Is there any introduction to abelian varieties of CM type?any reference?Like how to construct a abelian varieties given a CM field E?What is the properites of the Mumford Tate group of the abelian varieties of CM type?
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$\begingroup$ See math.stanford.edu/~conrad/vigregroup/vigre04.html $\endgroup$– Sam LichtensteinCommented Apr 28, 2010 at 15:03
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$\begingroup$ Can you tell me the name of Katz' paper $\endgroup$– TOMCommented Apr 28, 2010 at 15:13
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$\begingroup$ @TOM: I can't rmember. Is it the one about p-adic L-functions for CM fields? $\endgroup$– Kevin BuzzardCommented Apr 28, 2010 at 15:17
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IIRC I learnt a lot from Katz' papers from the 1970s. Of course the basic construction is the same as the elliptic curve case: you take C^g, quotient out by the lattice coming from E via its g embeddings into C, and then you have to prove that the quotient is an abelian variety, which involves writing down a non-degenerate Riemann form. This isn't hard, but I think I first saw it in one of Katz' papers. Oh---I should say that before I read Katz I read the section on abelian varieties over C in Cornell-Silverman (although there will be other references for this stuff).
answered Apr 28, 2010 at 15:05
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1$\begingroup$ Kevin, of course E really has 2g embeddings into C, so when you speak of the g embeddings maybe should bring out the choice of a "CM type". $\endgroup$– BCnrdCommented Apr 28, 2010 at 15:10
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$\begingroup$ +1 BCnrd; yes I implicitly chose a type didn't I! $\endgroup$ Commented Apr 28, 2010 at 15:15