Glasnik Matematicki, Vol. 53, No. 2 (2018), 239-264.

ON POINCARÉ SERIES OF HALF-INTEGRAL WEIGHT

Sonja Žunar

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
e-mail: szunar@math.hr

Abstract.   We use Poincaré series of K -finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL2(ℝ) to construct a spanning set for the space of cusp forms Sm(Γ,χ) , where Γ is a discrete subgroup of finite covolume in the metaplectic cover of SL2(ℝ) , χ is a character of Γ of finite order, and m5/2+ℤ≥0 . We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any f Sm(Γ,χ) . Using this last result, we construct a Poincaré series ΔΓ,k,m,ξ,χ Sm(Γ,χ) that corresponds, in the sense of the Riesz representation theorem, to the linear functional f ↦ f(k)(ξ) on Sm(Γ,χ) , where ξℂℑ(z)>0 and kℤ≥0 . Under some additional conditions on Γ and χ , we provide the Fourier expansion of cusp forms ΔΓ,k,m,ξ,χ and their expansion in a series of classical Poincaré series.

2010 Mathematics Subject Classification.   11F12, 11F37

Key words and phrases.   Cusp forms of half-integral weight, Poincaré series, metaplectic cover of SL2(ℝ)

Full text (PDF) (free access)

DOI: 10.3336/gm.53.2.03

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