The Fourier coefficients of metaplectic theta series on GL(2) over rational function fields

TL;DR

This paper investigates the Fourier coefficients of metaplectic theta series over rational function fields, revealing a hidden automorphism group acting on these coefficients that is not apparent from explicit formulas.

Contribution

It introduces an operation of the automorphism group on the Fourier coefficients of metaplectic theta series, uncovering symmetries not seen in existing explicit formulas.

Findings

Automorphism group acts on Fourier coefficients

New symmetry operation not visible in explicit formulas

Enhances understanding of theta series structure over function fields

Abstract

The idea of the metaplectic theta function was introduced by Tomio Kubota in the 1960s. These theta functions are constructed as residues of Eisenstein series and are only known completely in the case of double covers and, up to the ambiguity of a constant, for triple covers. In 1992 Jeff Hoffstein published formulae by which these can be computed in certain cases over a rational function field. The author gave an alternative approach in 2007. Both of these methods give the coefficients in a closed form. The rational function field is unusual in that it has a large automorphism group. In this paper we show that this group has an operation on the coefficients. This operation is not visible from the explicit formulae.