On Hecke eigenvalues of Siegel modular forms in the Maass space

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 $S{p}_2(\mathbb{Z})$. 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 $${\lambda }_F(n)\le n^{k-1}\text{exp}\left(c_1\sqrt{\frac{\log n}{\log \log n}}\right),$$ 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 $\{\frac{{\lambda }_F(n)}{n^{k-1}}{\}}_{n\in \mathbb{N}}$ 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.

10 pages, To appear Forum Math