A symplectic restriction problem

TL;DR

This paper studies the behavior of degree 2 Siegel modular forms when restricted to a specific subspace, providing asymptotic formulas and connecting to deep conjectures in number theory.

Contribution

It introduces a new relative trace formula for pairs of Heegner periods and establishes an asymptotic formula related to mass equidistribution and Lindelof hypothesis.

Findings

Asymptotic formula for the norm of Siegel modular forms on a restricted subspace

Evidence supporting the mass equidistribution conjecture

Connection to Lindelof hypothesis for Koecher-Maass series

Abstract

We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito-Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelof hypothesis for the corresponding Koecher-Maass series. The ingredients include a new relative trace formula for pairs of Heegner periods.