# Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

## TL;DR

This paper explicitly expresses average Siegel theta series as linear combinations of Eisenstein series by identifying cusp representatives and attaching corresponding Eisenstein series, generalizing Siegel's classical results.

## Contribution

It provides explicit formulas for representing average Siegel theta series as linear combinations of Eisenstein series for arbitrary level and character.

## Key findings

- Explicit cusp representatives for degree n Siegel upper half-space
- Representation of theta series as linear combinations of Eisenstein series
- Generalization of Siegel's classical results to arbitrary level and character

## Abstract

We find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of $\Gamma_0(\stufe)$. To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $\stufe$ and Dirichlet character $\chi_L$ modulo $\stufe$ as a linear combination of Siegel Eisenstein series.