Estimates for Rankin–Selberg L-Functions and Quantum Unique Ergodicity

By Peter Sarnak. 2001

175 citations recorded by the source corpus.

Abstract Subconvex bounds in the weight aspect for Rankin–Selberg L -functions associated to two cusp forms are established. These bounds are applied to prove the equidistribution of mass conjecture for CM forms.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f92e39acf316fb9de7faed75aed3a5d31a61029f

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In Algebra and Number Theory

1. V. Blomer
2. P. Michel
3. Akshay Venkatesh
4. Gergely Harcos
5. R. Munshi
6. Peter Sarnak
7. M. Young
8. Kannan Soundararajan

In Mathematical Physics

1. V. Blomer
2. P. Michel
3. Akshay Venkatesh
4. Gergely Harcos
5. Peter Sarnak
6. R. Munshi
7. M. Young
8. Kannan Soundararajan

Citations in the corpus

• Uniform Bound for Hecke L-Functions (2005, methodology): “After finishing the present work, we found that Sarnak had developed in [25] an approach to the subconvexity bound of Rankin–Selberg L-functions.”
• Weak subconvexity for central values of $L$-functions (2008, background): “Rudnick and Sarnak ([29], see also [25, 31]) have conjectured that as k → ∞, the measure μf approaches the uniform distribution measure 3 π dxdy y2 on the fundamental domain X = Γ\H.”
• Subconvexity for Rankin-Selberg L-Functions of Maass Forms (2002, methodology): “when g is holomorphic, where we can choose σ =1 / 2+ θ + e .W heng is a Maass cusp form, we can split Dg(s, 1, 1 ,h )i nto two terms as in the proof of Theorem 4.4 in Sarnak [ S2 ], such that the first term is bounded by |h| 1/2+θ+e−Re s (1 + |Ims|) 3 , while the second term is bounded by |h| 1−Re s”
• The spectral decomposition of shifted convolution sums (2007, background): “Jutila [Ju1] and Sarnak [Sa1, Sa2 ] independently considered an approximation of the Dirichlet series (7), along with a spectral decomposition, that could be continued to the half plane ℜs > 1/2 with a bound O(h”
• Rankin-Selberg $L$-functions in the level aspect (2002, background): “The main application of the convexity-breaking bound of [ Sa ] is the proof of this conjecture when the primitive forms are of CM type (in the sense of [Ri]).”
• Shifted convolution sums and subconvexity bounds for automorphic L-functions (2004, methodology): “[23] for a somewhat different method, and the references given there, and see [9] for an excellent survey).”
• Hybrid bounds for twisted L-functions (2008, methodology): “Good [Go1], [Go2] and many others (see, for example, [Ju2], [Ju3], [JM], [ Sa ], [LLY]) used a spectral decomposition for the Dirichlet series P lðnÞlðn þ hÞðn þ hÞ� s. This approach has certain di‰cul-”
• A new bound K2/3+ɛ for Rankin-Selberg ℒ-functions for Hecke congruence subgroups (2010, methodology): “We follow the argument in Sarnak [29, Appendix] to give a spectral decomposition of Dg(s, ν1, ν2, h) for holomorphic g.”
• An additive problem in the Fourier coefficients of cusp forms (2001, background): “ls. Using his estimates for triple products of eigenfunctions [S1] (see also [P, B-R]) he recently established quite strong bounds for Df(a,b;h) at least when the shift parameter his relatively small [S2]. This method has the advantage of generalizing naturally to number fields [P-S, C-PS-S] but suffe”
• The Second Moment of Twisted Modular L-Functions (2014, methodology): “folding trick to express the horocycle integral directly as a triple product involving Poincar´e series which can again be decomposed spectrally. This is the strategy followed by Good [Go] and Sarnak [Sa]. Finally, as a third option, one can use carefully chosen vectors in the representation space o”
• Functoriality for the classical groups (2004, background): “Let us formulate estimates towards Ramanujan in the following terms [46].”
• A general Voronoi summation formula for GL(n,Z) (2009, background): “hday 1. Introduction The Voronoi summation formulas for GL(2) and GL(3) have had numerous applications to problems in analytic number theory, perhaps most notably to recent subconvexity results (e.g. [1,6,15,16,20]). The formulas provide an identity between sums of the form X m∈Z am e 2πimα f(m) = X”
• Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem (2009, methodology): “They used a very eective spectral method based on bounds for triple products [Sa1, Sa2 ].”