The second moment of the Siegel transform in the space of symplectic lattices

TL;DR

This paper derives a new formula for the second moment of the Siegel transform over symplectic lattices, extending classical results and providing strong bounds on lattice point discrepancy for generic symplectic lattices.

Contribution

It generalizes Rogers' classical second moment formula to the space of symplectic lattices using spectral theory of Eisenstein series.

Findings

New second moment formula for symplectic lattices

Strong bounds on lattice point discrepancy

Extension of classical results to symplectic setting

Abstract

Using results from spectral theory of Eisenstein series, we prove a formula for the second moment of the Siegel transform when averaged over the subspace of symplectic lattices. This generalizes the classical formula of Rogers for the second moment in the full space of unimodular lattices. Using this new formula we give very strong bounds for the discrepancy of the number of lattice points in an Borel set, which hold for generic symplectic lattices.