On Selberg's eigenvalue conjecture

Luo, W.; Rudnick, Z.; Sarnak, P.

doi:10.1007/bf01895672

Cited by 141 publications

(56 citation statements)

References 18 publications

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“…Although estimate (3.5) has been improved in [GJ78,LRS95,I96,KS03], the Selberg conjecture (3.6) is still open. We refer to the surveys [S95,S03,BB13] for more detailed discussions.…”

Section: Definition 32 a Group G Has Kazhdan Property (T ) If One Omentioning

confidence: 99%

Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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Abstract. We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

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“…Although estimate (3.5) has been improved in [GJ78,LRS95,I96,KS03], the Selberg conjecture (3.6) is still open. We refer to the surveys [S95,S03,BB13] for more detailed discussions.…”

Section: Definition 32 a Group G Has Kazhdan Property (T ) If One Omentioning

confidence: 99%

Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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“…[68] for an exposition) and [2], respectively. For the Selberg conjecture 1 5 + ε, all ε > 0, was the best one proved in [45]. (3.10), (3.12), (4.24) and (4.25) were obtained a few months after that [33,34,36] as soon as the automorphy of Sym 4 π was also ready at hand [32].…”

Section: Q the Appearance Of Sym M π Does Not Need To Be As A Discrmentioning

confidence: 99%

Functoriality and small eigenvalues of Laplacian on Riemann surfaces

Shahidi

2004

Surveys in Differential Geometry

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“…If π v is a principal series representation defined by (quasi)characters ν, ν ′ , then by [GJ], e(ν) < 1/4 and e(ν ′ ) < 1/4. (Strictly speaking, when v is archimedean, one finds in [GJ] only the assertion that these exponents are ≤ 1/4, but one can eliminate the possibility of exponent 1/4 by using a simple version of the argument of [LRS1].) At the finite places, one can even replace 1/4 by 1/5 by using [Sh2]; see also [LRS2].…”

Section: Construction Ofmentioning

confidence: 99%

“…By Gelbart-Jacquet ( [GJ]), one has λ(π i,v ) < 1/4 for any i. (For F = Q, one even has the archimedean estimate: λ(π i,v ) < 5/28 by [LRS1].) So λ < 1/2 in this case, and by the remarks above, Ψ(f v,s ; W v ) is bounded in any vertical strip {1/2 ≤ ℜ(s) ≤ b} of finite width.…”

Section: Proof Of Lemmamentioning

confidence: 99%

Modularity of the Rankin-Selberg L-Series, and Multiplicity One for SL(2)

Ramakrishnan¹

2000

The Annals of Mathematics

199

177

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In memory of my father Sundaram Ramakrishnan (SRK) all agree at all the places. (Recall that we needed to know that they were very nearly the same to get Theorem M in the first place.) This may be of independent interest.The final application of our main result is the proof of the Tate conjecture for 4-fold products V of modular curves, asserting in particular that the order of pole at s = 2 of the L-function over any solvable (normal) number field K of the Galois module W ℓ := H 4 et (V Q , Q ℓ ) equals the rank of the group of K-rational codimension 2 Tate cycles on V Q (see Theorem 4.5.1). Moreover we show, in line with the works of Ribet ([Ri1]) and V. K. Murty ([Mu]) on the Jacobians of modular curves, that the latter number can also be computed with the Tate cycles replaced by the algebraic cycles modulo homological equivalence if the level of at least one of the curves is square-free. We refer to Chapter 5 for a precise statement.We would like to express our gratitude to Ilya Piatetski-Shapiro for his continued interest in this project, and for kindly writing down, with J. Cogdell, the form of the converse theorem for GL(4) which we need ([CoPS]). Thanks are also due to T. Ikeda for writing down his calculations of the archimedean factors of the triple product L-functions ([Ik2]), to S. Rallis for useful remarks on these L-functions, to F. Shahidi for explaining his approach to the same via Langlands's theory of Eisenstein series and for commenting on an earlier version, to my colleague T. Wolff for helpful conversations on an analytic lemma we use in Section 3.4, and to many others, including H. Jacquet, R. P. Langlands, J. Rogawski and P. Sarnak, who have shown encouragement and interest. Special thanks must go to J. Cogdell for reading the earlier and the revised versions thoroughly and making crucial remarks. Part of the technical typing of this paper was done by Cherie Galvez, whom we thank. Finally, we would like to express our appreciation to the following: the National Science Foundation for support through the grants DMS-9501151 and DMS-9801328, Université Paris-sud, Orsay, where we spent a fruitful month during September 1996, the DePrima Mathematics House in Sea Ranch, CA, for inviting us to visit and work there during August 1996 and 1998, MATSCIENCE, India, for hospitality in February 98, and -last, but not the least -the MSRI, Berkeley, for (twice) providing the right climate to work in; this project was started (in 1994) and essentially ended there. Notation and preliminaries 2.1.Let Q denote the algebraic closure of Q in C. For any subfield K of Q, let Gal(Q/K) denote the Galois group of Q over K, together with the profinite topology. For any number field F with ring of integers O F , let Σ(F ) (resp. Σ ∞ (F ), resp. Σ 0 (F )) denote the places (resp. archimedean places,

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On Selberg's eigenvalue conjecture

Cited by 141 publications

References 18 publications

Quantitative ergodic theorems and their number-theoretic applications

Quantitative ergodic theorems and their number-theoretic applications

Functoriality and small eigenvalues of Laplacian on Riemann surfaces

Modularity of the Rankin-Selberg L-Series, and Multiplicity One for SL(2)

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