Construction of Poincaré-type Series by Generating Kernels
Association for Women in Mathematics Series, Springer International Publishing Ag, ss.233-274, 2021
- Yayın Türü: Kitapta Bölüm / Araştırma Kitabı
- Basım Tarihi: 2021
- Doi Numarası: 10.1007/978-3-030-77700-5_8
- Yayınevi: Springer International Publishing Ag
- Sayfa Sayıları: ss.233-274
- Anahtar Kelimeler: Automorphic kernel, Multiplier system, Real weight Laplacian, Sup-norm bound, Wave distribution
- Boğaziçi Üniversitesi Adresli: Evet
Özet
Let Γ ⊂ PSL2(ℝ) be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let Γ ˜ be its cover in SL2(ℝ). Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight k∈ ℝ under the action of Γ ˜. The space of such functions admits the action of the hyperbolic Laplacian Δk of weight k. Following an approach of by Jorgenson et al. (Adv Math 288:887–921, 2016) (where k = 0), we use the spectral expansion associated to Δk to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincaré-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of Δk.