Fourier Coefficients and Algebraic Cusp Forms on $\mathrm{U}(2,n)$

TL;DR

This paper develops a theory of Fourier coefficients for non-holomorphic automorphic forms on the quaternionic Lie group U(2,n), linking theta lifts from U(1,1) to construct algebraic cusp forms.

Contribution

It introduces a new framework for scalar Fourier coefficients on U(2,n) and demonstrates their algebraicity via theta lifts from U(1,1).

Findings

Established a theory of Fourier coefficients for non-holomorphic automorphic forms.

Constructed examples of algebraic non-holomorphic cusp forms on U(2,n).

Linked Fourier coefficients to algebraic numbers through theta lifts.

Abstract

We establish a theory of scalar Fourier coefficients for a class of non-holomorphic, automorphic forms on the quaternionic real Lie group $\mathrm{U}(2,n)$. By studying the theta lifts of holomorphic modular forms from $\mathrm{U}(1,1)$, we apply this theory to obtain examples of non-holomorphic cusp forms on $\mathrm{U}(2,n)$ whose Fourier coefficients are algebraic numbers.