The Distribution of Fourier Coefficients of Weak Maass Forms
Abstract
The theory of weak Maass forms has been studied extensively in recent years, resulting in many striking results. Examples include the fundamental work of Bruinier and Ono which relates the Fourier coefficients of half-integral weight harmonic (weak) Maass forms to periods and central values/central derivatives of modular L–functions.
In this work, we investigate the distribution of Fourier coefficients of a generic family of weak Maass forms. For integral weight forms, we prove a quantitative Sato-Tate distribution for normalized Fourier coefficients of these forms of integral weight k and prime level p as p → ∞. As a direct application, we prove similar results for harmonic Maass forms of integral weight k ≤ 0 and prime level p along with the results for weakly holomorphic modular forms of integral weight k ≥ 2 and prime level p. The proofs involve geometrical method related to bounding the analytic conductor of a suitable ℓ-adic Fourier sheaf and approximating the normalized Fourier coefficients of the weak Maass forms by normalized Kloosterman sums.
For half-integral weight forms, we prove that these coefficients are quantitatively equidistributed with respect to the pushforward of the Haar measure on the unitary group U(1). Similarly, we prove quantitative equidistribution for normalized Fourier coefficients of these forms of half-integral weight k ≤ 1/2 and level 4p as p → ∞ along with the results for weakly holomorphic modular forms of half-integral weight k ≥ 3/2 and level 4p. The proofs involve analytic methods and approximating the normalized Fourier coefficients of the weak Maass forms by normalized Salié sums. As a crucial part of our analysis, we prove quantitative vertical equidistribution of Salié sums, and a uniform bound for sums of half-integral weight (opposite sign) Kloosterman sums with θ-multiplier.
Subject
Weak Maass formsHarmonic Maass forms
Weakly holomorphic forms
Sato-Tate law
Equidistribution
Kloosterman sums
Sali\'e sums
Citation
Tsai, Wei-Lun (2020). The Distribution of Fourier Coefficients of Weak Maass Forms. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /191895.