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Paramodular groups and theta series

Siegfried Böcherer, Rainer Schulze-Pillot

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.

Paramodular groups and theta series

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.

Paper Structure

This paper contains 10 sections, 27 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Paramodular groups and their modular forms
  3. Hecke algebras for paramodular groups
  4. Garrett's double coset decomposition for paramodular groups
  5. Theta series for the paramodular group
  6. Siegel's main theorem
  7. The basis problem for cusp forms of square free level
  8. Pullback of Eisenstein series
  9. Nonvanishing
  10. Basis problem

Key Result

Theorem 2.5

Let $R$ be a principal ideal domain with field of fractions $F$ and $\Lambda\subseteq V$ an $R$-lattice on the $2m$-dimensional $F$-vector space $V$ with nondegenerate alternating bilinear form $\langle, \rangle$, assume $\langle \Lambda,\Lambda\rangle\subseteq R$ and that the level of $\Lambda$ div can be split off orthogonally in $\Lambda$, i.e., one has $\Lambda=M \perp \Lambda'$ for some submo

Theorems & Definitions (68)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 58 more