On Fourier coefficients of Maass waveforms for 𝑃𝑆𝐿(2,𝑍)

Hejhal, Dennis A.; Arno, Steven

doi:10.1090/s0025-5718-1993-1199991-8

Cited by 13 publications

(10 citation statements)

References 36 publications

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“…The magnitude is also a basic matter for questions in quantum chaos, [Sr1,Wp]. Numerical investigations of D. A. Hejhal [HA,Hj2] have provided evidence for a generalized Ramanujan-Petersson conjecture: the coefficients |a(n)| should be O λ, (|n| ). The focus of the present investigation is a set of relations among the Fourier coefficients of a cuspidal automorphic eigenfunction.…”

Section: −πλmentioning

confidence: 99%

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Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

Wolpert

2003

Trans. Amer. Math. Soc.

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Abstract. A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane K(z) = y 1/2 K ir (2πmy)e 2πimx , z = x+iy, λ = 1 4 +r 2 the eigenvalue, s = 2πmλ −1/2 and K ir the Macdonald-Bessel function. The phase velocity of K on {|s|Imz ≤ 1} is a double-valued vector field, the tangent field to the pencil of geodesics G tangent to the horocycle {|s|Imz = 1}. For A ∈ SL(2; R) a multiterm stationary phase expansion is presented in λ for K(Az)e 2πin Rez uniform in parameters. An application is made to find an asymptotic relation for the Fourier coefficients of a family of automorphic eigenfunctions. In particular, for ψ automorphic with coefficients {an} and eigenvalue λ it is shown for the special range n ∼ λ 1/2 that an is O(λ 1/4 e πλ 1/2 /2 ) for λ large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound O(|n| 1/2 λ 1/4 e πλ

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Section: −πλmentioning

confidence: 99%

“…The Fourier coefficients are of interest, [Bm,BDHI,HA,Iw1,Iw2,Ms,Pt1,Pt2,Tr,Sr1,Sb1,Vk] The order of magnitude of the Fourier coefficients and their sums are basic issues, even for nonarithmetic groups [DI,Iw2,Pt1,Pt2,Sr1,Sb2]. Hecke gave the argument for the elementary bound valid for Γ cofinite with a cusp at infinity |a(n)|e…”

Section: Introductionmentioning

confidence: 99%

Asymptotic relations among Fourier coefficients of automorphic eigenfunctions

Wolpert

2003

Trans. Amer. Math. Soc.

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“…An attempt to get around these instabilities was carried out by Stark [Sta84], Hejhal and Arno [HA93], and Steil [Ste92,Ste94]. They used the eigenvalue equations…”

Section: Introductionmentioning

confidence: 99%

Maaß cusp forms for large eigenvalues

Then

2004

Math. Comp.

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“…On the other hand, the Sato-Tate conjecture, first formulated independently by Sato and Tate in the context of elliptic curves and reformulated by Serre [1968], predicts this behavior of the Fourier coefficients of "typical" GL 2-eigenforms. (See [Hejhal and Arno 1993] for some numerical evidence for the Sato-Tate conjecture.) in a fixed number field.…”

Section: Introductionmentioning

confidence: 99%