Hilbert modular forms and the Ramanujan conjecture
Identifieur interne : 000C99 ( Istex/Corpus ); précédent : 000C98; suivant : 000D00Hilbert modular forms and the Ramanujan conjecture
Auteurs : Don BlasiusSource :
- Aspects of Mathematics [ 0179-2156 ] ; 2006.
Abstract
Abstract: This paper completes the proof, at all finite places, of the Ramanujan Conjecture for motivic holomorphic Hilbert modular forms which belong to the discrete series at the infinite places. In addition, the Weight-Monodromy Conjecture of Deligne is proven for the Shimura varieties attached to GL(2) and its inner forms, and the conjecture of Langlands, often today called the local-global compatibility, is established at all places for these varieties. This latter conjecture gives, for a finite place v of the field of definition F, an automorphic description of the action of decomposition group Γv of the Galois group Gal $$ (\bar F/F) $$ on the l-adic cohomology of the the variety, at least if l is distinct from the residue characteristic of v. In particular, the Hasse-Weil zeta functions of these varieties are computed at all places.
Url:
DOI: 10.1007/978-3-8348-0352-8_2
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In addition, the Weight-Monodromy Conjecture of Deligne is proven for the Shimura varieties attached to GL(2) and its inner forms, and the conjecture of Langlands, often today called the local-global compatibility, is established at all places for these varieties. This latter conjecture gives, for a finite place v of the field of definition F, an automorphic description of the action of decomposition group Γv of the Galois group Gal $$ (\bar F/F) $$ on the l-adic cohomology of the the variety, at least if l is distinct from the residue characteristic of v. 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Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden</CopyrightHolderName><CopyrightYear>2006</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>12120</SeriesID><BookID>978-3-8348-0352-8</BookID><BookTitle>Noncommutative Geometry and Number Theory</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author><AuthorName DisplayOrder="Western"><GivenName>Don</GivenName><FamilyName>Blasius</FamilyName></AuthorName></Author></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para TextBreak="No">This paper completes the proof, at all finite places, of the Ramanujan Conjecture for motivic holomorphic Hilbert modular forms which belong to the discrete series at the infinite places. In addition, the Weight-Monodromy Conjecture of Deligne is proven for the Shimura varieties attached to <Emphasis Type="Italic">GL</Emphasis>(2) and its inner forms, and the conjecture of Langlands, often today called the <Emphasis Type="Italic">local-global compatibility</Emphasis>, is established at all places for these varieties. This latter conjecture gives, for a finite place <Emphasis Type="Italic">v</Emphasis> of the field of definition <Emphasis Type="Italic">F</Emphasis>, an automorphic description of the action of decomposition group Γ<Subscript>v</Subscript> of the Galois group <Emphasis Type="Italic">Gal</Emphasis><InlineEquation ID="IEq1"><InlineMediaObject><ImageObject Color="BlackWhite" FileRef="978-3-8348-0352-8_2_Chapter_TeX2GIFIEq1.gif" Format="GIF" Rendition="HTML" Type="Linedraw"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$$
(\bar F/F)
$$</EquationSource></InlineEquation> on the <Emphasis Type="Italic">l</Emphasis>-adic cohomology of the the variety, at least if <Emphasis Type="Italic">l</Emphasis> is distinct from the residue characteristic of <Emphasis Type="Italic">v</Emphasis>. In particular, the Hasse-Weil zeta functions of these varieties are computed at all places.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Hilbert modular forms and the Ramanujan conjecture</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Hilbert modular forms and the Ramanujan conjecture</title></titleInfo><name type="personal"><namePart type="given">Don</namePart><namePart type="family">Blasius</namePart><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Vieweg</publisher><place><placeTerm type="text">Wiesbaden</placeTerm></place><dateIssued encoding="w3cdtf">2006</dateIssued><dateIssued encoding="w3cdtf">2006</dateIssued><copyrightDate encoding="w3cdtf">2006</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: This paper completes the proof, at all finite places, of the Ramanujan Conjecture for motivic holomorphic Hilbert modular forms which belong to the discrete series at the infinite places. 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