Near holomorphy of some automorphic forms at critical points
type: Item
instance of: scholarly article
title: Near holomorphy of some automorphic forms at critical points
zbMATH Open document ID: 0708.11028
DOI: 10.1007/BF01233431
author: Antonia Wilson Bluher
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143838
review text: In this long and subtle article some results of \textit{G. Shimura} [Math. Ann. 278, 1-28 (1987; Zbl 0636.10023)] on near holomorphy of a distinguished automorphic form f on \({\mathcal Z}\times {\mathbb{C}}\) are extended. Here \({\mathcal Z}\) is a (bounded) symmetric domain arising from symmetric bilinear forms S over totally real number fields F. The form f in question always has an integral representation involving a Hilbert modular form of a common subfield E of the F's, an Eisenstein series and a certain theta series associated to the S's which allows to get a meromorphic continuation to the whole plane and guarantees real analyticity in \({\mathcal Z}\). The following questions are at the core of investigations: (1) finiteness and holomorphy of \(w\mapsto f(w,s_ 0).\) (2) more generally, near holomorphy of \(w\mapsto f(w,s_ 0)\) which amounts to say to be annihilated by a power of a certain non holomorphic differential operator. (3) nearly holomorphy of the residue in case of a simple pole. As to the importance of the questions above ``an affirmative answer is essential to the study of the arithmetic properties of \(f(w,s_ 0)''\). All these questions are tackled by Shimura (loc.cit.) in the case \(s=0\). His method is extended to nonpositive integers in the right half plane \(s\geq -b/2\) for a certain (half) integer b associated to f. Furthermore under additional assumptions on the common subfield E, the involved Hilbert modular form and the theta series better results are obtained by the author for a product of f by a certain L-series.
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 4162992
zbMATH Keywords: near holomorphy
zbMATH Keywords: integral representation
zbMATH Keywords: Hilbert modular form
zbMATH Keywords: Eisenstein series
zbMATH Keywords: theta series
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: real analyticity
MaRDI profile type: Publication
cites work: The special values of the zeta functions associated with cusp forms
cites work: The special values of the zeta functions associated with Hilbert modular forms
cites work: The arithmetic of certain zeta functions and automorphic forms on orthogonal groups
cites work: On certain zeta functions attached to two Hilbert modular forms. I: The case of Hecke characters
cites work: Algebraic Relations Between Critical Values of Zeta Functions and Inner Products
cites work: On differential operators attached to certain representations of classical groups
cites work: On Eisenstein series of half-integral weight
cites work: On the Eisenstein series of Hilbert modular groups
cites work: On a class of nearly holomorphic automorhic forms
cites work: On Hilbert modular forms of half-integral weight
cites work: Nearly holomorphic functions on Hermitian symmetric spaces
cites work: On the critical values of certain Dirichlet series and the periods of automorphic forms
cites work: Q5640160
description: scientific article; zbMATH DE number 4162992