On the analytic continuation of certain Poincaré series of elliptic modular groups
type: Item
instance of: scholarly article
title: On the analytic continuation of certain Poincaré series of elliptic modular groups
zbMATH Open document ID: 0646.10020
author: Ulrich Christian
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: Es seien \(\psi\) eine Kongruenzgruppe zur elliptischen Modulgruppe; \(N=\left( \begin{matrix} a&b\\ c&d\end{matrix} \right)\in \psi\); weiter \(z=x+iy\) aus der oberen Halbebene; \(N<z>=(az+b)/(cz+d)\); \(N\{z\}=cz+d\); \(d\omega _ z=dxdy/y^2\); \(k,g\in\mathbb{N}\); \(s_1,s_2\in\mathbb{C}\); \(\mathcal F(\psi)\) ein Fundamentalbereich von \(\psi\). Es wird gezeigt, daß die Poincarésche Reihe \[ K(\psi,k,g,w,z,s_1,s_2)=\sum_{N\in \psi}(-\bar w+N<z>)^{-k}(N\{z\})^{- g}| -\bar w+N<z>|^{-s_1}| N\{z\}| ^{-s_2} \] sich als Funktion von \(s_1, s_2\) meromorph auf den \(\mathbb{C}^2\) fortsetzen läßt. Ist \(F(z)\) eine Spitzenform zu \(\psi\) vom Gewicht \(g\), so besteht die Integralgleichung \[ F(z)=2^{g- 2}\frac{i^g}{\pi}(g-1)\int_{\mathcal F(\psi)} K(\psi,g,g,w,z,0,0)v gF(w)\,d\omega_w\quad (g\ge 2), \] \[ F(z)=\frac{i}{4\pi}\int_{\mathcal F(\psi)} \mathrm{Res}_{s=0}K(\psi,1,1,w,z,s,s) vF(w)\,d\omega_w\quad (g=1). \] Dieses wird bewiesen, indem man die Reihe \(K(\psi,k,g,w,z,s_1,s_2)\) mit dem Resolventenkern des Differentialoperators \[ -\Delta_g= -y^2(\partial^2/\partial x^2+\partial^2/\partial y^2)+igy(\partial /\partial x) \] in Beziehung bringt. {Vgl. das Referat der Report-Version im Zbl 0646.10020.}
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 32N10
zbMATH DE Number: 4053726
zbMATH Keywords: elliptic modular group
zbMATH Keywords: Poincaré series
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: automorphic form
zbMATH Keywords: resolvent kernel
zbMATH Keywords: cusp form
zbMATH Keywords: integral equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227922
OpenAlex ID: W2000292660
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DOI: 10.2748/TMJ/1178227922
description: scientific article; zbMATH DE number 4053726