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

TL;DR

This paper develops a theory for quaternionic Eisenstein series on the unitary group U(2,n), explicitly computes their Fourier expansions, and proves their Fourier coefficients are rational with bounded denominators, advancing understanding of quaternionic modular forms.

Contribution

It introduces a new family of quaternionic Eisenstein series on U(2,n) and establishes their Fourier coefficients are rational with uniformly bounded denominators, a first in the field.

Findings

Fourier coefficients are rational in a certain sense.

Bounded denominators depend only on specific parameters.

First explicit family with rational Fourier coefficients in quaternionic Eisenstein series.

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 $n\equiv 2 \operatorname{mod} 4$. 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.