A converse theorem for metaplectic Eisenstein series on the double and triple covers of \(\mathrm{SL}_2(\mathbb{Q}(\sqrt{-D}))\) (Q2816459)

The paper under review proves a converse theorem for the metaplectic Eisenstein series on the \(n\)-fold metaplectic cover of \(\mathrm{SL}_2(F)\), where \(F\) is an imaginary quadratic number field containing the \(n\)th roots of unity. Roughly speaking, the author's main result asserts that a nice family of double Dirichlet series arises from Mellin transforms of linear combinations of metaplectic Eisenstein series. Moreover the Dirichlet series satisfy the same functional equation of the Eisenstein series. This result is similar to the previous converse theorems relating double Dirichlet series to the Mellin transforms of Eisenstein series of half-integral weight, due to \textit{N. Diamantis} and \textit{D. Goldfeld} [Am. J. Math. 133, No. 4, 913--938 (2011; Zbl 1232.11060); J. Math. Soc. Japan 66, No. 2, 449--477 (2014; Zbl 1311.11043)]. The author expects that the converse theorem can be extended to all number fields \(F\) of class number one and all coverings of a degree \(n\geq 1\), and outlines some applications.