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The sup-norm problem on the Siegel modular space of rank two
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 4, August 2016
- pp. 999-1027
- 10.1353/ajm.2016.0032
- Article
- Additional Information
Let $F$ be a square integrable Maa{\ss} form on the Siegel upper half
space ${\mathcal H}$ of rank $2$ for the Siegel modular group ${\rm
Sp}_4({\Bbb Z})$ with Laplace eigenvalue $\lambda$. If, in addition, $F$
is a joint eigenfunction of the Hecke algebra and $\Omega$ is a compact
set in ${\rm Sp}_4({\Bbb Z})\backslash{\mathcal H}$, we show the bound
$\|F|_{\Omega}\|_{\infty} \ll_{\Omega} (1+\lambda)^{1-\delta}$ for some
global constant $\delta>0$. As an auxiliary result of independent interest
we prove new uniform bounds for spherical functions on semisimple Lie
groups.