Computing the Laplace eigenvalue and level of Maass cusp forms

JOURNAL OF NUMBER THEORY

卷:173

Article

Savala, Paul¹ 

[1] Whittier Coll, Dept Math, Whittier, CA 90608 USA

关键词: Maass forms for congruence subgroups;    Resonance;    Voronoi summation formula;    Laplace eigenvalue;   

DOI  :  10.1016/j.jnt.2016.09.015

来源: Elsevier

【 摘 要 】

Let f be a primitive Maass cusp form for a congruence subgroup Gamma(0) (D) subset of SL(2,Z) and lambda(f) (n) its n-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many lambda(f) (n) one can often solve for the level D, and in some cases, estimate the Laplace eigenvalue to arbitrarily high precision. This is done by analyzing the resonance and rapid decay of smoothly weighted sums of lambda(f)(n)e(alpha n(beta)) for X <= n <= 2X and any choice of alpha epsilon R, and beta > 0. The methods include the Voronoi summation formula, asymptotic expansions of Bessel functions, weighted stationary phase, and computational software. These algorithms manifest the belief that the resonance and rapid decay nature uniquely characterizes the underlying cusp form. They also demonstrate that the Fourier coefficients of a cusp form contain all arithmetic information of the form. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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