Restriction norms for Siegel Modular Forms

Abstract

We consider the L²-norm of Siegel modular forms when restricted to the imaginary axis in an average sense. We establish an asymptotic formula with a power-saving error for cusp forms of degrees 1 and 2 in the weight aspect. We also prove an asymptotic formula for cusp forms of degree 1 in the level aspect. The result are consistent with the Mass Equidistribution Conjecture for Siegel modular forms and the Lindelöf Hypothesis for some twisted Koecher-Maass series. Along the way, we perform a careful analysis of the Kitaoka formula of degree 2. Finally, we prove a non-trivial bound for symplectic Kloosterman sums appearing in the Kitaoka formula of degree 3.