(c), hence S.(~s ) ~ N 1/~ X-1/2(log N) ~ and S.(~) ~ N x/2 T 1/~ X 1/4 log T, where s,(+) = X c -1 s(n, n; c) + --. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSLs(Z)\H tt So we conclude that Y~ exp(-- tilT) XUJ N a/2 T v2 X v4 log T + N a/2 X- a/~ log2 N + N- a/2 T ~ log2 T + T log ~ T T 6/4 X 1/8 log 2 T, on taking N ---- T ~/~ X-v4. Next we show that the above estimate is still valid if we replace the smooth weight function exp(-- t/T) by the characteristic function X(t) of [1, T]. Take a smooth func- tiong(~) such that supp(g) _ [1/2, T + 1/2], 0 ~< g(~) ~< 1, andg(~) = 1 when ~ ~ [1, T]. Since it is known that t 1> 1 and [{t~:T<t 5~< T+ 1}]<T, we have Y~ X"i = Gg(tj) X uj + O(T). q~<T t i Let g(~) be the Fourier transform of g(~)exp(~/T): g(x) =f+_2g(~)exp(~/T)e(~x)d~. The easy estimate g(x) ~ min T, gives f [g(x)ldx~logT. For [x[ >/ 1, from partial integration 1 f+~ d g(x) -= 2~ix ~ (g(~) exp(~/T)) e(~x) d~, we infer that I e(--yx) g(x) dx 1 f+f ff-~ (g(~) exp(~/T)) fl+~176 dxd~ 1 1 log(T + Iy [) ~--+ + lYt + 1 IT--yl + 1 T Thus, from the Fourier inversion formula g(x) exp(x/T) =I+~g(~)e(--~x)d~, 236 WENZHI LUO AND PETER SARNAK we deduce that g(x) exp(x/T) = s g(~) e(-- ~x) d~ 1 1 log(T + Ix D) +O ix[+l+lT_xl+l + T " Therefore Ng(t,) X'q = g(~) (s (Xe-~"~)"~ ' exp(-- t/T)) d~ tj --1 (~ e- ,i/T e- tilt + log(T + tj) e- ti/Tt T ] ~ TS/* X1/8(l~ T)=" -4-0 --~ b IT-hi + 1 Thus, XUJ ~ T 5/* Xl/8(log T) 2. (58) Now [I1, lemma 1] X(112) + i ~j + O (X log9 X), (59) tIPr(X ) =X+ E tj<~ (1/2) + itj where ~r(X) = ~]~{v}<xAP and AP = logNP o if { P} is a power of the primitive hyperbolic class { Po }. Hence from (58), (59) and partial summation, we deduce that lri,(X ) = X -~- O(X 7/10 log S X) on taking T = X$/1~ Finally Theorem 1.4 follows by partial summation. REFERENCES [CD] Y. COLIn DE VER~dmE, Ergodicit6 et fonctions propres du laplacien, Com. Math. Phys., 102 (1985), 497-502. [D-If] J.-M. DESaOmLLERS, H. IWAmEC, Kloosterman sum and Fourier coefficients of cusp forms, Invent. Math., 70 (1982), 219-288. [D-I2] J.-M. DSSHOmLLERS, H. IWAmSC, The non-vanishing of Rankin-Selberg zeta-functions at special points, Selberg trace formula and related topics, Contemp. Math., 53, Amer. Math. Soc., Providence, RI, 1986, 59-95. A. ERDf~LYI et al., Higher Transcendental Functions, vol. 2, McGraw-Hill, 1953. [ERI [G-R] I. S. GR~SHTEIN, I. M. RYZmK, Tables of Integrals, Series and Products, Academic Press, New York and London, 1965. [H1] D. A. HEJHA% The Selberg Trace Formula for PSL(2, R), Vol. 1, Springer Lecture Notes, 548, Springer- Verlag, 1976. [H2] D. A. HEJHAL, The Selberg Trace Formula for PSL(2, R), Vol. 2, Springer Lecture Notes, t001, Springer- Verlag, 1983. D. A. t~JHAL, Eigenvalues for the Laplacian for Hecke triangle groups, Memoirs ofAMS, Vol. 469, 1992. [H3] [H-R] D. A. HEJ~L, D. RACKNER, On the topography of Maass wave forms, Exper. Math., I (1992), 275-305. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H ~ 237 [HN-L] j. HOFFSTEIN, P. LOCKHART, Coefficients of Maass forms and the Siegel zero, appendix by D. Goldfeld, J. Hoffstein, D. Lieman, An affective zero free region, Annals of Math., 140 (1994), 161-181. [11] H. IWAmEC, Prime geodesic theorem, J. Reine Angew. Math., 349 (1984), 136-159. [I2] H. IwAsmc, Small eigenvalues for I'0(N), Acta Arith., LVI (1990), 65-82. [13] H. IWANmC, The spectral growth of automorphic L-functions, f. Reine Angew. Math., 428 (1992), 139-159- [l-S] H. IwAm~C, H. SARNAI<, L ~~ norms of eigenfunctions of arithmetic surfaces, To appear in Annals of Math. [14] H. IWANmC, Non-holomorphic modular forms and their applications, Modular forms, Durham conference, edited by R. Rankin, 1984, 157-196. D. JAKOBSO~, Quantum ergodieityfor Eisenstein series on PSL2(Z)\PSL~(R), Preprint, Princeton, 1994. [J] [K-N] L. KOIPERS, H. NmDEm~SlTER, Uniform distribution of sequences, New York, Wiley, 1974. [KU] N. V. KOZNETSOV, Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture, Sums of Kloosterman sums, Mat. Sb., 111 (1980), 334-383. [MEU] T. M~URMAN, On the order of the Maass L-function on the critical line, Number Theory, Vol. I, Budapest, 1987, Colloq. Math. Soc. Janos Bolyai, 51 (1990), 325-354. [RA] S. RAV.ANUJAN, Some formulae in the arithmetic theory of numbers, Messenger of Math., 45 (1916), 81-84. [RAN] B. RANDOL, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. AMS, 233 (1977), 241-247. Z. RUDmCK, P. SARNAK, The behavior of eigenstates of arithmetic hyperbolic manifolds, Com. Math. [R-s] Phys., 161 (1994), 195-213. P. SARNAK, Arithmetic quantum chaos, The RA Biyth Lecture, University of Toronto, 1993. [SA1] [SA2] P. SARNAK, Some Applications of modular forms, Cambridge Univ. Press, 1990. [SE] A. SELBERO, Collected Papers, Vol. 1, Springer-Verlag, 1989, 626-674. [SH] A. I. SCHNIRELMAN, Ergodic properties of eigenfunctions, Usp. Math. Nauk., 9.9 (1974), 181-182. [SHI] G. SHIMURA, On the holomorphy of certain Dirichlet series, Proe. London Math. Soc. (3), 31 (1975), 79-98, G. STEIL, l~lber die Eigenwerte des Laplace Operators und die Hecke Operatoren fiir SL(2, Z), preprint, [ST] [T] E. C. TITCH~ARSH, The Theory of the Riemann Zeta Function, Oxford, 1951. [W] A. WEIL, On some exponential sums, Proc. Nat. Acad. Sci. USA, 34 (1948), 204-207. [Z1] S. ZELDITCH, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. Jnl., 55 (1987), 919-941. [z2] S. Z~LDITCI-I, Selberg trace formulae and equidistribution theorems, Memoirs of AMS, Vol. 96, No. 465, Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, New Jersey 08544-1000 USA Manuscrit refu le ler mars 1994. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Quantum ergodicity of Eigenfunctions on PSL2(Z)/H 2

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Springer Journals
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Copyright © 1995 by Publications Mathématiques de L’I.H.É.S.
