The Weil Proof and the Geometry of the Adelès Class Space
Identifieur interne : 002793 ( Istex/Corpus ); précédent : 002792; suivant : 002794The Weil Proof and the Geometry of the Adelès Class Space
Auteurs : Alain Connes ; Caterina Consani ; Matilde MarcolliSource :
- Progress in Mathematics ; 2009.
Abstract
Summary: This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclusion of the idèles class group of a global field in the noncommutative adèles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.
Url:
DOI: 10.1007/978-0-8176-4745-2_8
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We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclusion of the idèles class group of a global field in the noncommutative adèles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.</p></abstract><textClass><keywords scheme="Book-Subject-Collection"><list><label>SUCO11649</label><item><term>Mathematics and Statistics</term></item></list></keywords></textClass><textClass><keywords scheme="Book-Subject-Group"><list><label>SCM</label><label>SCM11000</label><label>SCM25001</label><label>SCM21006</label><label>SCP19013</label><label>SCM11019</label><item><term>Mathematics</term></item><item><term>Algebra</term></item><item><term>Number Theory</term></item><item><term>Geometry</term></item><item><term>Mathematical Methods in Physics</term></item><item><term>Algebraic Geometry</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2010-11-21">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-11-28">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/C27A84DD768D360668F338A0E2E29C1867B33BC6/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus springer-ebooks not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Birkhäuser Boston</PublisherName><PublisherLocation>Boston</PublisherLocation></PublisherInfo><Series><SeriesInfo SeriesType="Series" TocLevels="0"><SeriesID>4848</SeriesID><SeriesTitle Language="En">Progress in Mathematics</SeriesTitle><SeriesAbbreviatedTitle>Progress in Mathematics
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Sciences</OrgDivision><OrgName>New York University</OrgName><OrgAddress><Street>Mercer St. 251</Street><City>New York</City><Postcode>10012</Postcode><State>New York</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="AffID2"><OrgDivision>Eberly College of Science, Dept. Mathematics</OrgDivision><OrgName>Pennsylvania State University</OrgName><OrgAddress><City>University Park</City><Postcode>16802</Postcode><State>Pennsylvania</State><Country>USA</Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Chapter ID="b978-0-8176-4745-2_8" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" Language="En" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><ChapterID>8</ChapterID><ChapterDOI>10.1007/978-0-8176-4745-2_8</ChapterDOI><ChapterSequenceNumber>8</ChapterSequenceNumber><ChapterTitle Language="En">The Weil Proof and the Geometry of the Adelès Class Space</ChapterTitle><ChapterFirstPage>339</ChapterFirstPage><ChapterLastPage>405</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer Science+Business Media, LLC</CopyrightHolderName><CopyrightYear>2009</CopyrightYear></ChapterCopyright><ChapterHistory><RegistrationDate><Year>2009</Year><Month>10</Month><Day>8</Day></RegistrationDate><OnlineDate><Year>2010</Year><Month>11</Month><Day>21</Day></OnlineDate></ChapterHistory><ChapterContext><SeriesID>4848</SeriesID><BookID>978-0-8176-4745-2</BookID><BookTitle>Algebra, Arithmetic, and Geometry</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1_8" CorrespondingAffiliationID="Aff1_8"><AuthorName DisplayOrder="Western"><GivenName>Alain</GivenName><FamilyName>Connes</FamilyName></AuthorName><Contact><Email>alain@connes.org</Email></Contact></Author><Author AffiliationIDS="Aff2_8"><AuthorName DisplayOrder="Western"><GivenName>Caterina</GivenName><FamilyName>Consani</FamilyName></AuthorName><Contact><Email>kc@math.jhu.edu</Email></Contact></Author><Author AffiliationIDS="Aff3_8"><AuthorName DisplayOrder="Western"><GivenName>Matilde</GivenName><FamilyName>Marcolli</FamilyName></AuthorName><Contact><Email>matilde@caltech.edu</Email></Contact></Author><Affiliation ID="Aff1_8"><OrgName>Collège de France</OrgName><OrgAddress><Street>3, rue d’Ulm</Street><City>Paris</City><Postcode>F-75005</Postcode><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff2_8"><OrgDivision>Mathematics Department</OrgDivision><OrgName>Johns Hopkins University</OrgName><OrgAddress><City>Baltimore</City><State>MD</State><Postcode>21218</Postcode><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3_8"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>California Institute of Technology 1200 E California Blvd</OrgName><OrgAddress><City>Pasadena</City><State>CA</State><Postcode>91101</Postcode><Country>USA</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1_8" Language="En" OutputMedium="All"><Heading>Summary</Heading><Para TextBreak="No">This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclusion of the idèles class group of a global field in the noncommutative adèles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.