Transcription of Complex Multiplication - James Milne
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Complex Multiplication - James Milne
Complex Multiplication Milne April 7, 2006. These are preliminary notes1 for a modern account of the theory of Complex multipli- cation. A shortened (minimal) version will be included in my book on Shimura varieties , and a complete longer version may one day be published separately. April 7, 2006. First version posted on the web; 113 pages. Please send comments and corrections to me at Available at c 2006. Milne . Copyright 1 This should be taken seriously: there are omissions, repetitions, clumsy statements and proofs, and incon- sistencies in notation. Contents Contents 3. I Analytic theory 9. 1 CM-algebras and CM-types .. 9. Review of semisimple algebras and their modules 9; CM-algebras 11; CM-types. 12; The reflex field of a CM-pair 14; The reflex norm. 15; Classification of the primitive CM-pairs 19; Positive involutions and CM-algebras 21. 2 Complex abelian varieties .. 22. Complex tori 22; The cohomology of Complex tori 24; Hermitian forms and alternating forms 26; Riemann forms 26; Abelian varieties 28.
Preface The theory of complex multiplication is not only the most beautiful part of mathematics but also of all science. D. Hilbert2 Abelian varieties with complex multiplication3 are special in that they have the largest possible endomorphism rings.
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