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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10.1007/BF02699377
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QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\I-I 2 W~.NZm LUO and PF.TER SARNAK 1 To Wolfgang Schmidt on the Occasion of His 60th Birthday 1. Introduction Schnirelman [SH], Colin de Verdi~re [CD] and Zelditch [Z1] have proven the quantum analogue of the geodesic flow on a compact Riemannian manifold Y being ergodic. Let A denote the Laplacian on Y and % an orthonormal basis of L*-eigenfunctions of A. The corresponding eigenvalues are denoted by ),j. One forms the probability measures d~tj(z) := I%(z)I" dV(z), dV being the volume element (actually they consider a microlocalization of these measures to S~(Y), the unit cotangent bundle). If the geo- desic flow on S~(Y) is ergodic they show the existence of a full density subsequence )'Jk (i.e. one satisfying Y~ik<~X 1 "~Y~xj<X 1) for which ~jk(A)~Vol(A)/Vol(Y) for all nice sets A (e.g. geodesic balls). Zelditch [Z2] has extended this result to noncompact surfaces such as the modular surface X = PSI_~(Z)\H 2. He shows that if h e C~~ the space of smooth functions on X with compact supports, and j~ h(x) dV(x) = 0, then (1) X I<h,~>l ~ xj.~< x log 7." Here A ~, B means I AI~< CB where C depends only on -r. Selberg [SE] has shown that in this case, ~xs~< x 1 ~ ),/12 and from this one can easily deduce the quantum ergodicity using (1). Recent works of Hejhal-Rackner [H-R] and Rudnick-Sarnak JR-S] suggest that at least for this X much more is true. Namely, that there are no exceptional subsequences, that is ~t~ ~ dV as j -+ oo. This phenomenon might be called Quantum Unique Ergo- dicity. Hejhal-Rackner confirm this numerically while Rudnick-Sarnak show that certain natural candidates for such singular limits (that is, measures concentrated on closed geodesics) do not occur. 1. The research of the first author was supported by NSF Grant DMS 9304580, while he was a member at the Institute for Advanced Study during 1993-1994. The second author was partially supported by NSF Grant DMS 9102082. 208 WENZHI LUO AND PETER SARNAK This paper is concerned with this individual equidistribution conjecture for X. While we fall short of proving it we obtain a number of results in that direction. Firstly we prove the conjecture for the continuous part of the spectrum of X--that is we show that the Eisenstein series become individually equidistributed. Secondly for the discrete spectrum (cusp forms) we show that if exceptional subsequences occur they must be very sparse. We also introduce the discrepancy--a well-known measure of equidistri- bution for sequences--to quantify the measure of equidistribution of the ~Ts. This enables us to show that except for a sparse set ofj's the ~j's become equidistributed at a certain rate. For more background on this problem see the Lectures [SA]. Along the way we establish a conjecture of Iwaniec concerning the average size of Rankin-Selberg L-func- tions on their critical lines. The latter may be used to obtain new bounds for the remainder term in the Prime Geodesic Theorem (see below). We turn to a precise description of our results. The spectrum of A on L2(X) consists of three types, see Hejhal [H2]: (A) %(z) = ~/3/~, the constant function; (B) q~l(z), ?2(z), ..., an orthonormal basis of cusp forms, Aq~ + ),j % = 0; (C) E z,~ + it , t >1 0, the unitary Eisenstein series which furnish the continuous spectrum, AE -t--( 1 + t2) E = 0. We will assume that the basis % is chosen to be simultaneously eigenfunctions of the Hecke algebra. This choice is possible and in fact determines the %'s up to a scalar and hence determines ~j uniquely. It is quite likely that there is only one o.n.b. of %'s anyway, since the numerical evidence points to the spectrum being simple [H3, ST]. We define ~t ---- E z, -~ + it Note that ~x~(X) = o% so that there b( ) is no canonical normalization of ~x t . Our first result is that ~t become individually equidistributed. Theorem 1.1. -- Let A, B be compact Jordan measurable subsets of X, then lim Ez,(A) _ VoI(A) ,-~oo tzt (B) Vol(B) " The renormalizafion is actually needed since we in fact show that as t -+ o% ~,(A) ~ 48 Vol(A) log t. (2) 7~ Jakobson [J] has recently extended Theorem 1.1 to the microlocalizations ~t of ~, to S~(X). QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H ~ 209 To describe our main result concerning ~j we introduce some norms on functions on X. For H e C~(X) let II H It~,~ be defined by: 0~1+~, (3) II H [[k,. = max sup l y k ~+~.<~ ,~ (0~) ~ (Oy)~ (~) , where F is the usual fundamental domain for X in H. For H an integrable function on X we denote by H the mean value Vol(X~---) H(z) dV(z). Theorem 1.9,. -- For ~,> 0 and H e C~(X), X(1/2) + ~ ' f H(z) dtLj(z ) -- H 2 '~ [[ H 1[~,8 x/<x the implied constant depending on ~ only. The upper bound here is essentially the square root of that in (1) and in fact is sharp (i.e. it cannot be replaced by any exponent less than 1/2). Theorem 1.2 asserts that on average H(z) d~,~(z) -- H is of size X71/4. We expect that this is true indi- vidually (see [SA]). As a corollary to Theorem 1.2 we can address a question of Zel- ditch [Z], as to the size of an exceptional subsequence. He showed in general that such a subsequence must be of zero density. The following asserts that an exceptional subse- quence must be very thin. Corollary 1.3. -- Let Jk be a subsequence of j's corresponding to a subset S C N and for which ~Jk -+ ~ 4= 3dV/rc. Then for any ~> 1/2, IS c~ [1, N]] = O~(N~). We also establish Theorem 1.2 for H an individual Eisenstein series, that is xj<x "'2 +it'~s) <~([ t l + 1)" X"/2'+*" By Cauchy's inequality and in view of the integral