Paramodular groups and theta series

TL;DR

This paper investigates paramodular groups of any degree with square-free level, focusing on their Hecke algebra, boundary components, and the construction of theta series that generate cusp forms, supported by a new geometric proof of Garrett's decomposition.

Contribution

It introduces paramodular theta series that generate cusp forms for square-free levels and large weights, along with a novel geometric proof of Garrett's double coset decomposition.

Findings

Paramodular theta series generate cusp forms at large weights.

A new geometric proof of Garrett's double coset decomposition.

Analysis of the Hecke algebra and boundary components for paramodular groups.

Abstract

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling and pullback of Eisenstein series method. For this we give a new geometric proof of Garrett's double coset decomposition which works in our more general situation.