Metaplectic Eisenstein Series

A metaplectic Eisenstein series is an automorphic form obtained by summing a function on a finite central group extension over cosets associated with a parabolic subgroup. It transforms under the central subgroup of roots of unity by a specified group character. When the character is nontrivial, the series is called genuine and does not descend to a function on the underlying group (Kazhdan and Patterson 1984).

In the cubic case over the imaginary quadratic field , let be the ring of Eisenstein integers, with . Let be the subgroup of the special linear group generated by matrices congruent to the identity matrix modulo 3 together with . The cubic group character is trivial on . On the matrices congruent to the identity matrix modulo 3, it is the cubic residue symbol when and 1 when (Patterson 1977, Dunn and Radziwiłł 2024).

Write for a point of three-dimensional hyperbolic space, with a complex number and . If is the height and is the stabilizer of infinity in , the series at the cusp at infinity is

where , for the bottom row of , and the overline denotes the complex conjugate. The weight is well defined on the right cosets because on . Reindexing the sum gives for . The analytic continuation in has a simple pole at . Its complex residue is proportional to Kubota's cubic theta function (Deligne 1980, Yoshimoto 1987).