Sieving for mass equidistribution

Holowinsky, Roman

doi:10.4007/annals.2010.172.1499

Cited by 50 publications

(79 citation statements)

References 13 publications

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“…The next two lemmas combine to give an estimate for |F k | 2 , φ by using a smoothed incomplete Eisenstein series and bounds for a shifted convolution problem. This mirrors the route taken by Holowinsky [7]. Applying Proposition 2.1 of [14], which follows from expanding |F k | 2 , and keeping track of the dependencies on m and h one has…”

Section: 2supporting

confidence: 58%

“…Under the assumption of the Generalized Lindelöf Hypothesis effective error terms have been obtained by Watson [26] and Young [28]. For the unconditional result our arguments essentially follow those of Holowinsky and Soundararajan [7,24,8], except for one modification which we have borrowed from Iwaniec's course notes on QUE. We have also used some ideas of Matt Young [28] and the final optimization uses a trick from Iwaniec's course notes on QUE.…”

Section: Effective Quementioning

confidence: 75%

“…To handle the off-diagonal terms in (4.16) we use a version of Shiu's bound (as in Holowinsky's work, see Theorem 1.2 of [7]). This gives (4.17)…”

Section: 2mentioning

confidence: 99%

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Small scale distribution of zeros and mass of modular forms

2018

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We study the behavior of zeros and mass of holomorphic Hecke cusp forms on SL 2 (Z)\H at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as k → ∞. We show that the zeros equidistribute within such balls as k → ∞ as long as the radii shrink at a rate at most a small power of 1/ log k. This relies on a new, effective, proof of Rudnick's theorem on equidistribution of the zeros and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper.We also examine the distribution of the zeros near the cusp of SL 2 (Z)\H. Ghosh and Sarnak conjectured that almost all the zeros here lie on two vertical geodesics. We show that for almost all forms a positive proportion of zeros high in the cusp do lie on these geodesics. For all forms, we assume the Generalized Lindelöf Hypothesis and establish a lower bound on the number of zeros that lie on these geodesics, which is significantly stronger than the previous unconditional results.

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Section: 2supporting

confidence: 58%

Section: Effective Quementioning

confidence: 75%

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Small scale distribution of zeros and mass of modular forms

2018

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“…Proof. The proof closely follows the argument of Holowinsky [11], with appropriate modifications at the prime p = 2. Consider the Eisenstein series where c q (n) is a Ramanujan sum (see [21,Section 2.2] and [15, Section 4.2]).…”

Section: Extending the Length Of Summationmentioning

confidence: 68%

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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“…A stronger notion of quantum ergodicity, the quantum unique ergodicity (QUE) proposed by Rudnick-Sarnak [30] demands that all high energy eigenfunctions become completely flat, and it supposedly holds for negatively curved compact Riemannian manifolds. One case for which QUE was rigorously proved concerns arithmetic surfaces, thanks to tools from number theory and ergodic theory on homogeneous spaces [29,23,24]. For Wigner matrices, a probabilistic version of QUE was settled in [7].…”

mentioning

confidence: 99%

Universality for a class of random band matrices

Bourgade

Erdős

Yau

et al. 2017

Advances in Theoretical and Mathematical Physics

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We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ∼ N . All previous results concerning universality of non-Gaussian random matrices are for meanfield models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.

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Sieving for mass equidistribution

Cited by 50 publications

References 13 publications

Small scale distribution of zeros and mass of modular forms

Small scale distribution of zeros and mass of modular forms

Quantum unique ergodicity for half-integral weight automorphic forms

Universality for a class of random band matrices

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