Higher regulators, Hilbert modular surfaces, and special values of \(L\)-functions
type: Item
instance of: scholarly article
title: Higher regulators, Hilbert modular surfaces, and special values of \(L\)-functions
zbMATH Open document ID: 0962.11024
DOI: 10.1215/S0012-7094-98-09202-X
author: Guido Kings
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The paper consists of an introduction to set the stage and to announce the main result (and some of its shortcomings), six sections and references. The first section gives a brief overview of the construction of abelian schemes over Hilbert modular surfaces. In the second section the main object of study, the fundamental relative motive \(\mathcal V^{p,q}\), is introduced. Its various realizations are discussed. In particular, there is a comparison result between the Betti realization and intersection cohomology. The third section deals with the Betti and de Rham realizations. The fourth section gives an overview of \(L\)-functions. The results of Brylinski-Labesse and Chai are discussed. The exposition is quite detailed and technical. In the fifth section the main result (cf. \textit{infra}) is presented. The proof of this result takes quite some place. As usual an important role is played by Beilinson's Eisenstein map. This is explained in the sixth section. Let \(F/{\mathbb Q}\) be a real quadratic field with ring of integers \(\mathcal O_F\). Let \(G/{\mathbb Q} \hookrightarrow\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\) such that the maps \(G\rightarrow{\mathbb G}_m\), \({\mathbb G}_m\hookrightarrow\text{Res}_ {F/{\mathbb Q}}{\mathbb G}_{m,F}\) and \(\text{det}:\text{Res}_{F/{\mathbb Q}}\text{GL}_{2,F}\rightarrow \text{Res}_{F/{\mathbb Q}}{\mathbb G}_{m,F}\) lead to a Cartesian square. The group \(G\) gives rise to a Shimura variety \(S({\mathbb C})\) with a \(G({\mathbb A}_f)\) action. Let \(V^{p,q}=\text{Sym}^pV_2^{\vee}\otimes\text{Sym}^qV _2^{\vee}\), where \(V_2^{\vee}\) is the dual of the standard representation of \(\text{GL}_2\). It is known that intersection cohomology with respect to the Baily-Borel compactification decomposes into \(G({\mathbb A}_f)\)-isotypic components \(H^2_{\text{ét}}(\pi_f)\) which are \(\text{Gal} (\overline{\mathbb Q}/{\mathbb Q})\)-modules. An important result due to J. L.~Brylinski and J.-P.~Labesse says that, for almost all …
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 19F27
zbMATH DE Number: 1425091
zbMATH Keywords: higher regulators
zbMATH Keywords: Hilbert modular surface
zbMATH Keywords: motivic cohomology
zbMATH Keywords: absolute Hodge cohomology
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09202-x
OpenAlex ID: W2027883985