representation 4rd P(s) fF E(z, s) I q~j(z)l ~ dV(z), L(u s | us, s) = p2(s/2O P(s/2 + itj) P(s/2 -- its) the last implies the following "mean Lindel6f" conjecture of Iwaniec [I1]: L UsNU~, + it (5) 2: 2 <~(Itl-4- 1)4X 1+~, xi< x cosh(=ts) where L(u~| uj, s) is the Ranldn-Selberg L-function (see w 2 and w 3 below) and ),j = (1/4) + t~. We apply (5) to counting prime geodesics on X. Let ,~r(x) -- [{P I N(P) < x}l, 27 210 WENZHI LUO AND PETER SARNAK where P runs over the primitive conjugacy classes of F = PSL2(Z) and N(P) is the corresponding norm [H1]; ~r(x) counts the number of prime geodesics on X of length at most log x. Theorem 1.4. ,~r(X) = li(x) + O(x (7'~~ +'), for ~ > 0. Here as usual, li(x)= dr/log t. For a general discrete cofinite F C PSL,(R) the best-known bound for the remainder term is O(xS/4/log x) [RAN, SE], while for F = PSLz(Z), Iwaniec in the paper quoted above established the bound O,(x135/4sl+*). In w 6 we will give an outline of a proof of Theorem 1.4. It is based on (5) and Iwaniec's method in [I1]. It should be pointed out that the expected remainder term here is O~(xllm+*). In the analogy with the Riemann zeta function and primes, the analogue of the Riemann Hypothesis for Selberg zeta function is true. Even so the abundance of eigenvalues puts the O,(x lira+ ~) bound completely out of reach. Indeed the O(x s/*) bound is reasonably straightforward but anything beyond that invoIves capturing cancellation in the sums over the eigenvalues. To quantify equidistribution of the ~j's it seems best to avoid issues of subsequences (as has been traditional in this problem) and to investigate directly the discrepancy. Various notions of discrepancy have been introduced in connection wich equidistribution of sequences [K-N]. The spherical cap discrepancy D(~) is defined by (6) D(~) = sup ~(B) 3Vol,B_(~ BCX 7~ where the supremum is over all injective geodesic balls in X. It is clear that if D(~) -+ 0 then Vtj become equidistributed and the size of D(Vtj) gives the rate. The choice of balls in this geometry seems natural enough. A somewhat bold conjecture that emerges from Theorem 1.2 (see also [SA] and the question of Colin de Verdi~re [CD]) is that for ~> 0, D(Vtj) ~X7 11/41+~. As pointed out in [SA] this equidistribution rate if true would be optimal. Theorem 1.5. Z I D(~j)12 x '2~ + xj~<x The main achievement in Theorem 1.5 is that the exponent of X is less than 1. We have made no effort to reduce it further which is certainly possible by these methods. On the other hand the optimal exponent 112 seems out of reach by these methods. From Theorem 1.5 it follows that with exception of a very sparse set, the [zr become equidistributed at a rate which is a negative power of k s. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSLz(Z)\H ~ 211 We end the introduction with some comments about the proofs. As pointed out in [SA] these questions of equidistribution are in part related to obtaining non-trivial bounds for Ranking-Selberg L-functions on their critical lines. Concer- ning Theorem 1 an involved but pleasing computation which exploits each factor of ( 1 ) ( 1 ) E z, ~ + it E z, -~ + it differently, leads to two features: (1) The entire problem in this case reduces to estimating L-functions. (2) The Rankin-Selberg L-functions in question conveniently factor into Euler products of degree at most two. We can therefore appeal to known estimates on the Riemann zeta function and L-functions of cusp forms (Meurman [MEU]) to prove the Theorem. For the case of cusp forms, viz Theorem 2 we no longer have any direct relation to L-functions. Our method is to first establish Theorem 2 for certain families of h's called incomplete Poincar~ series. This allows us to exploit that ~j is a Hecke eigenform and to represent [ q~(z)13 with expressions involving the Fourier coefficients in quadratic polynomials. Eventually this allows us to use the Fourier coefficient--Kloosterman sum connection, that is the Petersson--Kuznetsov trace formula [KU], to convert the problem to estimating exponential sums. To do so effectively Well's bound on Kloosterman sums is used as a key arithmetical ingredient. Also crucial in our analysis are the recent bounds of Iwaniec [I2] and Hoffstein-Lockhart [HN-L] for Fourier coefficients of cusp forms in the j aspect. Theorem 1.5 is derived from Theorem 1.2 using only Fourier analysis and geometry. The key point here is the structure of the spectral development of the characteristic function of a geodesic ball. All of the above results may be proven for congruence subgroups of PSL~(Z) (save for the possibility of small eigenvalues Xj < 1/4 intervening in Theorem 1.5). However inasmuch as we use heavily Poincar~ and Eisenstein series (and related Fourier coefficients) we do not know how to establish these results for compact arithmetic surfaces. 2. Eisenstein Series This section is devoted to the proof of Theorem 1.1. The Eisenstein series E(z, s) for X are defined by 1 y" (7) E(z, s) = ~] y(yz)" = ~ ,~o ~r~\r ,~ ~lcz +dl ~"' where ~R(s) > 1, z = x + ~y, F = PSL.