{"product_id":"abelian-varieties-classic-text-on-advanced-algebraic-geometry","title":"Abelian Varieties - Classic Text on Advanced Algebraic Geometry","description":"\u003cp\u003eIn this review of Abelian Varieties the bottom line is clear: this is a specialist mathematical monograph best suited to graduate students and researchers who need a focused, classical treatment of topics not fully developed elsewhere. Lang's book has been out of print but remains valuable because it covers concrete constructions and reciprocity results that complement later scheme theoretic works; for readers wanting an accessible account of the Picard and Albanese constructions and explicit applications to Kummer theory, this edition delivers precisely that.\u003c\/p\u003e\u003ch2\u003eKey Features\u003c\/h2\u003e\u003cul\u003e\n\u003cli\u003e\n\u003cstrong\u003eHistorical perspective:\u003c\/strong\u003e Lang situates the subject in the developments prior to scheme theory, which helps readers understand motivations behind later formalism.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003ePicard variety construction:\u003c\/strong\u003e The book gives a hands-on construction of the Picard variety useful for those studying line bundles and moduli questions.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eAlbanese variety treatment:\u003c\/strong\u003e The explicit treatment of the Albanese variety clarifies duality and mapping properties often used in research.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eReciprocity and Kummer applications:\u003c\/strong\u003e The reciprocity law for correspondences and its application to Kummer theory supply concrete tools for arithmetic and geometric investigations.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChow theory for traces and images:\u003c\/strong\u003e Coverage of the K\/k-trace and image provides results relevant to practitioners working over nonclosed fields.\u003c\/li\u003e\n\u003c\/ul\u003e\u003ch2\u003eWho It's For\u003c\/h2\u003e\u003cp\u003eAbelian Varieties is ideal for graduate students, number theorists, and algebraic geometers who need detailed constructions and formulae not emphasized in purely scheme theoretic texts. It is especially helpful for researchers who want to see explicit arguments about Picard and Albanese varieties and the use of correspondences in Kummer theory.\u003c\/p\u003e\u003cp\u003eReaders seeking a modern, scheme-first textbook may prefer Mumford or later expositions for a different perspective; Lang's book is complementary rather than a replacement and assumes some comfort with classical algebraic geometry language.\u003c\/p\u003e\u003ch2\u003ePros \u0026amp; Cons\u003c\/h2\u003e\u003cp\u003e\u003cstrong\u003ePros\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eConcise, focused coverage of topics like the Picard and Albanese constructions that many newer texts omit.\u003c\/li\u003e\n\u003cli\u003eContains concrete reciprocity results and applications to Kummer theory valuable for arithmetic geometry.\u003c\/li\u003e\n\u003cli\u003eUseful historical and technical perspective that makes the transition to scheme theoretic treatments clearer.\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003e\u003cstrong\u003eCons\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\u003cli\u003eThe exposition predates modern scheme language, so readers seeking a fully scheme theoretic approach will need to consult supplementary sources.\u003c\/li\u003e\u003c\/ul\u003e\u003ch2\u003eSpecifications\u003c\/h2\u003e\u003ctable\u003e\n\u003ctr\u003e\n\u003ctd\u003eTitle\u003c\/td\u003e\n\u003ctd\u003eAbelian Varieties\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eAuthor\u003c\/td\u003e\n\u003ctd\u003eS. Lang\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eAvailability\u003c\/td\u003e\n\u003ctd\u003eReprinted by Springer-Verlag after being out of print\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eTopics covered\u003c\/td\u003e\n\u003ctd\u003ePicard variety, Albanese variety, reciprocity for correspondences, Kummer theory, K\/k-trace\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003ePerspective\u003c\/td\u003e\n\u003ctd\u003eClassical algebraic geometry with explicit constructions\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eIntended audience\u003c\/td\u003e\n\u003ctd\u003eGraduate students and researchers in algebraic geometry and number theory\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003c\/table\u003e\u003ch2\u003eOur Verdict\u003c\/h2\u003e\u003cp\u003eAbelian Varieties is a worthwhile purchase for those who need the specific constructions and reciprocity results Lang presents; it complements scheme theoretic texts and remains good value for researchers seeking explicit arguments about Picard and Albanese varieties and Kummer applications.\u003c\/p\u003e\u003ch2\u003eFrequently Asked Questions\u003c\/h2\u003e\u003cp\u003e\u003cstrong\u003eIs this book modern scheme theoretic treatment?\u003c\/strong\u003e\u003cbr\u003eAnswer. No, it predates the scheme focus; it provides classical constructions that complement later scheme theoretic works.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eWho benefits most from reading this book?\u003c\/strong\u003e\u003cbr\u003eAnswer. Graduate students and researchers in algebraic geometry and number theory who want explicit constructions and reciprocity results.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eDoes the book cover Kummer theory?\u003c\/strong\u003e\u003cbr\u003eAnswer. Yes, it includes applications of the reciprocity law for correspondences to Kummer theory.\u003c\/p\u003e","brand":"S. 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