On Poincar\'e series of half-integral weight

TL;DR

This paper constructs and analyzes Poincaré series of half-integral weight cusp forms on metaplectic groups, providing explicit formulas, non-vanishing results, and connections to linear functionals and Fourier expansions.

Contribution

It introduces a method to generate a spanning set of cusp forms using Poincaré series of matrix coefficients for metaplectic covers, including explicit formulas and non-vanishing criteria.

Findings

Constructed a spanning set for cusp forms using Poincaré series.

Derived formulas for Petersson inner products of these cusp forms.

Provided Fourier expansions and series representations of the constructed forms.

Abstract

We use Poincar\'e series of $ K $-finite matrix coefficients of genuine integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ to construct a spanning set for the space of cusp forms $ S_m(\Gamma,\chi) $, where $ \Gamma $ is a discrete subgroup of finite covolume in the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $, $ \chi $ is a character of $ \Gamma $ of finite order, and $ m\in\frac52+\mathbb Z_{\geq0} $. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any $ f\in S_m(\Gamma,\chi) $. Using this last result, we construct a Poincar\'e series $ \Delta_{\Gamma,k,m,\xi,\chi}\in S_m(\Gamma,\chi) $ that corresponds, in the sense of the Riesz representation theorem, to the linear functional $ f\mapsto f^{(k)}(\xi) $ on $ S_m(\Gamma,\chi) $, where $ \xi\in\mathbb C_{\Im(z)>0} $ and $ k\in\mathbb Z_{\geq0} $. Under some additional conditions on $ \Gamma $ and $ \chi $, we provide the Fourier expansion of cusp forms $ \Delta_{\Gamma,k,m,\xi,\chi} $ and their expansion in a series of classical Poincar\'e series.