</Para></Abstract><KeywordGroup Language="En" OutputMedium="All"><Heading>Key words</Heading><Keyword>noncommutative geometry</Keyword><Keyword>cyclic homology</Keyword><Keyword>thermodynamics</Keyword><Keyword>factors</Keyword><Keyword>Riemann zeta function</Keyword><Keyword>correspondences</Keyword><Keyword>Lefschetz trace formula</Keyword><Keyword>adèles</Keyword><Keyword>idèles</Keyword><Keyword>Frobenius</Keyword></KeywordGroup><ArticleNote Type="Motto"><SimplePara><Emphasis Type="Italic">2000 Mathematics Subject Classifications</Emphasis>: 11M36, 58B34, 11M26, 46L55</SimplePara></ArticleNote></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>The Weil Proof and the Geometry of the Adelès Class Space</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>The Weil Proof and the Geometry of the Adelès Class Space</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">Alain</namePart><namePart type="family">Connes</namePart><affiliation>Collège de France, 3, rue d’Ulm, F-75005, Paris, France</affiliation><affiliation>E-mail: alain@connes.org</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Caterina</namePart><namePart type="family">Consani</namePart><affiliation>Mathematics Department, Johns Hopkins University, 21218, Baltimore, MD, USA</affiliation><affiliation>E-mail: kc@math.jhu.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Matilde</namePart><namePart type="family">Marcolli</namePart><affiliation>Department of Mathematics, California Institute of Technology 1200 E California Blvd, 91101, Pasadena, CA, USA</affiliation><affiliation>E-mail: matilde@caltech.edu</affiliation><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>Birkhäuser Boston</publisher><place><placeTerm type="text">Boston</placeTerm></place><dateIssued encoding="w3cdtf">2010-11-21</dateIssued><copyrightDate encoding="w3cdtf">2009</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Summary: This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclusion of the idèles class group of a global field in the noncommutative adèles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.</abstract><relatedItem type="host"><titleInfo><title>Algebra, Arithmetic, and Geometry</title><subTitle>Volume I: In Honor of Yu. I. Manin</subTitle></titleInfo><name type="personal"><namePart type="given">Yuri</namePart><namePart type="family">Tschinkel</namePart><affiliation>Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, 10012, New York, New York, USA</affiliation><affiliation>E-mail: tschinkel@cims.nyu.edu</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Yuri</namePart><namePart type="family">Zarhin</namePart><affiliation>Eberly College of Science, Dept. Mathematics, Pennsylvania State University, 16802, University Park, Pennsylvania, USA</affiliation><affiliation>E-mail: zarhin@math.psu.edu</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Contributed volume" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2009</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11649">Mathematics and Statistics</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="SCM">Mathematics</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM11000">Algebra</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM25001">Number Theory</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM21006">Geometry</topic><topic authority="SpringerSubjectCodes" authorityURI="SCP19013">Mathematical Methods in Physics</topic><topic authority="SpringerSubjectCodes" authorityURI="SCM11019">Algebraic Geometry</topic></subject><identifier type="DOI">10.1007/978-0-8176-4745-2</identifier><identifier type="ISBN">978-0-8176-4744-5</identifier><identifier type="eISBN">978-0-8176-4745-2</identifier><identifier type="BookTitleID">150868</identifier><identifier type="BookID">978-0-8176-4745-2</identifier><identifier type="BookChapterCount">16</identifier><identifier type="BookVolumeNumber">269</identifier><identifier type="BookSequenceNumber">269</identifier><part><date>2009</date><detail type="volume"><number>269</number><caption>vol.</caption></detail><extent unit="pages"><start>339</start><end>405</end></extent></part><recordInfo><recordOrigin>Springer Science +Business Media, LLC, 2009</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Progress in Mathematics</title></titleInfo><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2009</copyrightDate><issuance>serial</issuance></originInfo><identifier type="SeriesID">4848</identifier><recordInfo><recordOrigin>Springer Science +Business Media, LLC, 2009</recordOrigin></recordInfo></relatedItem><identifier type="istex">C27A84DD768D360668F338A0E2E29C1867B33BC6</identifier><identifier type="ark">ark:/67375/HCB-J0SVB06L-W</identifier><identifier type="DOI">10.1007/978-0-8176-4745-2_8</identifier><identifier type="ChapterID">8</identifier><identifier type="ChapterID">b978-0-8176-4745-2_8</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer Science +Business Media, LLC, 2009</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Springer Science+Business Media, LLC, 2009</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/C27A84DD768D360668F338A0E2E29C1867B33BC6/metadata/json</uri></json:item></metadata><annexes><json:item><extension>xml</extension><original>true</original><mimetype>application/xml</mimetype><uri>https://api.istex.fr/document/C27A84DD768D360668F338A0E2E29C1867B33BC6/annexes/xml</uri></json:item></annexes></istex></record>Pour manipuler ce document sous Unix (Dilib)
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