Mass equidistribution for Saito-Kurokawa lifts

TL;DR

Under GRH, the paper proves that the mass and zero divisors of Saito-Kurokawa lifts become uniformly distributed on the Siegel modular variety as their weights grow large.

Contribution

It establishes mass and zero divisor equidistribution for Saito-Kurokawa lifts assuming GRH, extending understanding of their asymptotic behavior.

Findings

Mass equidistribution on Siegel modular variety as weight increases

Zero divisors of lifts also equidistribute under GRH

Results depend on the assumption of the Generalized Riemann Hypothesis

Abstract

Let $F$ be a holomorphic cuspidal Hecke eigenform for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of $F$ equidistributes on the Siegel modular variety as $k\longrightarrow \infty$. As a corollary, we show under GRH that the zero divisors of Saito--Kurokawa lifts equidistribute as their weights tend to infinity.