Motives for Hilbert modular forms
type: Item
instance of: scholarly article
title: Motives for Hilbert modular forms
zbMATH Open document ID: 0829.11028
DOI: 10.1007/BF01232663
author: Don Blasius
author: Jonathan Rogawski
published in: Inventiones Mathematicae
publication date: 1994-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/144142
review text: In this paper the authors attach motives to collections \(\pi_1, \ldots, \pi_n\) of holomorphic Hilbert modular forms over a totally real field \(F\) of discrete series type at infinity. The method they used is based on the construction of certain sub-motives of the cohomology of fiber systems over Picard modular surfaces. Let \(U(3)\) be a unitary group in three variables defined relative to a quadratic extension \(E = KF\) that determines a Picard modular surface, where \(K\) is an imaginary quadratic extension of \(\mathbb{Q}\). Then the sub-motives of interest are attached to endoscopic automorphic forms on \(U(3)\) by passing from \(\pi_1, \ldots, \pi_n\) to Picard modular surfaces through a base change to \(GL(2)_E\), a twisting and a descent to \(U(2)_{E/F}\), an extension to \(U(2) \times U(1)\), an endoscopic transfer to quasi-split \(U(3)\), and finally a transfer to an inner form of \(U(3)\). Producing motives themselves involves the construction of algebraic correspondences, and this is achieved by verifying the Hodge conjecture for the generic fibers of the fiber systems of abelian varieties carried by the Picard modular surfaces. As an application the authors obtain compatible systems of \(\ell\)-adic representations of \(\text{Gal} (\overline F/F)\) of degree \(2^n\) whose \(L\)-function coincides with the tensor product \(L\)-functions \(L(s, \pi_1 \otimes \cdot \otimes \pi_n)\) at almost all places.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 549642
zbMATH Keywords: Shimura varieties
zbMATH Keywords: \(\ell\)-adic representations
zbMATH Keywords: motives
zbMATH Keywords: Hilbert modular forms
zbMATH Keywords: Picard modular surfaces
zbMATH Keywords: endoscopic automorphic forms
zbMATH Keywords: Hodge conjecture
MaRDI profile type: Publication
Wikidata QID: Q101493158
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description: scientific article; zbMATH DE number 549642