Experimental Mathematics

The Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newforms

Josep González and Jorge Jiménez-Urroz

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Abstract

The Sato–Tate conjecture has been recently settled in great generality. One natural question now concerns the rate of convergence of the distribution of the Fourier coefficients of modular newforms to the Sato–Tate distribution. In this paper, we address this issue, imposing congruence conditions on the primes and on the Fourier coefficients as well. Assuming a proper error term in the convergence to a conjectural limiting distribution, supported by experimental data, we prove the Lang–Trotter conjecture, and in the direction of Lehmer’s conjecture, we prove that $\tau (p) = 0$ has at most finitely many solutions. In fact, we propose a conjecture, much more general than Lehmer’s, about the vanishing of Fourier coefficients of any modular newform.

Article information

Source
Experiment. Math., Volume 21, Issue 1 (2012), 84-102.

Dates
First available in Project Euclid: 31 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.em/1338430816

Mathematical Reviews number (MathSciNet)
MR2904910

Zentralblatt MATH identifier
1256.11032

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms

Keywords
Sato–Tate distribution Lang–Trotter conjecture Lehmer’s conjecture

Citation

González, Josep; Jiménez-Urroz, Jorge. The Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newforms. Experiment. Math. 21 (2012), no. 1, 84--102. https://projecteuclid.org/euclid.em/1338430816


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