Hecke operators on half-integral weight Siegel Eisenstein series

TL;DR

This paper constructs a basis for half-integral weight Siegel Eisenstein series of level 4N, applies Hecke operators explicitly, and diagonalizes a subspace to achieve a multiplicity-one result, advancing understanding of their structure.

Contribution

It introduces a basis for these Eisenstein series, applies Hecke operators explicitly, and proves a multiplicity-one theorem for a specific subspace.

Findings

Explicit basis construction for half-integral weight Siegel Eisenstein series.

Hecke operators are applied directly and their images are expressed explicitly.

Diagonalization yields a multiplicity-one result for a subspace.

Abstract

We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of $\Gamma_0(4)$, commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of $\Gamma_0(4)$, obtaining a multiplicity-one result.