Equidistribution of Fourier coefficients of integral weight Maass-Poincaré series
One of the breakthrough results in mathematics without any doubt is the proof of the Sato-Tate Conjecture recently. One of the evidences to show its importance, is of course, the great consequences and applications of the Sato-Tate Conjecture. There are dozens of papers in the literature related to these topics. The paper under review concerns the case of integral weight Maass-Poincaré series. It is well known that Fourier coefficients of weak Maass forms have many interesting connections with diverse arithmetic objects. Hence it is not a surprise that it stands at the core of the research recently. In the paper under review, the authors show that the normalized Fourier coefficients of a generic family of Maass-Poincaré series of integral weight \(k\) and prime level \(p\) become quantitatively equidistributed with respect to the Sato-Tate measure as \(p \to \infty\). As a consequence, they deduce similar results for harmonic Maass forms of integral weight \(k \leq 0\) and level \(p\), and weakly holomorphic modular forms of integral weight \(k \geq 2\) and level \(p\). By using SageMath, the authors illustrate their results where the reader can see the equidistribution geometrically. The proof of the main result is a mixture of algebraic and analytic tools.
scientific article; zbMATH DE number 7597097
scientific article