Fourier coefficients of sextic theta series (Q2796024)

Historically, modular forms emerged as functions attached to algebraic groups. Concretely, \(\mathrm{SL}_2\) as a group over \(\mathbb{Q}\) or a real quadratic field played a crucial role in the early developments of the theory of modular forms by, for example, Hecke. To make theta series, long known at that time, fit into the framework of algebraic groups, it was necessary to define ``multiplier systems'', which essentially encoded the transformation behavior of one fixed modular form -- typically one of the Jacobi theta functions or the eta function. Weil then observed that one can mitigate the need for a multiplier system by passing to a double cover of \(\mathrm{SL}_2\), which is not an algebraic group over \(\mathbb{Z}\). Till today both approaches coexist in the classical community, while the geometric and representation theoretic approach to modular forms embarked almost completely on Weil's perspective. Examples of contemporary research on this topic comprise [\textit{S. Friedberg} and \textit{D. Ginzburg}, J. Number Theory 146, 134--149 (2015; Zbl 1366.11075); \textit{B. Brubaker} and \textit{S. Friedberg}, Geom. Funct. Anal. 25, No. 4, 1180--1239 (2015; Zbl 1372.11061); \textit{M. H. Weissman}, Trans. Am. Math. Soc. 368, No. 5, 3695--3725 (2016; Zbl 1369.14058)].NEWLINENEWLINENEWLINERephrasing the theory of half-integral weight modular forms in terms of double covers of algebraic groups makes it natural to consider higher metaplectic covers. Kubota laid the foundations of a theory of modular forms on such covers by defining and studying Eisenstein series. In analogy with the case of half-integral weight Eisenstein series, they have a simple pole with respect to the spectral variable. The residues are analogues of theta functions, and one expects that their Fourier expansions enjoy rich arithmetic properties.NEWLINENEWLINENEWLINEIt is a difficult problem, however, to determine Fourier coefficients of theta functions on \(n\)-fold covers of \(\mathrm{SL}_2\). As opposed to the case of usual theta series, relations from Hecke operators do not sufficiently restrict them. A list and discussion of what cases have been considered is contained in the paper's introduction. The body of the paper extends this list by \(n=6\), for which the authors carry out numerical computations. They arrive at three conjectures relating all Fourier coefficients to each other which are not already connected by Hecke relations.