Rationality of quaternionic Eisenstein series on $\mathrm{U}(2,n)$

Abstract

Let $\mathbf{G}=\mathrm{U}(2,n)$ be the unitary group associated to a Hermitian space over a quadratic imaginary number field $E$. We assume that 2 is unramified in $E$, and the Hermitian space splits at all finite places and has signature $(2,n)$, where $n\equiv 2\mathrm{mod}4$. A theory of Fourier expansions of quaternionic modular forms on $\mathbf{G}$ is developed by Hilado, McGlade, and Yan. In this paper, we define a family of degenerate Heisenberg Eisenstein series $E_{\ell }$ for $\ell >n$ on $\mathbf{G}$, which is a weight $\ell$ quaternionic modular form, and we explicitly compute their Fourier expansions. We prove that the Fourier coefficients of $E_{\ell }$ are rational in a certain sense, and that their denominators are uniformly bounded by an integer depending only on $\ell ,n$, and $E$. This provides the first family of quaternionic Eisenstein series whose Fourier coefficients are known to be rational or algebraic.

41 pages, in English