Modular forms on indefinite orthogonal groups of rank three

TL;DR

This paper develops a theory of modular forms on indefinite orthogonal groups of rank three, establishing their properties, Eisenstein series, and rational Fourier coefficients, extending classical and exceptional group results.

Contribution

It introduces a new framework for modular forms on SO(3,n+1), simplifying prior theories on quaternionic exceptional groups and proving algebraic properties of associated Eisenstein series.

Findings

Existence of absolutely convergent Eisenstein series as modular forms.

Proven algebraic Fourier coefficients for these Eisenstein series.

Rational Fourier expansion of the 'next-to-minimal' modular form on quaternionic E8.

Abstract

We develop a theory of modular forms on the groups $\mathrm{SO}(3,n+1)$, $n \geq 3$. This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of $\mathrm{SO}(3,n+1)$, there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on $\mathrm{SO}(2,n)$. As an application, we prove that the so-called "next-to-minimal" modular form on quaternionic $E_8$ has rational Fourier expansion, under a mild local assumption.