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

Luo, Wenzhi; Rudnick, Zeév; Sarnak, Peter

doi:10.3934/jmd.2009.3.271

Cited by 14 publications

(28 citation statements)

References 47 publications

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“…where <, > is the invariant scalar product in U This result compares nicely with the classic variance for the distribution of geodesic cycles (or Heegner points) in the modular surface SL(2, Z)\H, as computed in [12]. It manifests once again the general principle that the variances of these arithmetic distributions should be proportional to the scalar product of the eigenfunctions twisted by their associated central L-values.…”

Section: Introductionsupporting

confidence: 59%

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A note on the distribution of integer points on spheres

Luo

2009

Math. Z.

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Section: Introductionsupporting

confidence: 59%

“…First for ν = μ, by Rankin-Selberg convolution applied to the pair θ(P, ·), θ (Q, ·) (on the group 0 (8) instead of 0 (4) for the latter notational convenience) and Tauberian theorem, we have as in §5 of [12],…”

Section: Proof Of the Theoremmentioning

confidence: 99%

A note on the distribution of integer points on spheres

Luo

2009

Math. Z.

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“…Proof. The proof of (8.1) relies on some computations of Matthes [29,30] (further refined in a lemma of Luo-Rudnick-Sarnak [27]). Let…”

Section: The Summation Formula and Proof Of Quementioning

confidence: 99%

Quantum unique ergodicity for half-integral weight automorphic forms

Lester¹,

Radziwiłł²

2020

Duke Math. J.

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We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for Γ 0 (4) lying in Kohnen's plus subspace and for halfintegral weight Hecke Maaß cusp forms for Γ 0 (4) lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on Γ 0 (4)\H as the weight tends to infinity.

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“…(vii) The quantum variance V (ψ) is the same as the variance for the fluctuations of the arithmetic measures on the modular surface obtained by collecting all closed geodesics. See the work of Luo, Rudnick and Sarnak [28].…”

Section: This Variance Summentioning

confidence: 99%

Quantum Variance of Maass-Hecke Cusp Forms

Zhao

2009

Commun. Math. Phys.

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In this thesis we study quantum variance for the modular surface X = Γ\H where Γ = SL 2 (Z) is the full modular group. This is an important problem in mathematical physics and number theory concerning the mass equidistribution of Maass-Hecke cusp forms on the arithmetic hyperbolic surfaces. We evaluate asymptotically the quantum variance, which is introduced by S. Zelditch and describes the fluctuations of a quantum observable. We show that the quantum variance is equal to the classical variance of the geodesic flow on S * X, the unit cotangent bundle of X, but twisted by the central value of the Maass-Hecke L-functions.Our approach is via Poincare series and Kuznetsov trace formula, which transfer the spectral sum into the sum of Kloosterman sums. The treatment of the nondiagonal terms contributions is subtle and forms the core of this thesis. It turns out that the continuous spectrum part is not negligible and contributes to the main term, but in general, it is small in the cuspidal subspace. We then make use of Watson's explicit triple product formula to determine the leading term in the asymptotic formula for the quantum variance and analyze its structure, which leads to our main result.ii Dedicated to my parents iii ACKNOWLEDGMENTS I express my deepest thanks and gratitude to my advisor, Professor Wenzhi Luo, for his patience, guidance and constant encouragement which made this work possible. I am very grateful for his enlightening discussions and suggestions as well as the intellectual support he has provided.

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

Cited by 14 publications

References 47 publications

A note on the distribution of integer points on spheres

A note on the distribution of integer points on spheres

Quantum unique ergodicity for half-integral weight automorphic forms

Quantum Variance of Maass-Hecke Cusp Forms

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