Computing the Laplace eigenvalue and level of Maass cusp forms

TL;DR

This paper develops methods to determine the level and estimate the Laplace eigenvalue of Maass cusp forms using finitely many Fourier coefficients, leveraging resonance phenomena and advanced analytical techniques.

Contribution

It introduces algorithms that use resonance analysis of Fourier coefficients to recover the level and eigenvalue of Maass cusp forms from limited data.

Findings

Can determine the level D from finitely many Fourier coefficients.

Allows high-precision estimation of the Laplace eigenvalue.

Shows Fourier coefficients encode all arithmetic information of the cusp form.

Abstract

Let $f$ be a primitive Maass cusp form for a congruence subgroup $\Gamma_0(D) \subset $ SL($2,\mathbb{Z}$) 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 $X \leq n \leq 2X$ and any choice of $\alpha \in \mathbb{R}$, 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.