The Weil Proof and the Geometry of the Adelès Class Space
Identifieur interne : 000511 ( Main/Merge ); précédent : 000510; suivant : 000512The Weil Proof and the Geometry of the Adelès Class Space
Auteurs : Alain Connes [France] ; Caterina Consani [États-Unis] ; Matilde Marcolli [États-Unis]Source :
- 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
Links toward previous steps (curation, corpus...)
- to stream Istex, to step Corpus: 002793
- to stream Istex, to step Curation: 002793
- to stream Istex, to step Checkpoint: 000462
Links to Exploration step
ISTEX:C27A84DD768D360668F338A0E2E29C1867B33BC6Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">The Weil Proof and the Geometry of the Adelès Class Space</title>
<author><name sortKey="Connes, Alain" sort="Connes, Alain" uniqKey="Connes A" first="Alain" last="Connes">Alain Connes</name>
</author>
<author><name sortKey="Consani, Caterina" sort="Consani, Caterina" uniqKey="Consani C" first="Caterina" last="Consani">Caterina Consani</name>
</author>
<author><name sortKey="Marcolli, Matilde" sort="Marcolli, Matilde" uniqKey="Marcolli M" first="Matilde" last="Marcolli">Matilde Marcolli</name>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">ISTEX</idno>
<idno type="RBID">ISTEX:C27A84DD768D360668F338A0E2E29C1867B33BC6</idno>
<date when="2009" year="2009">2009</date>
<idno type="doi">10.1007/978-0-8176-4745-2_8</idno>
<idno type="url">https://api.istex.fr/document/C27A84DD768D360668F338A0E2E29C1867B33BC6/fulltext/pdf</idno>
<idno type="wicri:Area/Istex/Corpus">002793</idno>
<idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">002793</idno>
<idno type="wicri:Area/Istex/Curation">002793</idno>
<idno type="wicri:Area/Istex/Checkpoint">000462</idno>
<idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">000462</idno>
<idno type="wicri:Area/Main/Merge">000511</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">The Weil Proof and the Geometry of the Adelès Class Space</title>
<author><name sortKey="Connes, Alain" sort="Connes, Alain" uniqKey="Connes A" first="Alain" last="Connes">Alain Connes</name>
<affiliation wicri:level="3"><country xml:lang="fr">France</country>
<wicri:regionArea>Collège de France, 3, rue d’Ulm, F-75005, Paris</wicri:regionArea>
<placeName><region type="region" nuts="2">Île-de-France</region>
<settlement type="city">Paris</settlement>
</placeName>
</affiliation>
<affiliation></affiliation>
</author>
<author><name sortKey="Consani, Caterina" sort="Consani, Caterina" uniqKey="Consani C" first="Caterina" last="Consani">Caterina Consani</name>
<affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country>
<wicri:regionArea>Mathematics Department, Johns Hopkins University, 21218, Baltimore, MD</wicri:regionArea>
<placeName><region type="state">Maryland</region>
</placeName>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country>
</affiliation>
</author>
<author><name sortKey="Marcolli, Matilde" sort="Marcolli, Matilde" uniqKey="Marcolli M" first="Matilde" last="Marcolli">Matilde Marcolli</name>
<affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country>
<wicri:regionArea>Department of Mathematics, California Institute of Technology 1200 E California Blvd, 91101, Pasadena, CA</wicri:regionArea>
<placeName><region type="state">Californie</region>
</placeName>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country>
</affiliation>
</author>
</analytic>
<monogr></monogr>
<series><title level="s">Progress in Mathematics</title>
<imprint><date>2009</date>
</imprint>
</series>
</biblStruct>
</sourceDesc>
</fileDesc>
<profileDesc><textClass></textClass>
<langUsage><language ident="en">en</language>
</langUsage>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml: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.</div>
</front>
</TEI>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Mathematiques/explor/BourbakiV1/Data/Main/Merge
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000511 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Merge/biblio.hfd -nk 000511 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Mathematiques |area= BourbakiV1 |flux= Main |étape= Merge |type= RBID |clé= ISTEX:C27A84DD768D360668F338A0E2E29C1867B33BC6 |texte= The Weil Proof and the Geometry of the Adelès Class Space }}
![]() | This area was generated with Dilib version V0.6.33. | ![]() |