(Z) and Foo ={z~z+n,n~Z}. The Fourier development of E(z, s) is well known [SA2]: 2Y 1/~" ~ n 8- ,/2 (8) E(z,s) =y" + ~(s)y 1-" + ~-~,=1 al_2,(n ) K,_,/2(2~ny ) cos(2rcnx), 212 WENZHI LUO AND PETER SARNAK where ~(s) = ~-,/2 P(s/2) ~(s), q~(s) -- ~(2s -- 1), ~(2s) ~(~) = y~ d ~ a[. and K is the Bessel function. In order to prove the equidistribution of ~t we consider its inner products with various functions spanning Lz(X). We begin with inner products with Maass cusp forms %.. Set (1)(1 %(z) E z,~+it E z,~--it yZ . (9) J,(t)-=fx~d~=fx To investigate this we first consider (10) L(s) = ~s(~) E ~, ~ + it E(~, ~) y--r Note that all of the above integrals converge rapidly since % is a cusp form. Now for ~R(s) > 1 we can "unfold" the integral in (10) using the definition (7) to get (11) Is(s ) = q~s(z) E z, ~ § it y~ --o y2 Since E(z, s) = E(-- ~, s) it follows that I~(s) =- 0 if q~ is an odd cusp form (the cusp forms are of two types %(-- 5) -~r ~= 1,-- 1) so we may assume that % is even. In this case it has a Fourier development ao (12) %(z) =yl/2 ,~ p~(1) Xs(n ) K,9(2T:ny ) cos(2~nx). Here (1/4) + t~ = X s and the coefficients Xs(n ) satisfy the multiplicative relations which are a consequence of %(z) being a Hecke eigenform. (We will ignore the normalization pj(1) since in the discussion of this section j is fixed.) These amount to oo Xs(n ) _ II(1 -- xj(p)p -~ +p-~")-~ (13) L(%,s):= Y~ n ~ So o~ (yl/2 ,~=1 Xs(n ) K,j(2rcny) cos(2~nx)) 2112 co ) 4(1 -[- 2it) ~ nu ~-2"(n) K~,(2rcny) cos(2rrnx) y" dxdy n ~ 1 y2 oo ( ~=lX (n) K~tj(2rcny ) n~ t a-~t(n) K,t(27:ny))Y" dy__ if; 4(1 § 2it) K~tj(27:y ) K~t(2rcy) f dy__ ~(1 + 2it) .~1 ,e ] Y QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H 2 which on evaluation of the integral ([G-R]) yields it) R(,) r(s+i j+it) 2 2 2 2--S (14) 4(1 4- 2it) P(s) /Z ~t X~(n) ~- 2it (/z ) where R(s) = n= 1 /Z s Now R(s) may be expressed in terms of the L-function L(%, s) in (13): (15) R(s) ----- L(%, s -- it) L(%, s 4- it) ~(2s) To prove (15) first we factor R(s) (we write X(n) for X~(n) in short), R(s) = H R~(s) io where oo (16) R~(s) = Z x(p j) p*~t ._2.(pa) p-~, J=0 Now ~_~.(p~) = Z p-~"~ = ~=o 1 -- p-2. SO R~(s) = ~ x(p j) p~jt 1 -- p-~.(J+l~ 5=0 1--p-~" P-J~ co oo --p-~" ]~ x(p~)p -~l'+"') 1 -- p-~" ( ).= ZoX(p~ p-Ol,-u, i=0 p-- 2it ) 1( 1 1 -- x(p) p-~,+-I + p-21~+m 1 -- p-2U 1 -- x(p) p-I,-.) + p-2~.-m -- 1 -- p--28 (1 -- x(p)p -'*-"' + p-2,.-.,) (1 -- x(p)p -''+"' +p-Z("+"')" This proves (15). Using this in combination with (14), (11) and (9) we get P + P 4- it~ _ it) = 271:-- 2it 214 WENZHI LUO AND PETER SARNAK One can check using the functional equation for L(%.,s) that the RHS of (17) is real as it has to be in view of (9). An interesting point here is that J~(t) = 0 if L(q%,l/2) =0. We are now ready to investigate the behavior of Jj(t) as t -+ oo. Using Stirling's formula (I F(a -+- it)] ~e-=ltl/21t ]o-(~m), we see that the r factors in (17) will yield a factor ,--- c [ t I- a/2 as t ~ oo (here t~ is fixed). This together with the arithmetic estimates: (A) (logt) -a ~ 1~(1 + it) l ~ logt, (1 )[ (B) L ?~,-~+it <j,,It ~>0, yields Proposition 2.1. IJ~(t)l <~,~ It[ -'~/~'+*, for all r O. The estimate (A) above is well-known in the theory of the Riemann zeta function (IT]). Actually later we will need an improvement of (A) due to Weyl. Indeed this method of Weyl of estimating exponential sums is used crucially by Meurman [MEU] (1) in his proof of the estimate (B) above. The standard convexity bound for L % ~ -t- it l ( 1 )] i~lm+~ is L q%,~ + it <j,~ It . Clearly this would not suffice to show J~(t)-+0 as t -+ oo. However any improvement (in the exponent) of the standard bound would suffice for our purpose. This completes the analysis of the inner products of ~xt with cusp forms. We turn now to inner products of ~t with incomplete Eisenstein series. Let h(y) ~ C~~ +) be a rapidly decreasing function at 0 and 0% that is, for any positive integer N, h(y) = ON(y ~) when 0 <y ~< 1, and h(y) = ON(y -~) when y > 1. Let H(s) be its Mellin transform (18) H(s) = h(y)y_S dy --o Clearly H(s) is entire in s and is of Schwartz class in t for each vertical line a + it. The inversion formula gives = H(s)y' ds (19) h(y) 1 I ~+~w for any a e R. For such an h we form the convergent series y, h(Y(TZ)) = 1 f H(s) E(z,s) ds; (2O) F~(z) vGr~\F ~ J~(s)=2 QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H * 215 F h belongs to C~176 and is rapidly decreasing in the cusp (i.e. as y -+ oo). Hence we may form 1 H(s) E(z,s)ds E z,~+it y~ I( 2~i (,) = 2 _ 1 H(s) y*ds E z,-~+it y, fo'~ Ii ( 1 ) 2dxdy 2hi (*) = 1 H(s)Sd s yl/2+. + ~ + it yi/~-*t 2hi (~) = 2 f0i, ( + 14(1 -& 2it)I' = I K"(2nnY)[2 " Now q~(i[1 +it)~=\ 1, so that the first term above contributes (as t-+m) (21) 2 fo o h(y) d_y + a rapidly decreasing function of t depending on h. The second term which we denote by I~(t) is Oo t/v I K. (2ny)]~y-:- ds. I2(t) = nil ~(1 + 2it)12 (~,=~ ' :: ].L 3' The series can be evaluated as was first done by Ramanujan [RA] l a_2,(n) 1~ _ ~(s) ~(s -- 2it) ~(s + 2it) .=1 n ~ ~(2s) The y-integral is evaluated in terms of F-functions as before. We obtain (note that t is fixed while s is the variable for the integral) In(t) = in 14(1 ~- 2it)t ~(2s) F(s) ds (s) = g _ _ 2 ~ B(s) ds, say. in ]4(1 + 2it)] ~ ~(,)=~ 216 WENZHI LUO AND PETER SARNAK Now shift the integral to ~R(s) = 112, 4hi + 2 [ B(s) ds + O(t-~~ In(t) in 1~(1 + 2it)[ 2Res"=lB(s) in 1~(1 § 2it)[ 2 3~(s)=112 The O-term comes from the contribution of poles at s = 1 i 2it. To justify shifting the contour we use Stifling formula to estimate the F-factors and the fact that H(a + it) is rapidly decreasing in t. In fact using this and Weyl's bound (22) ~(2 + it)<~t a/6,+", we find that B(s) ds 4~ t u/8)+' t -~/~ = t-(116)+~ (23) in 1~(1 + 2it)] 2 (s)= 1/2 This corresponds to the bound in Proposition 2.1. The residue term is more complicated. Write B(s) as ~(s) G(s) where G(s) is holomorphic at s = 1. Then ( G' ) Ress= 1B(s)= G(1) 2,(+~-(1) , where ,( is Euler's constant. Now a simple cal- culation gives G(1) = --24 H(1) 7~ G' H' ~' ~' ~-(1) =~ (1) -t-C+~(1--2it) +-~(1 + 2it) and r'(s_ i,) r' +r,2 + + C being independent of t. According to the Weyl-Hadamard-de la Vall6e Poussin bound [T] ~' log t (24) -~ (1 -+- it) ~ log log~ and Stirling's approximation -~ + it = log t -}- O(I), we have =-- (logt ] Ress=i B(s) 48~ H(1) log t + O k~] (25) QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL,(Z)\I'I ~ 217 as t-+ oo. Note that ye 9 H(1) = fo ~ This leads to Proposition 2.2. -- Let F E C=(X) be of the form F~ as in (20). Then -~- y2 ] log t as t ---~ oo. With Proposition 2.1 and 2.2 we are ready to establish the following Proposition which by standard approximation arguments implies Theorem 1.1: Proposition 8.3. -- Let F e C00(X) (i.e. F is a continuous function of compact support in X); then -~ yZ ] log t as t ----~ oo. Proof. -- It is easy to see that the functions of the form F h as above together with the cusp forms ~0~ are dense in C0(X ) (the space of continuous functions vanishing in the cusp). Let F E Co0(X ) and ~ ~ 0; then we can find G = G 1 + G2 with G 1 a finite sum of cusp forms and G 2 in the space of incomplete Eisenstein series with corresponding h E C~0(R+), such that [[ G -- F ][~ < ~. If H = G -- F then H is rapidly decreasing in the cusp and so we can find an h I/> 0 which is rapidly decreasing and for which Hl(z ) = E h~(y(yz) /> ] H(z)[, "rE Poo\P and fx H~(z) dV(z) < 5~. Hence by positivity of d~t we have ~<-- lim 1 f H(z) d~t(z ) Jx From this the Proposition follows easily. To end this section we remark that while ] < F, ~z t > [ is expected to be of size t-1/2 for F in the space of cusp forms (and smooth), the above shows for F an incomplete Eisenstein series and of mean zero, that ( F, ~z t >]log t ~ 0, but this convergence is very slow. 28 218 WENZHI LUO AND PETER SARNAK 8. Incomplete Eisensteln Series and Poincar6 Series In this section we will establish Theorem 1.2 in a very special but important case, i.e., H(z) is either an incomplete Eisenstein series or an incomplete Poincar6 series. For these functions, we have the advantage of being able to "unfold" the integral d~j(z) appeal automorphic theory fx H(z)and then to L-function and the Petersson- Kuznetsov formula. In the next section we will see that Theorem 1.2 holds in general by an approximation argument. Proposition 8.1. -- Let h(x) be a smooth function on (0, oo), supported in (xo, oo), x o > O, such that for some C~,k >>. 1, I h"'(x) l -< c,,~ x -~, i, k ~> o. Let Ph, o(Z) = Y~ h(y(vz)). Then for any a > O, T >1 1, we have X I<Pho, iUj(Z)]=>-P~o] 2 ~,ClllC T 1+*, , , , 8,8 tj<T 1 I~ P~,o(~) uv(~). where Ph, o -- Vol(F\H) \H Proof. --By unfolding the integral and using (12) we have, with pj(n) = 0j(1) Xj(n), fo o luj(z)12( ~ h(y(yz)))dV(z)= ~ ]0j(k)[ 2 K (r) Set hl(x ) = h(x-1), and hi(x) x 8-1 dx = h(x) x -8- ~ dx. (26) G(s) = [~ 3o Then G(s) is entire and by Mellin inversion 1 [ G(s) x_Sds, a>0. (27) hi(x) = ~ Jco~ It is clear from the definition of G(s), by partial integration, that C1,1 (28) G(s) ~~ l) ... (s-t+ 1)[ QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H 2 219 for l/> O, sr and 1 >> ~=9t(s)>/ %>0. By Mellin transform and a well-known formula (see (35)), it follows that K~i(r ) h r _dr _ ! [" G(s) 2=i ~,(2= I k I)' J, where ~ > 1. Then we have [ us(z ) { 2 Ph, o(Z) dV(z) (29) fr\a _ if8rd__ o, G(s) L(us| co oo where L(u s| = ~] [ps(n) l 2n -8=los( 1)12 ~lX]( n) n-*" We move the line of integration to ~(s) = 1/2 and pass the simple pole of the inte- Ph, o(z) dV(z), in view of grand at s= 1 with residue Ph,0--Vol(r\H) 1 Ir \H dy = 3=_ 1 G(1), which Res,= 1 L(u s | us, s) = 12= .2 cosh(=tj) and Ph,0 = 3r:-1 1] ~ h(y)y follows from the unfolding method. In order to finish the proof, we need to understand the behavior of L(u s | uj, s) uniformly in j and s, which is of independent interest. Let L(2~(u s) stand for the second symmetric power L-function [SHI] attached to the Maass-Hecke form us(z), and Rs(s ) = ~(2s) ~ Z](n) n-8, the Rankin-Selberg convolution L-function. We have oo ~o Rj(s) = ~(s) L(e'(us, s), ~: X](n) n -8 = ~(s) Z ),/n 2) n-8. n=l n=l Hence co oo L(2)(us, s) = ~(2s) Z Xs(n ~) n -8 = ~= .=1 los(n) n-8, can) = Z xs(k2), 12k = n for 9t(s) > 1. It is well known from the work of Shimura [SHI] that L(2)(u~ ~., s) is entire and 220 WENZI-II LUO AND PETER SARNAK is invariant under the change of variable s -+ 1 -- s. Define and let w = (1/2) + ito. Consider the integral (x > O) X--S LI2~(uj, s + w) F(s + l) ds, (3o) --1 f 2~i o~ where l is a positive integer, and ~ = (1/2) + I/log tj. Clearly (30) equals co y~ cj(r,) F(nx), r = 1 .t0 1 U(s+l) t- ds= e -~*-~d~. where F(t) = ~ ol s Moving the line of integration in (30) to --., we pass the simple pole of the integrand at s = 0 with residue F(l)L~21(u~, w), and get 1 I L'~'(u~, s + w) r(s + l) x-~ ds. r(/) L~~ w) + f~ ,-o~ s Now the integral equals -- 1 ~ L~21(u x ~ , j,--s+w) r(--s+l)--ds 2~i Jlo~ s 11 ~ o ~i~ s + ~ 0~/s + ~/~ (s ~+ ~) ~Is + ,/, ~ Moreover 0j(s + ~) - t~'~+~,-~(~'+;-~) -~ (l + o(I s + ~ I ~+~) t71) 0A- s + w) ~1 (1 + oil s + t;1). In [13] itis shown that 2~,<~ X~(n) < tj N, hence L~2~(uj, s + ~) = Rj(s + ~)/~(s + ~) < t~, where ~ is an arbitrarily small positive number. We conclude that the above integral equals ( =3 1,0,~,+~ xo t~ (31) _ ~.0 t;~.o ~ ~(~) F w, .=~ ~---~ xC + o(I w ,), QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H 2 221 where (32) V(w, t) = P(-- s + l) -- ds. al 2 Hence, we have derived the "approximate functional equation" r(l) L(2)(us, w) = ~ q(n) .=1 n w F(nx) + ~0 q~"" Z cj(n) F w, ( ~] + o(I w I '~'~'+~ x ~ t]). .=17 xt U Now we integrate both side of the above expression from ~81~ i to e~al2/t~ with respect to the measure dx/x, which yields, for ts ~ T (we denote T < tj ~< 2T by t~ ,~ T), 5, F -- n= 1 7/, w ~Tlti Y fTIt~ ~lc~(n ) ( ~_y) dy [is/z,+aT_am+a). @ ~3ito l~2ito ---=- F w, -- + 0 (I w = n w Tier Y We observe that: 1. F(w, t) ~t -~ I w [~, t > 0. This is easily seen by moving the line (a) in (32) to (a). 