On Hermitian Eisenstein series of degree 2

Abstract

We consider the Hermitian Eisenstein series $E_k^{(\mathbb{K})}$ 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 $E_4^{{(\mathbb{K})}^2}=E_8^{(\mathbb{K})}$, 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 $E_k^{(\mathbb{K})}$, $k=4,6,8,10,12$ are algebraically independent whenever $\mathbb{K}\ne \mathbb{Q}(\sqrt{-3})$. 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.

15 pages