Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series

TL;DR

This paper analyzes the action of Hecke operators on degree 2 Siegel Eisenstein series, constructing eigenforms and relations without relying on Fourier coefficients, thus advancing understanding of their algebraic structure.

Contribution

It introduces a method to evaluate Hecke operator actions on Siegel Eisenstein series without Fourier coefficients and constructs explicit eigenforms and relations.

Findings

Constructed a basis of simultaneous eigenforms for the full Hecke algebra.

Computed eigenvalues of the Hecke operators on these eigenforms.

Derived Hecke relations among Eisenstein series for square-free level and trivial character.

Abstract

We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level and arbitrary character, without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations in the case that $\stufe$ is square-free and the character is trivial, we generate a basis for the space of Eisenstein series.