Mass equidistribution for Hecke eigenforms

By R. Holowinsky, Kannan Soundararajan. 2008

203 citations recorded by the source corpus.

We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors see arxiv.org:math/0809.1640 and arxiv.org:math/0809.1635.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/86c9186e3fd14eb2612b884f0f5ef1c8f393c6f1

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Nodal Domains of Maass Forms I (2012, background): “dly. We examine here the formulation of QUE in terms of the Fourier coefﬁcients. For holomorphic cusp forms of even weight k (L2-normalized with respect to the Petersson inner-product) it is shown in [LS03] that QUE is equivalent to: For m 2Z and h 2C¥ 0 (0;¥) ﬁxed (82) G(k) k ¥ å n=1 r f (n)r f (n+”
Equidistribution Of Cusp Forms In The Level Aspect (2010, methodology): “cture spelled out in 2002 by Kowalski, Michel, and VanderKam [20] in the spirit of a conjecture of Rudnick and Sarnak [30] made in 1994. Our proof follows the strategy of Holowinsky and Soundararajan [15] who showed in 2008 that newforms of level 1 and large weight have equidistributed mass. The new”
Weak subconvexity for central values of $L$-functions (2008, methodology): “For example, in the application to the Rudnick-Sarnak conjecture any log power saving together with Holowinsky’s work would suffice to show that |〈φFk, Fk〉| → 0, for φ a fixed Maass cusp form.”
The Eigenvector Moment Flow and Local Quantum Unique Ergodicity (2013, background): “One case for which QUE was rigorously proved concerns arithmetic surfaces, thanks to tools from number theory and ergodic theory on homogeneous spaces [19,20,25].”
Nair–Tenenbaum bounds uniform with respect to the discriminant (2011, background): “One of these is the recent proof [6] of Quantum Unique Ergodicity by Soundararajan and Holowinsky, which combines different approaches by its two authors.”
Distribution of mass of holomorphic cusp forms (2012, background): “ of Ffor 2 <p≤ ∞. A ﬁrst guess might be that the mass of Fshould be nicely distributed on Γ\Hsuch that Fhas no essential peaks. Indeed, the mass equidistribution distribution conjecture, proved in [HSo], tells us that the measure |F(z)|2dxdy/y2 tends to the uniform measure (π/3)dxdy/y2 (in the se”
The quantum unique ergodicity conjecture for thin sets (2013, methodology): “product L-functions via Watson’s formula [W], or alternatively by shifted convolution sums via Poincare series. These two methods are apparently not equivalent and indeed Holowinsky and Soundararajan [HS] exploit both approaches in their proof of the mass equidistribution conjecture. We shall presen”
A Dynamical Approach to Random Matrix Theory (2017, background): “One case for which QUE was rigorously proved concerns arithmetic surfaces, thanks to tools from number theory and ergodic theory on homogeneous spaces [81, 82, 100].”
Real zeros of holomorphic Hecke cusp forms (2011, background): “25, 2018 1 arXiv:1103.3262v1 [math.NT] 16 Mar 2011 2 Amit Ghosh and Peter Sarnak One of the striking consequences of the recent proof of the holomorphic QUE conjecture by Holowinsky and Soundararajan [HS] is that Z(f) is equidistributed in X as k!1(see also [Ru] and the report [Sa-2] for a discussio”