On Hermitian Eisenstein series of degree 2

TL;DR

This paper studies Hermitian Eisenstein series of degree 2 over imaginary quadratic fields, revealing their algebraic independence, special identities, and their role in generating spaces of cusp forms as the field varies.

Contribution

It determines the influence of the quadratic field on Eisenstein series, establishes algebraic independence for most fields, and describes their generation of cusp form spaces.

Findings

Identity $E^{(K)}_4^2 = E^{(K)}_8$ holds only for $K= Q( oot 4 oot -3)$.

Eisenstein series are algebraically independent for all $K eq Q( oot 4 oot -3)$.

These forms generate the cusp form space in the Maass Spezialschar as a module over the Hecke algebra.

Abstract

We consider the Hermitian Eisenstein series $E^{(\mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $\mathbb{K}$ and determine the influence of $\mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(\mathbb{K})}^2}_4 = E^{{(\mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $\mathbb{K} = \mathbb{Q} (\sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(\mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $\mathbb{K}\neq \mathbb{Q}(\sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maass space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-\infty$. Finally, we show that these forms generate the space of cusp forms in the Maass Spezialschar as a module over the Hecke algebra as $\mathbb{K}$ varies over imaginary-quadratic number fields.