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In his book, "Introduction to the arithmetic theory ..." (Ch 7, section 8) Shimura constructs the Zeta function of an abelian variety with CM and expresses it as a product of L-functions.

Since I found his proof of the main theorem of complex multiplication for elliptic curves very messy, (along with other proofs - and it's not clear that this one can be made simpler) I'd like to ask:

Are there other sources where the zeta function of an abelian variety is constructed (elliptic curve case is in Lang's Elliptic functions), or would you suggest learning this from Shimura?

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There's a beautiful treatment in Section 7 of the paper of Serre-Tate (Annals 1968) assuming only the Shimura-Taniyama formula (ibid. p512). A proof of the Shimura-Taniyama formula can be found Tate's Bourbaki talk 352 using p-divisible groups, or in Milne's CM notes 8.1. – anon Mar 1 '13 at 2:21
Brian Conrad's notes on complex multiplication give an algebraic proof of the main theorem of complex multiplication, including the Taniyama-Shimura formula and the application to L-functions of CM abelian varieties. They will be published as appendix A of the book 'CM Liftings' with Chai and Oort. There's a preprint available on his website – Zavosh Mar 2 '13 at 7:50

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