On Hecke eigenvalues of Siegel modular forms in the Maass space

TL;DR

This paper establishes new bounds and properties of Hecke eigenvalues for Siegel modular forms in the Maass space, including omega-results, upper bounds, limit points, and positivity, advancing understanding of their growth and distribution.

Contribution

It provides improved omega-results, upper bounds, and insights into the limit points and positivity of Hecke eigenvalues for Maass forms in Siegel modular forms of genus two.

Findings

Proved an omega-result showing eigenvalues grow at least as fast as a specific exponential function.

Established an improved upper bound on eigenvalues involving exponential of square root of log n.

Showed the sequence of normalized eigenvalues has infinitely many limit points and all eigenvalues are positive.

Abstract

In this article, we prove an omega-result for the Hecke eigenvalues $\lambda_F(n)$ of Maass forms $F$ which are Hecke eigenforms in the space of Siegel modular forms of weight $k$, genus two for the Siegel modular group $Sp_2(\Z)$. In particular, we prove $$ \lambda_F(n)= \Omega(n^{k-1}\text{exp} (c \frac{\sqrt{\log n}}{\log\log n})), $$ when $c>0$ is an absolute constant. This improves the earlier result $$ \lambda_F(n)= \Omega(n^{k-1} (\frac{\sqrt{\log n}}{\log\log n})) $$ of Das and the third author. We also show that for any $n \ge 3$, one has $$ \lambda_F(n) \leq n^{k-1}\text{exp} \left(c_1\sqrt{\frac{\log n}{\log\log n}}\right), $$ where $c_1>0$ is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence $\{\frac{\lambda_F(n)}{n^{k-1}}\}_{n \in \N}$ and show that it has infinitely many limit points. Finally, we show that $\lambda_F(n) >0$ for all $n$, a result earlier proved by Breulmann by a different technique.