ThisbookgrewoutofnotesfromseveralcoursesthatthefirstauthorhastaughtoverthepastnineyearsattheCaliforniaInstituteofTechnology,andearlierattheJohnsHopkinsUniversity,CornellUniversity,theUniversityofChicago,andtheUniversityofCrete.Ourgeneralaimistoprovideamodernapproachtonumbertheorythroughablendingofcomplementaryalgebraicandanalyticperspectives,emphasizingharmonicanalysisontopologicalgroups.OurmoreparticulargoalistocoverJohnTate'svisionarythesis,givingvirtuallyallofthenecessaryanalyticdetailsandtopologicalpreliminaries——technicalprereq-uisitesthatareoftenforeigntothetypical,morealgebraicallyinclinednumbertheorist.MostoftheexistingtreatmentsofTate'sthesis,includingTate'sown,rangefromtersetocryptic;ourintentistobemoreleisurely,morecomprehen-sive,andmorecomprehensible.Tothisendwehaveassembledmaterialthathasadmittedlybeentreatedelsewhere,butnotinasinglevolumewithsomuchdetailandnotwithourparticularfocus.