On Hecke eigenvalues of Siegel modular forms in the Maass space

Abstract

In this article, we prove an Omega result for the Hecke eigenvalues λF⁢(n){\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⁢p2⁢(ℤ){Sp_{2}({\mathbb{Z}})}. In particular, we prove λF⁢(n)=Ω⁢(nk-1⁢exp⁡(c⁢log⁡nlog⁡log⁡n)),\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\exp\biggl{(}c\frac{\sqrt{\log n}}{\log% \log n}\biggr{)}\biggr{)}, when c>0{c>0} is an absolute constant. This improves the earlier result λF⁢(n)=Ω⁢(nk-1⁢(log⁡nlog⁡log⁡n))\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\biggl{(}\frac{\sqrt{\log n}}{\log\log n}% \biggr{)}\biggr{)} of Das and the third author. We also show that for any n≥3{n\geq 3}, one has λF⁢(n)≤nk-1⁢exp⁡(c1⁢log⁡nlog⁡log⁡n),\lambda_{F}(n)\leq n^{k-1}\exp\biggl{(}c_{1}\sqrt{\frac{\log n}{\log\log n}}% \biggr{)}, where c1>0{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 {λF⁢(n)/nk-1}n∈ℕ{\{\lambda_{F}(n)/n^{k-1}\}_{n\in{\mathbb{N}}}} and show that it has infinitely many limit points. Finally, we show that λF⁢(n)>0{\lambda_{F}(n)>0} for all n , a result proved earlier by Breulmann by a different technique.