2. F(w,t)~z(t/]w[) -l+(lm, t>0. This is also easily seen by moving the line (~) in (32), but this time to (l- 1/2). Thus, if t/Iw I > T ~ then for any N, F(w, t) ~ T-N(] w lit) ~" by choosing l = IN/a] § 4. Hence, by Cauchy's inequality, we have ~--1 F Z I L(~>(u~,w)l ~ ~ I] ;/ = y T<tj~2T T<tj~<2T /2 Z =F w, -- +O( Tl+alw 12e T<ti~2T. n=l n w y Now ffwe choose l = [10/a] -{- 4, and note that [cj(n)/n~l  n (1/2,+8 since I ),~(k)l < k 1/~ [SA2], we have s = F w, ~lw[ 2. n~(mlwl)X+e n w Also ~>~TI+~ n w 222 WENZHI LUO AND PETER SARNAK Using [D-Ill X ] 52 a. vj(n)[ 2< (T 2+N) T ~ X [a,~l 2 , ty~<T n~<N n~<N the lower bound [I2], vj(1) >> t~ -~ (where o~(n) = cosh(~t/2) v~(n), vj(n) = vj(1) Xj(n)), the definition of ej(n), the above remarks on F, and Cauchy's inequality, we deduce that (33) Z I L'2'(uj, w)l 2 < TZ+~lw [ ~+~. T< t]~2T Finally, by the upper bound v~(1) ~ t~. [HN-L], and the crude bound ~(w) ---- O(I w W2), we have proven Theorem 3.2. (34) I L(u~| w)[ ~ < T~+~ I w l '11/~'+~ tj<T cosh 2~tj This establishes (4) and the conjecture of Iwaniec (5). We return to (29). By Cauchy's inequality, (fr~Hlu~(z) [2 ph, o(Z ) d~(z)--Ph, of o(1/2) But according to Theorem 3.2 and Stirling's formula, r Idsl 52 I 'G(s)"L(uj| tj~T ~sl~<T/10 Tl+e Cs,8 Also, we have trivially, again using Theorem 3.2 and (28), I [ tj ~ T ~s I/> T/10 Cs,8 T 1+~. This completes the proof of Proposition 3.1. Proposition 3.3. -- Let h(x) be a smooth function on (0, ~), supported in (Xo, oo), x o > O, such that for some C~, k >1 1, ]h~)(x)[~< C~,kx -k, i,k>/ 0. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H s 223 Let Pj,.~(z) = ~2 h(y(yz)) e(mx(yz)). ve roo\r Then for any ~ > O, m 4: O, T >>. 1, we have x I < p~,~, I ~(~)I ~ > I ~ <~ m2(C~,2 + Co~,o) T'+*" tj<~ Proof. -- Without loss of generali W we may assume that the first Fourier coef- ficient p~(1) of uj(z) is real. It follows from the unfolding method that I uj(z)12 ( E h(y(yz)) e(mx(yz))) dV(z) Y E F~\ r flail ;o o ( ---- Y, pj(k) pj(k + m) K.i(r) K~q l +-~ r h --r k*0, --m = pj(1) Y~ N p~(k 2 -I- km]d) K.j(r) Kui 1 + ~ r h dl,~k*O,--,~/e 7~dlk[ r Here we use, for k> 0, pj(k)= pj(1)Xj(k), O~(-- k) = % pj(k), ~5 =  1, and the multiplicafivity of Hecke eigenvalues dl (n, r -~ " By a standard dyadic partition it suffices to cstimatc thc sums with t~ ~ T, k ~ K, K > 0. The cases where dK >i AT (A is sufficiently large) can be ignored, since from [G-R] Kit (x) = io o e-X cooh~ cos t-~ d-r, we have K.(x) ~x -le-~% x/>0. So the contribution from these terms is exponentially small, in view of 05(1) ~ cosh(r:t/2) t~., X~(k) ~ k 1/2, and h(y) is supported in y >~Y0 > 0. Henceforth, we assume dK ~< AT. Similarly, we can assume m ~ T, because otherwise r I 1 + re~d* [ > AT. Recall [G-R] x -x K~(ax) K~(bx) dx (35) Io ~ 1--X--~+v l_~,l_b~) 2 ' 2 ' ' 9t(a + b) > 0, ~lX< 1 -- {9t~[ -- 191v [, 224 WENZHI LUO AND PETER SARNAK where oo F(,, ~,y, z) = ~] ~(~ + 1) ... (~ + n-- 1) ~(~ + 1) ... (~ + n-- 1) z" ,=0 "((7 § 1) ... (V + n -- 1) nl is the hypergeometric series. Taking 9~= l--a, ~t =v----its, a=b = 1 or a = b = I 1 + m/dk [, and using ab ~< (a e + b e)/2 and Stirling's formula we obtain ti~TY~ 11 k~x ~ 95(1) p~(k~ + km/d) Kitj(r ) Kit j 1 + ~ r h 2~ d[ k 1 , ~o 9 C 2 K e T'. 0,0 Here we have applied the following result due to Kuznetsov [KU, D-I2]: (36) Y~ I v~(/)I e e-9/T = 27:-2 T 2 -~- O((T -k l l/e) d(l)). ti~<T Hence we can assume K >> T 1/2, and m ~ K 2/3. With the above reduction, we proceed to prove Proposition 3.3. As before, set hl(x ) : h(x-1), and define G(s) as in (26). We use the Mellin transform to obtain (with a : e) fo (r) r K~@r) K~ti( [ 1 + m/dklr) h ~ ~- _ 1 f G(s) fo"rS_lK~q(r) K~ti([1 +m/dklr) drds. 2~i o) (2~ dk) 8 To deal with the inner integral we apply the formula (35) and [G-R p. 1040] r '-1 K~ti(r ) K,ti((1 + mldk) r) dr : 2 -s+~ r l'rs/'~-x(1 -- "r) "/2-~ 1 + -~- -4- x dr. If we can show (37) Y~ I IF_, a k v~(k 2 -4- km/d) f(k, t~)]2 ~ mT2+, K, tj~T k~K QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H I 225 where a k=a~(m,d,'r,s) = k 2 1 +~+'~ m ]"i 1+~ f(k, t~) = f(ra, ~, k, "~, q) = 1 + -~- +, k ak// then from Cauchy's inequality, (28), Stirling formula, and by considering [ ~s[~< T/10 and [~s[> T/10 separately, we obtain Proposition 3.3. In the following we will prove (37). It suffices to treat the ease where [t;--T[~<T 1-'. We infer that E e -C"i-*'m-'' [kExa~ v~(t? + kmld)f(k, t,)l ~ = Y, ak ~z Z v~(k ~ + kin/d) ~(P + lm/d) h(t~) + O(1), k,l~K j where k(t) = e-(" - ~,/~a-,,, f(k, t) f(l, -- t) + e- '" + T,l~t-.,t f(k, -- t) f(l, t). Applying Kuznetsov's formula [KU] to the inner sum, we deduce that ~ vj(k 2 + km/d) ~j(l ~ + lm/d) h(t~) " I/ ~ l,I = -~- t tanh(rct) h(t) dt -- - ~ i~;( 1 + 2it) r_ ~,(k ~ + kmld) d.(/~ + trald) dt + - Y~ c -1 S(k ~ + km/d, l ~ + lm/d; c) J2,, cosh(~zt---~ dr. 713 r ~o Here S(m, n; c) is the Kloosterman sum and d I d 2 ~ n Since '_' t tanh(~t) h(t) dt ~ T s+', d,(k' + km/d) d,(l ~ + lm/d) dt ~ m" T 1+', Io h(t) ]~(1 + 2it)i" 29 226 WENZHI LUO AND PETER SARNAK their contribution is at most m ~ T ~ +" K. It remains to estimate the sum of Kloosterman sums which we shall do for each modulus e separately. If c > K 2 T -z +', we estimate the integral transform of Bessel's function 2i L,, 4~/(k~ + krn/d ) (l 2 + lm/d) t cosh(~t) dt 7r j_Q ~ C by moving the line of integration to ~t = -- B, where B is a sufficiently large integer, depending upon r From Poisson integral formula [G-R] (z/2)~ f[ J~(z) = cos(z cos 0) sin 2~ OdO and Stirring formula for F(s), it is easy to see that the resulting integral is very small. The residues of the integrand at the relevant poles are also very small. Therefore it remains to consider terms with e ~< K ~ T -1 + ~. We have h(t) j~,, ~/(k~ + km[d) (/a-k- lm/d) t dt cosh(~t) ao C f T -[- TI-z 1~ T (4=~r = Y. u J2, cosh(~t~ dt -4- O(T-1). ~=+I jT_TI_$ log2 T C We need the following Van der Corput's 1emma [T]: Lemma 3.4. -- (1) If f '(x)>l ~ > 0, or if(x)<<. -- ~ < 0, x e [a, b], then e ul') dx ~ -. ff 1 tz (2) Iff"(x) >>. r> O, orf"(x)<~ -- r< O, x e [a, b], then e u(*) dx ~ ~. ff 1 Now (see [ER]) 1 1 e -~ ' (1 + O(r-1)), J,,(x) - (r + TM + x -r to,(x) = r~-U-+ x ~ + r log X QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSI.