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

Henry H. Kim, Yi Shan

TL;DR

The paper constructs a family of degenerate Heisenberg Eisenstein series $E_ u$ on the unitary group $ ext{U}(2,n)$ (with $n$ even) as weight-$oldsymbol{V}_ u$ quaternionic modular forms, and proves their Fourier coefficients are rational with uniformly bounded denominators. It provides a detailed local–global analysis: non-archimedean ranks via Siegel series across split, inert, and ramified primes, and archimedean ranks via Fourier transforms and hypergeometric representations, yielding explicit formulas for rank-2 and rank-1 coefficients and a transparent description of the constant term. The results establish the first family of quaternionic Eisenstein series with known rationality/algebraicity of Fourier coefficients, and they frame a conjectural Saito–Kurokawa-type lift from $ ext{SL}_2$ cusp forms to $ ext{U}(2,n)$, with predicted $L$-function factors and CAP behavior. Together, these findings illuminate the arithmetic structure of quaternionic modular forms on unitary groups and connect Eisenstein-coefficient rationality to theta-lift phenomena and base-change $L$-functions.

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.

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

TL;DR

The paper constructs a family of degenerate Heisenberg Eisenstein series on the unitary group (with even) as weight- quaternionic modular forms, and proves their Fourier coefficients are rational with uniformly bounded denominators. It provides a detailed local–global analysis: non-archimedean ranks via Siegel series across split, inert, and ramified primes, and archimedean ranks via Fourier transforms and hypergeometric representations, yielding explicit formulas for rank-2 and rank-1 coefficients and a transparent description of the constant term. The results establish the first family of quaternionic Eisenstein series with known rationality/algebraicity of Fourier coefficients, and they frame a conjectural Saito–Kurokawa-type lift from cusp forms to , with predicted -function factors and CAP behavior. Together, these findings illuminate the arithmetic structure of quaternionic modular forms on unitary groups and connect Eisenstein-coefficient rationality to theta-lift phenomena and base-change -functions.

Abstract

Let be the unitary group associated to a Hermitian space over a quadratic imaginary number field . We assume that 2 is unramified in , and the Hermitian space splits at all finite places and has signature , where . A theory of Fourier expansions of quaternionic modular forms on is developed by Hilado, McGlade, and Yan. In this paper, we define a family of degenerate Heisenberg Eisenstein series for on , which is a weight quaternionic modular form, and we explicitly compute their Fourier expansions. We prove that the Fourier coefficients of are rational in a certain sense, and that their denominators are uniformly bounded by an integer depending only on , and . This provides the first family of quaternionic Eisenstein series whose Fourier coefficients are known to be rational or algebraic.
Paper Structure (24 sections, 33 theorems, 179 equations)

This paper contains 24 sections, 33 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the unitary groups of real rank 2
  3. Hermitian spaces
  4. Unitary group and its Heisenberg parabolic subgroup
  5. Quaternionic Heisenberg Eisenstein series
  6. Quaternionic modular forms on
  7. The Heisenberg Eisenstein series
  8. Abstract Fourier expansion
  9. Rank 2 Fourier coefficients of
  10. Rank 2 Fourier coefficients: non-archimedean components
  11. Siegel Series for quadratic spaces
  12. Split case
  13. Inert case
  14. Ramified case
  15. Rank 2 Fourier coefficients: archimedean components
  16. ...and 9 more sections

Key Result

Theorem 1.1

(thm full Fourier expansion) Assume that the discriminant of $E$ is odd. When $\ell>n$, the Eisenstein series $E_\ell(g)$ has rational Fourier coefficients in the following sense: Moreover, the value of the $T$th Fourier coefficient $a_{T}(E_{\ell},g_{f})$ at $g_{f}=1$ is non-zero if and only if $T$ lies in a self-dual Hermitian $\mathcal{O}_{E}$-lattice $\mathbf{V}_{0}(\mathbb{Z})$ of $\mathbf{V

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • ...and 52 more