This can be a reprinting of the revised moment version (1974) of David Mumford's vintage 1970 booklet. It provides a scientific account of the fundamental effects approximately abelian forms. It comprises expositions of analytic tools appropriate over the floor box of complicated numbers, in addition to of scheme-theoretic tools used to accommodate inseparable isogenies whilst the floor box has optimistic attribute. A self-contained evidence of the life of the twin abelian sort is given. The constitution of the hoop of endomorphisms of an abelian type is mentioned. those are appendices on Tate's theorem on endomorphisms of abelian kinds over finite fields (by C. P. Ramanujam) and at the Mordell - Weil theorem (by Yuri Manin).David Mumford used to be provided the 2007 AMS Steele Prize for Mathematical Exposition. in keeping with the quotation: '"Abelian Varieties"...remains the definitive account of the subject...the classical thought is superbly intertwined with the fashionable conception, in a fashion which sharply illuminates both...[It] will stay for the foreseeable destiny a vintage to which the reader returns over and over'.

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