~(Z)\H z and 0 V~r~+x2+r Or %(x) = -- log 02 1 -- ~,(~) --- Or* ~ + x 2 If T/(K2/r > A -1, using part (1) of Lemma 3.4 and Weil's bound for Kloosterman sum [W] 1 1 [ S(m, n; c)[ <~ a(c) (m, n, c)~d, we deduce that the corresponding contribution is 1 K 9 T :+" max %/7 T 1 +' ~ -- ~ KT'. ~K~-~+~ ~ T K If T/(K2/c) ~< A-1, and c >> m, we use (1) of the Lemma to deduce that the corresponding contribution is max ~/c 1 TX +, ~ KT ". IfT/(K*/r ~< A -1, and r ~ m, we use (2) of the Lemma to deduce that the corresponding contribution is + g max ~/~ 1 T 1 +~/-~2 ~ %/~T1 ~ c Hence, summing over k, l, we obtain (37). 4. Appro~dmatlon The estimates for (P, ~ ) which were established in the previous section for incomplete Poincar6 series may be used to obtain similar bounds for a general smooth function F. To do this we need to approximate such functions by P's. Let F e C~176 Let Cz, C2,..., GT. be a decomposition of X into neighbourhoods, C z of the cusp, (3 2 of the point i, C 8 of the point p = e ~/s and C j, j = 4, ..., L containing no elliptic fixed points. If we choose a partition of unity subordinate to this decomposition, we can write F = ~'=1 Fj where Fj has the same smoothness properties as F and each Fj is supported in (3~. So for our purposes here we can assume to begin with that F is supported in such a neighbourhood Cli0" Let Ci be the lift to H of C~ into a fundamental K2/____fc 228 WENZHI LUO AND PETER SARNAK domain. Let ~ be the Fco periodic function on H which is equal to F in C~.,. Clearly F(x,y) is smooth and is supported iny/> 1/2. Also (38) F(z) -- Y, F(yz), Wio y~ rco\F where W~o = 1 ifjo + 2, 3 and otherwise is the order of the stabilizer. Expanding F(z) in a Fourier series in x gives cO (39) ~(z) = Z h.(y) e(mx), in=-- co hence co (40) F(z) = W~--~ ,~=-. ~: V~.,.(z). This will serve as our means of approximating F by the P's. From partial integration we see that, for ~t 1> 0 and k >t 0, 0J+~ F(z) , (41) Y~I h~'(y) [ ~< ([ m I + 1) -~ sup 3P [ .~>1,~ (ay) ~ (ex) ~ hence, for ~t I> 0, and L /> 0, (42) max C,.~(hm) ~< (1 m + 1 ])-~ I[ F I]r,,L+~. i,k~L The norm is the one introduced in (3). Now Z ( P,...... P-J ) I. W~o [ < F, ~ ) -- ~ I = [(( P~o,0 ~J ) -- P~o,o) + m:t:o Applying Gauchy's inequality to this with weight ~,, > 0, ~o = 1, gives I<F,~>--FI ~ '~ (Z~: ~) (I ( P,.o.O ~# > --P~.ol~ + Z ~.1 < Pn,....~j > Is). m m ::t: 0 Summing this for [ t~]~< T yields I= Y, I<F,~j>-~l ~ ty~<T Z ~. X I <Ph.,.~> D. ,n,o tj~T m q~<T Using (42) with ~t-----3, We are read 7 to apply the main result of section S. ~,, = ([ m I + 1) st" and the bounds of section 3, we get I 4. II F 11~..8 T 1 +" This established Theorem 1.2. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSI.~(Z)\H * 229 Finally we establish the above quantity for F = 9xk" We can estimate ~xk(z) from its Fourier expansion. It is easily seen that ~b.(z) is exponentially small for y ~ t~ xv, [I-S]. In fact in [I-S] a stronger estimate has been esta- and also that [[ ?z i ][oo ~--~ blished. However we use the crude bound which together with the Fourier expansion yields Corollary 3.1. -- Let kk ~e 0; then ti<~T Note that this bound is only of use when T is much larger than h since the trivial bound for the above sum is T2[ t, [- This Corollary is the analogue of Theorem 3.2 which is equivalent to *i< T ., ~ + it , ~ ~, (I t I + 1) 6T~+'- 5. Discrepancy This section is devoted to proving the upper bounds for the discrepancy D(~t~) as claimed in Theorem 1.5. The noncompactness of X leads to some technical compli- cations. To deal with these, choose 20 > 1 and set Vol(B) I D~o(~j) = sup ~t~(B) Vol(X) B(~, r) C X, ~ 6F, ~(~)~ Yo Vol(B) , and D,o(~Zj) ---- sup ~.j(B) Vol(X), so that D(~) = max(DU~ D,o(~)), and in particular (44) We will estimate D ~~ and D~o differently and we begin with D,o(bS). Let k(z, ~) be a point-pair invariant on H (see [SE]) and K(z, ~) = ~ver k(Yz, ~)" If Zm~,,~(z) is the characteristic function of the ball B (which we are assuming is injective in X) and if we define k,(z, ~) = 1 when d(z, ~) < r, and = 0 otherwise, then we have (45) X~(z) = K,(z, ~). 230 WENZHI LUO AND PETER SARNAK According to the spectral expansion [SE] we have, at least in the L~-sense, oo (46) K,(z, ~) = Z h,(t~) V~(z) r k=O +~ h,(t) E z,-~+i E ~,-~+it dr. 1i Here h,(t) is the Harish-Ghandra-Selberg transform [SEL] of k,. In order to use the expansion (46) to estimate the discrepancy we must smooth K, so as to deal with absolutely convergent series. For ~ > 0 let d?. be an approximate identity, that is, t~(z, ~) /> 0 is supported in a ball of radius ~, and H+,(z, ~) aV(z) = 1. We can and will also choose d?~(z, ~) so that +,(z, ~) ~ r and its Harish-Ghandra- Selberg transform h'" satisfies [ h~'(t)[ ~ 1 for I t]~< 1/~ and is rapidly decreasing for [ t[ >> l/e. Given the ball B = B(~, r) as above, let A 1 = B(~, rl) , A, = B(~, r2) where r 1 -= r -- 2r and r 2 -- r + 2~ (if r 1 < 0 then A 1 is taken to be the empty set and ?(Ax = 0). For a function F(z) defined on F\It we set (v 9 ~) (z) := f F(~) ~,(~, z) aV(~t, where ~,(z, ~) = Z d?,(yz, ~). y~F It is easily seen that with these choices (47) k,~ 9 ~,(z) ~< ?(A(z) ~< k,2 9 ~,(z). For l ----- 1, 2 the expansions of these functions take the form co (48) k,,, ~,(z) = ~__20h,,(t~) h'"(t~) ~(~) ~(~) 1;(1)(1) + ~ hn(t ) hC~'(t) E z, ~ + it E ~, -~ + it ,it. The mean value over X of the functions in (47) differ by a quantity which is O(r and we may therefore conclude that Vol(B) ] (49) ~(B) Vol(X) ~ ~ + ,=~ y~ I ~,01~ hn(tk) h'"(t,) ( ~, V, ) %(~) 1 " (.,-~+)E 1 it)dt +_~f_hn(t) h,,,(t)(~,E 1 it) (~,-~+ . QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSL~(Z)\H z 231 Now I Z h,~(t,) h("(t,) < ~, ,~ ) +~(~)1 ~ k,o <~ ( Z [ h,t(t~)I ~ ] h'"(t~)II ~(~)I ~) ( y' I < ~,, ~ > I ~ I ~'='(t~)I), k*o k*O which, according to the estimate on h ~, is ( Y~ I ~,,(t~)i~1 ~(~)i S) ( y~ I ( ~, ~ ) I~) 9 [tkl.