The Arithmetic of Fundamental Groups: PIA 2010 PDF
Preview The Arithmetic of Fundamental Groups: PIA 2010
Contributions in Mathematical and Computational Sciences Volume 2 • Editors HansGeorg Bock WilliJäger OtmarVenjakob Forothertitlespublishedinthisseries,goto http://www.springer.com/series/8861 • Jakob Stix Editor The Arithmetic of Fundamental Groups PIA 2010 123 Editor JakobStix MATCH-MathematicsCenterHeidelberg DepartmentofMathematics HeidelbergUniversity ImNeuenheimerFeld288 69120Heidelberg Germany [email protected] ISBN978-3-642-23904-5 e-ISBN978-3-642-23905-2 DOI10.1007/978-3-642-23905-2 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2011943310 MathematicsSubjectClassification(2010):14H30,14F32,14F35,14F30,11G55,14G30,14L15 (cid:2)c Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) toLucieEllaRoseStix • Preface to the Series Contributions toMathematical and Computational Sciences Mathematicaltheoriesandmethodsandeffectivecomputationalalgorithmsarecru- cialincopingwiththechallengesarisinginthesciencesandinmanyareasoftheir application. Newconceptsandapproachesarenecessaryinordertoovercomethe complexitybarriersparticularlycreatedbynonlinearity,high-dimensionality,mul- tiplescalesanduncertainty. Combiningadvancedmathematicalandcomputational methodsand computertechnologyis an essential keyto achievingprogress, often eveninpurelytheoreticalresearch. Thetermmathematicalsciencesreferstomathematicsanditsgenuinesub-fields, aswellastoscientificdisciplinesthatarebasedonmathematicalconceptsandmeth- ods, including sub-fields of the natural and life sciences, the engineering and so- cial sciences and recently also of the humanities. It is a major aim of this series to integratethe differentsub-fieldswithin mathematicsand the computationalsci- ences,andtobuildbridgestoallacademicdisciplines,toindustryandotherfields ofsociety,wheremathematicalandcomputationalmethodsarenecessarytoolsfor progress. Fundamentalandapplication-orientedresearchwillbecoveredinproper balance. The series will further offer contributions on areas at the frontier of research, providingboth detailed informationon topical research, as well as surveys of the state-of-the-artinamannernotusuallypossibleinstandardjournalpublications.Its volumes are intended to cover themes involving more than just a single “spectral line”oftherichspectrumofmathematicalandcomputationalresearch. TheMathematicsCenterHeidelberg(MATCH)andtheInterdisciplinaryCenter forScientificComputing(IWR)withitsHeidelbergGraduateSchoolofMathemat- icalandComputationalMethodsfortheSciences(HGS)areinchargeofproviding andpreparingthematerialforpublication.Asubstantialpartofthematerialwillbe acquiredinworkshopsandsymposiaorganizedbytheseinstitutionsintopicalareas ofresearch. Theresultingvolumesshouldbemorethanjustproceedingscollecting vii viii PrefacetotheSeries paperssubmittedinadvance.Theexchangeofinformationandthediscussionsdur- ingthemeetingsshouldalsohaveasubstantialinfluenceonthecontributions. This series is a venture posing challenges to all partners involved. A unique styleattractingalargeraudiencebeyondthegroupofexpertsinthesubjectareasof specificvolumeswillhavetobedeveloped. SpringerVerlagdeservesourspecialappreciationforitsmostefficientsupportin structuringandinitiatingthisseries. HeidelbergUniversity, HansGeorgBock Germany WilliJäger OtmarVenjakob Preface Duringthemorethan100yearsofitsexistence,thenotionofthefundamentalgroup hasundergoneaconsiderableevolution.ItstartedbyHenriPoincaréwhentopology asasubjectwasstillinitsinfancy. Thefundamentalgroupinthissetupmeasures thecomplexityofapointedtopologicalspacebymeansofanalgebraicinvariant,a discrete group, composedof deformationclasses of based closed loopswithin the space. In this way, for example, the monodromyof a holomorphicfunction on a Riemannsurfacecouldbecapturedinasystematicway. It was throughthe workof AlexanderGrothendieckthat, raising into the focus the roleplayedby thefundamentalgroupingoverningcoveringspaces, so spaces overthegivenspace,aunificationofthetopologicalfundamentalgroupwithGalois theory of algebra and arithmetic could be achieved. In some sense the roles have been reversed in this discrete Tannakian approach of abstract Galois categories: first,wedescribeasuitableclassofobjectsthatcapturesmonodromy,andthen,by abstractpropertiesofthisclassaloneandmoreoveruniquelydeterminedbyit, we find a pro-finite group that describes this category completely as the category of discreteobjectscontinuouslyacteduponbythatgroup. Butthedifferentincarnationsofafundamentalgroupdonotstophere.Thecon- ceptofdescribingafundamentalgroupthroughitscategoryofobjectsuponwhich thegroupnaturallyactsfindsitspro-algebraicrealisationinthetheoryofTannakian categoriesthat,whenappliedtovectorbundleswithflatconnections,ortosmooth (cid:2)-adicétalesheaves,ortoiso-crystalsor...,givesrisetothecorrespondingfunda- mentalgroup,eachwithinitsnaturalcategoryasahabitat. In more recent years, the influence of the fundamental group on the geometry of Kähler manifolds or algebraic varieties has become apparent. Moreover, the program of anabelian geometry as initiated by Alexander Grothendieck realised somespectacularachievementsthroughtheworkoftheJapaneseschoolofHiroaki Nakamura, AkioTamagawaandShinichiMochizukiculminatinginthe proofthat hyperboliccurvesoverp-adicfieldsaredeterminedbytheouterGaloisactionofthe absoluteGaloisgroupofthebasefieldontheétalefundamentalgroupofthecurve. Anaturalnexttargetforpiecesofarithmeticcapturedbythefundamentalgroup arerationalpoints,thegenuineobjectofstudyofDiophantinegeometry.Herethere ix
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