摘要:WereportrecentprogressonajointprojectwithYangCao.Ifanalgebraicvarietyoveranumberfieldverifiesstrongapproximationoffafinitesetofplaces,ithasbeenfirstconjecturedbyWittenbergthatthispropertyismaintainedundertheremovalofanysubvarietyofcodimensiontwo.Ifthisisthecase,thenwesaythatthevarietysatisfiesarithmeticpurity.Acloselyrelatedquestionisthedensityofintegralpointswhosemultivariablepolynomialvalueshavenocommongcd\'s.Weconfirmthearithmeticpurityforsemi-simplesimplyconnectedk-simpleisotropiclinearalgebraicgroups,andforaffinequadratichypersurfaces,usingdifferentmethods.Theyshowhowthefibrationmethodforrationalpointsandvarioussievemethods(e.g.affinealmostprimelinearsieve,Ekedahl’sgeometricsieve,Iwaniec’shalf-dimensionalsieve)matchtogether.

摘要：一族代数方程定义了一个代数簇，它们的解称为有理点。丢番图几何所研究的是如何利用代数簇的几何性质来刻画有理点的密度，其起源可追溯到古巴比伦时期。菲尔兹奖得主Faltings称此领域目前的猜想多于定理。对于反典范丛丰沛的代数簇，人们猜想有理点无穷多并且均匀分布。Manin及其合作者Batyrev和Tschinkel首先提出了关于反典范丛高度下有理点增长率的渐近公式的猜测。公式鲜明之处在于，多项式项指数为1，log项蕴含着代数簇Picard群的秩，以及前导常数与有理点在adelic空间的弱逼进密切关联。本综述报告将试图介绍近年来解析数论中各类求均阶的技巧如何被成功运用于Manin猜想，特别是对于Chatelet曲面的证明。