~ 11~, k# 0 I~kl .~l/r Using (46) we see that (50) x I ~,,(t~)i ~ I r~(~)I ~ r I I K,,(~, ~)I ~ aV(~). The right hand side can easily be estimated uniformly in ~, r for ~(~) ~< Yo and one finds it to be O(1 + sSyo). We have shown that ~.o l ~kl .~ 1/r k* 0 The same considerations hold verbatim for the Eisenstein series contribution. On taking supremum over all balls with ~(~)-<<Y0 we are led to (51) [ i)~o(~j)12 ~ ~2 + (1 + a"Yo) ( ~ I ( ~-,, % ) [ ~ Itkl ~l/*,k*o Next we turn to estimating D~~ which we do in a crude fashion. First we need an upper bound for ~q.<T 1%(z)]2. Ifk~(z, ~), ~ = I/T is an approximation to the identity as before (and for which h(t) >t O, which can be arranged), then Ko(z, z) = X k~(z, vz) >> E I ~,(z)I S. v~r ~j~r On the other hand the middle term above is easily estimated as being ~ T2+ Ty, where y = ~(z). It follows that (52) Y~ 1%(z)[2 ~ T 2 + Ty. q.<<T From the Fourier expansion of %(z), it is easily seen that I%(z)12 is exponentially small for y >> t t, ]. Hence in considering balls B(~, r) when computing DV0(~z~), with lt, l.< T, we can ignore those balls with center ~ satisfying ~(~) >> T. (Note also that the volume of such a ball is O(T-2).) For B = B(~, r) withy0 < ~(~) ~< T we have Vol(B) Vol(B) i~ a~ dy (53) ( ~;, B ) Vol(X) ~< Vol(X----~ + I ( ~,, B ) I ~< + I q~(z) y~ 0 232 WENZHI LUO AND PETER SARNAK Taking supremum over all of these balls and using the fact that D(~i) ~< I we conclude that dx 4y (54) I D~~ ~ .e ! + I ,~/z)I ~ f S; Yo o y2 " Gathering the bounds, we have (55) I n(~@ [ 2 ,~ r + (1 -t- *Syo) ( ~ I < bq, % ) t ~ I tkl '~ l/t, /g ::l= 0 ~dy I ~/z)I ~ f We now sum these inequalities for t~[~ T and apply Corollary 3.1, (43) and (52), where in the latter we use y ~< T, to get E t[' + Itl~ T '+'~ + -- Y~ D(~) ~ ~. T 2 ~ + (I +Yo ~) ,~, Yo tj ~ T t [ ~ llr for ~l > 0. That is T 2 D(btj) 2 ~ T ~ z 2 + (1 +Y0 z3) T~+* ~--19 ..{_ --. ts~<'r Yo Now optimize the choices of ~ = T -a, Yo = T% This yields the bound T 2-c2/211 for the RHS. This concludes the proof of Theorem 1.5. 6. Appendi~ In this section we will give an outline of the proof of Theorem 1.4 which follows closely the method in [I1]. Indeed using the argument below together with the analysis in w 9 of [I1], one can improve the exponent 7/10 slightly. However the bound O,(x ~2/s~+ ") which is mentioned following equation (12) of [I1] appears to remain out of reach. On the other hand, the exponent 2/3 can be deduced conditionally on the Lindel6f Hypo- thesis for the usual Dirichlet L-functions (see [I4], p. 189). We first show that XUi exp(-- t/T) ~ T 5/* XVS(log T) 2. (56) Let h(~) be a smooth function supported in [N, 2N] whose derivatives satisfy ]h~V'(~)[ ~N -~, forp=0,1,2 .... and = h(~) d~ = N. I~ co QUANTUM E, RGODICITY OF EIGENFUNCTIONS ON PSL2(Z)\H z 233 Thus, in review of [I1 lemma 8] and (5), we have h(n) [ vj(n)12 = ~ N + r(tj, N), 52 [ r(t~, N)[ ~ T ~ N~/~(log T) ~. tj~<T We deduce that ~h(n) (~ l vj(n)l ~ X ~*i exp(-- tilT)) = ~ ~ (~[ v~(n)[~'k(n)) X"i exp(-- tJT) ti _ _ 12 E X *ti exp(-- tilT) + O(T 2 N-Ira(log T)2). 7~ ~ q Therefore, we only need to treat (57) ~ ] vj(n)l 2 X"i exp(-- tilT) for n ~ [N, 2N]. Let q~(x) be a smooth function on [0, oo] such that [~(x)[ <x, x-+0, I #"(x)t < x -", x -~ oo, for p = 0, 1, 2, 3. Define ~o = ~ 1 E Jo(y) ~(y) , ~%(x) = ~XJo(~X) J0(~y) q~(y) dy d~, f fo ~.(x) = ~XJo(~X) Jo(~y) ~(y) dy ar~, /fo~ --o ~(t) - 2i sinh ,~t [L.(x) -- J_~.(x)] ~(x) ax '~ Io '~ With these definitions we have ~(~) = ~(~) + ~(~) 30 234 WENZHI LUO AND PETER SARNAK and the Kuznetsov formula [KU] reads. For ll, l~ t> 1, Y, c}(b) ~(t,5 + _2 I * ~(t) V j(/1) 4,(h) 4,(t,) at ' '~ 30 1~(1 + 2it)[ = = a,,,,, Oo + ~ c-' S(/1, 12; c) ~,~ (4~ ~i-~l, lc). For ~ we choose -- sinh ~(.) = x exp(ix cosh ~), 29 = log X + ~. Then [D-I2 lemma 7] sinh(~ -t- 2i~) t ~(t) = sinh ~t -- cosh [~ % = 2= ~ sinh ~ ~' -- sinh 29 txJo(tx) (cosh ~ ~ -- t~) -8/2 at, -- sinh 29 I 0~ t~)_8/~ ~H(x) - ~ j~ tXJo(tX) (cosh ~ ~ -- at. It is easy to see that ~(t$) = XUi e-ti/T + O(e-'~ti), and q~o ~ X- x[~, - 2 f~ ~(t) J0 1 ~(1 + 2it)I n (d~'(n))2 dt,~ T(log T) 2 d2(n). Furthermore, Jo(y) ~ min(1,y -~/2) and [ S(n, n; c) l <~ (n, v) 112 c v~ d(c), hence S.(~s ) ~ N 1/~ X-1/2(log N) ~ and S.(~) ~ N x/2 T 1/~ X 1/4 log T, where s,(+) = X c -1 s(n, n; c) + --. QUANTUM ERGODICITY OF EIGENFUNCTIONS ON PSLs(Z)\H tt So we conclude that Y~ exp(-- tilT) XUJ N a/2 T v2 X v4 log T + N a/2 X- a/~ log2 N + N- a/2 T ~ log2 T + T log ~ T T 6/4 X 1/8 log 2 T, on taking N ---- T ~/~ X-v4. Next we show that the above estimate is still valid if we replace the smooth weight function exp(-- t/T) by the characteristic function X(t) of [1, T]. Take a smooth func- tiong(~) such that supp(g) _ [1/2, T + 1/2], 0 ~< g(~) ~< 1, andg(~) = 1 when ~ ~ [1, T]. Since it is known that t 1> 1 and [{t~:T<t 5~< T+ 1}]<T, we have Y~ X"i = Gg(tj) X uj + O(T). q~<T t i Let g(~) be the Fourier transform of g(~)exp(~/T): g(x) =f+_2g(~)exp(~/T)e(~x)d~. The easy estimate g(x) ~ min T, gives f [g(x)ldx~logT. For [x[ >/ 1, from partial integration 1 f+~ d g(x) -= 2~ix ~ (g(~) exp(~/T)) e(~x) d~, we infer that I e(--yx) g(x) dx 1 f+f ff-~ (g(~) exp(~/T)) fl+~176 dxd~ 1 1 log(T + Iy [) ~--+ + lYt + 1 IT--yl + 1 T Thus, from the Fourier inversion formula g(x) exp(x/T) =I+~g(~)e(--~x)d~, 236 WENZHI LUO AND PETER SARNAK we deduce that g(x) exp(x/T) = s g(~) e(-- ~x) d~ 1 1 log(T + Ix D) +O ix[+l+lT_xl+l + T " Therefore Ng(t,) X'q = g(~) (s (Xe-~"~)"~ ' exp(-- t/T)) d~ tj --1 (~ e- ,i/T e- tilt + log(T + tj) e- ti/Tt T ] ~ TS/* X1/8(l~ T)=" -4-0 --~ b IT-hi + 1 Thus, XUJ ~ T 5/* Xl/8(log T) 2. 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