Eisenstein series on Shimura varieties
type: Item
instance of: scholarly article
title: Eisenstein series on Shimura varieties
zbMATH Open document ID: 0589.10030
DOI: 10.2307/2006963
author: Michael Harris
published in: Annals of Mathematics. Second Series
publication date: 1984-01-01T00:00:00Z
review text: In a previous paper [Invent. Math. 63, 305--310 (1981; Zbl 0452.10031)] the author proved rationality properties of the Fourier coefficients of holomorphic Eisenstein series attached to cusp forms on boundary components of Siegel's upper half plane of degree \(n\). The proof -- which did not use any explicit knowledge of the Fourier coefficients -- was based on the fact that these Eisenstein series (at least in the range of absolute convergence) are completely characterized by their Hecke eigenvalues. The purpose of the paper under review is to generalize that important result. The author starts from a bounded symmetric domain \(X\) and an arithmetic subgroup \(\Gamma\) of the group of holomorphic automorphisms of \(X\); to a cusp form on a boundary component of \(X\) one can associate a holomorphic Eisenstein series \(E(f)\) on \(X\). Suppose now that on \(X\) one has a theory of automorphic forms rational over \(K\) \((K\) a number field). Then the author proves that -- under some additional conditions -- the following principle holds: ''If \(f\) is rational over \(K\) and \(E(f)\) is in the range of absolute convergence then \(E(f)\) is rational over \(K\)''. The main idea of proof is the same as in the previous paper. However two new difficulties arise in this general context: The notion of rationality of automorphic forms on \(X\) has to be treated carefully. Here the author uses \textit{G. Shimura}'s theory of canonical models [Ann. Math. (2) 91, 144--222 (1970; Zbl 0237.14009); and ibid. 92, 528--549 (1970; Zbl 0237.14010)] as interpreted by \textit{P. Deligne} [Sémin. Bourbaki 1970/71, Exp. No. 389, Lect. Notes Math. 244, 123--165 (1971; Zbl 0225.14007)]. The second new point is that instead of elementary estimates of Hecke eigenvalues (which were used in the previous paper) the author uses \textit{I. G. Macdonald}'s work [''Spherical functions on a group of \(p\)-adic type'', Ramanujan Inst. 2, 79 p. (1971; Zbl 0302.43018)].
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 3946232
zbMATH Keywords: rationality properties
zbMATH Keywords: bounded symmetric domain
zbMATH Keywords: holomorphic Eisenstein series
zbMATH Keywords: rationality of automorphic forms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2006963
OpenAlex ID: W2317701080
description: scientific article; zbMATH DE number 3946232