Growth and nonvanishing of restricted Siegel modular forms arising as Saito-Kurokawa lifts

TL;DR

This paper investigates the behavior of restricted Siegel modular forms, specifically Saito-Kurokawa lifts, linking their analytic properties to $L$-functions via Ichino's formula.

Contribution

It provides new insights into the growth and nonvanishing of these forms by connecting their restriction behavior to $L$-functions through a key formula.

Findings

Established a relationship between restricted Siegel forms and $L$-functions.

Proved nonvanishing results for certain families of $L$-functions.

Analyzed the growth behavior of Saito-Kurokawa lifts on restricted domains.

Abstract

We study the analytic behavior of the restriction of a Siegel modular form to $\mathbb{H} \times \mathbb{H}$ in the case that the Siegel form is a Saito-Kurokawa lift. A formula of Ichino links this behavior to a family of $GL_3 \times GL_2$ $L$-functions.