Zeta dimension formula for Picard modular cusp forms of neat natural congruence subgroups
type: Item
instance of: scholarly article
title: Zeta dimension formula for Picard modular cusp forms of neat natural congruence subgroups
zbMATH Open document ID: 0945.11011
DOI: 10.1007/BF02942561
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2000-03-19T00:00:00Z
review text: Let \(K\) be an imaginary quadratic field, and let \(\mathfrak O_K\) be its ring of integers. If \(\Gamma_K = \mathbb U ((2,1), \mathfrak O_K)\) is the full Picard modular group of \(K\), then for each \(m \geq 3\) the natural congruence subgroup \(\Gamma_K (m)\) acts freely on the two-dimensional complex unit ball \(\mathbb B = \{ (z_1, z_2) \in \mathbb C^2 \mid |z_1|^2 + |z_2|^2 < 1 \}\). In this paper, the author derives an explicit polynomial formula for the dimensions of the spaces of cusp forms of weight \(n \geq 2\) for such a congruence subgroup \(\Gamma_K (m)\). The coefficients of these polynomials in the natural variables \(m\) and \(n\) are expressed by some Bernoulli numbers of the Dirichlet characters of \(K\) and by values of Euler factors of zeta functions. The proof is based on detailed knowledge about classification of Picard modular surfaces and combines algebraic geometric methods with number theoretic ones.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1416658
zbMATH Keywords: Picard modular groups
zbMATH Keywords: Picard modular surfaces
zbMATH Keywords: modular forms
zbMATH Keywords: zeta functions
author: Rolf-Peter Holzapfel
MaRDI profile type: Publication
cites work: Compactification of arithmetic quotients of bounded symmetric domains
cites work: Q3788124
cites work: Q3720687
cites work: Q3861597
cites work: Q3846750
cites work: The Parabolic Contribution to the Number of Linearly Independent Automorphic Forms on a Certain Bounded Domain
cites work: A Class of Minimal Surfaces in the Unknown Region of Surface Geography
cites work: Q3481844
cites work: Q2785518
cites work: Q3939864
cites work: Hermitian Forms Over Local Fields
cites work: Q5094823
cites work: Q5565587
cites work: Tamagawa numbers
cites work: Q4111656
cites work: Pathologies III
cites work: Hirzebruch's proportionality theorem in the non-compact case
cites work: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux
cites work: THE PROBLEM OF STRONG APPROXIMATION AND THE KNESER-TITS CONJECTURE FOR ALGEBRAIC GROUPS
cites work: Q5631207
cites work: Q5677789
cites work: Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen
full work available at URL: https://doi.org/10.1007/bf02942561
OpenAlex ID: W1969037455
description: scientific article; zbMATH DE number 1416658