Quantum ergodicity of eigenfunctions on \(\text{PSL}_ 2(\mathbb{Z}) \backslash H^ 2\). (English) Zbl 0852.11024

The quantum ergodicity in the title concerns distribution results for measures \(|f(z) |^2 d\mu(z)\) on the upper half plane \(H^2\) modulo the full modular group \(\Gamma= \text{PSL}_2 (\mathbb{Z})\), where \(f\) is either a Maass cusp form, or an Eisenstein series with its spectral parameter on the critical line. \(d\mu (z)\) denotes the invariant measure \(y^{-2} dx dy\).
Let \(u_1, u_2, \dots\) be a maximal orthogonal system of Maass cusp forms, with corresponding eigenvalues \(\lambda_1, \lambda_2, \dots\;\). It is shown that for smooth integrable functions \(F\) on \(X= \Gamma \setminus H^2\): \[ \sum_{\lambda_j\leq x} \;\Biggl|\int_X F|u_j |^2 d\mu- {\textstyle {3\over \pi}} \int_X F d\mu \Biggr|^2 \ll_\varepsilon C_F x^{1/2+ \varepsilon} \qquad (x\to \infty) \] for each \(\varepsilon> 0\). The constant \(C_F\) depends on supremum norms of partial derivatives of \(F\) up to order 8.
Let \(E(z, s)\) be the Eisenstein series of weight zero, with eigenvalue \(s- s^2\). As \(X\) has infinite mass for the measure \(|E(z, 1/2+ it)|^2 d\mu (z)\), the quantum ergodicity is formulated in terms of compact Jordan measurable subsets of \(X\). It is shown that for each such subset \(A\): \[ \int_A |E(z, 1/2+ it)|^2 d\mu (z)\sim {\textstyle {48 \over \pi}} \mu(A) \log t \qquad (t\to \infty). \] Other results in the paper are an improvement in the error term of the prime geodesic theorem for \(X\), and the estimate \[ \sum_{\lambda_j\leq x} \sup_B \Biggl|\int_B |u_j (z) |^2 d\mu (z)- {\textstyle {3\over \pi}} \mu (B) \Biggr|^2 \ll x^{20/ 21+ \varepsilon} \qquad (x\to \infty). \] \(B \subset X\) runs over the injective geodesic circles in \(X\).
The proofs use many techniques and results of the theory of real analytic modular forms: \(L\)-functions associated to modular forms, approximation by Poincaré series, Hecke operators, the Kuznetsov sum formula, and bounds on Fourier coefficients of Maass cusp forms.


11F37 Forms of half-integer weight; nonholomorphic modular forms
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F30 Fourier coefficients of automorphic forms
Full Text: DOI Numdam EuDML


[1] Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien,Com. Math. Phys.,102 (1985), 497–502. · Zbl 0592.58050
[2] J.-M. Deshouillers, H. Iwaniec, Kloosterman sum and Fourier coefficients of cusp forms,Invent. Math.,70 (1982), 219–288. · Zbl 0502.10021
[3] J.-M. Deshouillers, H. Iwaniec, 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. · Zbl 0595.10025
[4] A. Erdélyi et al., Higher Transcendental Functions, vol. 2, McGraw-Hill, 1953. · Zbl 0052.29502
[5] I. S. Gradshtein, I. M. Ryzhik,Tables of Integrals, Series and Products, Academic Press, New York and London, 1965.
[6] D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol. 1,Springer Lecture Notes,548, Springer-Verlag, 1976.
[7] D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol. 2,Springer Lecture Notes,1001, Springer-Verlag, 1983.
[8] D. A. Hejhal, Eigenvalues for the Laplacian for Hecke triangle groups,Memoirs of AMS, Vol.469, 1992. · Zbl 0798.11017
[9] D. A. Hejhal, D. Rackner, On the topography of Maass wave forms,Exper. Math.,1 (1992), 275–305. · Zbl 0813.11035
[10] 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. · Zbl 0814.11032
[11] H. Iwaniec, Prime geodesic theorem,J. Reine Angew. Math.,349 (1984), 136–159. · Zbl 0527.10021
[12] H. Iwaniec, Small eigenvalues for {\(\Gamma\)}0(N),Acta Arith., LVI (1990), 65–82.
[13] H. Iwaniec, The spectral growth of automorphic L-functions,J. Reine Angew. Math.,428 (1992), 139–159. · Zbl 0746.11024
[14] H. Iwaniec, H. Sarnak, Lnorms of eigenfunctions of arithmetic surfaces, To appear inAnnals of Math. · Zbl 0833.11019
[15] H. Iwaniec, Non-holomorphic modular forms and their applications,Modular forms, Durham conference, edited by R. Rankin, 1984, 157–196. · Zbl 0558.10018
[16] D. Jakobson,Quantum ergodicity for Eisenstein series on PSL2(Z)\(\backslash\)PSL2(R), Preprint, Princeton, 1994.
[17] L. Kuipers, H. Niederreiter,Uniform distribution of sequences, New York, Wiley, 1974. · Zbl 0281.10001
[18] N. V. Kuznetsov, Petersson’s conjecture for cusp forms of weight zero and Linnik’s conjecture, Sums of Kloosterman sums,Mat. Sb.,111 (1980), 334–383. · Zbl 0427.10016
[19] T. Meurman, 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.
[20] S. Ramanujan, Some formulae in the arithmetic theory of numbers,Messenger of Math.,45 (1916), 81–84.
[21] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces,Trans. AMS,233 (1977), 241–247.
[22] Z. Rudnick, P. Sarnak, The behavior of eigenstates of arithmetic hyperbolic manifolds,Com. Math. Phys.,161 (1994), 195–213. · Zbl 0836.58043
[23] P. Sarnak,Arithmetic quantum chaos, The RA Blyth Lecture, University of Toronto, 1993.
[24] P. Sarnak,Some Applications of modular forms, Cambridge Univ. Press, 1990. · Zbl 0721.11015
[25] A. Selberg,Collected Papers, Vol. 1, Springer-Verlag, 1989, 626–674. · Zbl 0675.10001
[26] A. I. Schnirelman, Ergodic properties of eigenfunctions,Usp. Math. Nauk.,29 (1974), 181–182. · Zbl 0324.58020
[27] G. Shimura, On the holomorphy of certain Dirichlet series,Proc. London Math. Soc. (3),31 (1975), 79–98, · Zbl 0311.10029
[28] G. Steil, Über die Eigenwerte des Laplace Operators und die Hecke Operatoren für SL(2,Z), preprint, 1993.
[29] E. C. Titchmarsh,The Theory of the Riemann Zeta Function, Oxford, 1951. · Zbl 0042.07901
[30] A. Weil, On some exponential sums,Proc. Nat. Acad. Sci. USA,34 (1948), 204–207. · Zbl 0032.26102
[31] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces,Duke Math. Jnl.,55 (1987), 919–941. · Zbl 0643.58029
[32] S. Zelditch, Selberg trace formulae and equidistribution theorems,Memoirs of AMS, Vol. 96, No. 465, 1992. · Zbl 0753.11023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.