In our talk we will report about work in progress with one of my PhD students on

sup-norm bounds for Siegel cusp forms of genus $n$ and weight $k$ on average.

The bounds obtained are straightforward generalizations of the respective optimal

bounds obtained in earlier joint-work with Jay Jorgenson in the classical case $n

=1$. As in the $1$-dimensional case, the quantity that needs to be bounded can

be related to the eigenspace for the smallest eigenvalue of the respective weight

$k$ Laplacian. This then allows to employ heat kernel estimates to arrive at the

desired bounds.