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For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. (This is the definition I saw.)

Now assume $A$ is simple. From sec. 1 in Ch. 1 in Lang's book "Complex Multiplication", if $A$ has complex multiplication, then $F = \mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. (If I understand Thm. 3.1, Lemma 3.2 and Thm. 3.3 correctly.)

But I also saw a paragraph on another book:

http://postimage.org/image/tuoj3u709/

I don't understand this example well. Does it has an imaginary quadratic field? Or these above theorems in Lang's book require the abelian variety being over a characteristic 0 field?

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Most people, include as a definition of complex multiplication that your abelian variety is defined over a field of characteristic zero. –  stankewicz Dec 10 '12 at 20:26
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@stankewicz: Tate proved that abelian varieties over finite fields always have complex multiplication over the ground field (in the sense of the endomorphism algebra over the ground field containing a CM algebra of rank twice the dimension, where "CM algebra" means "product of CM fields"). @unknown: That link is wrong: the endomorphism algebra of a ss elliptic curve over an alg. closed field is of rank at most 4, so it cannot be a quaternion algebra over a quadratic field. It is just a quaternion algebra over $\mathbf{Q}$ and contains an imaginary quadratic field (even infinitely many). –  user29720 Dec 11 '12 at 6:43
    
@kreck, I felt strange that it is written that "...be a quaternion algebra over a quadratic field...". It is good to know that it contains an imaginary quadratic field and the theorem Tate proved. Could you give the reference for it. Thank you very much. –  user565739 Dec 11 '12 at 6:53
    
@unknown: reference for which? I mentioned a couple of things. –  user29720 Dec 11 '12 at 7:28

1 Answer 1

There are a number of definitions of complex multiplication in the literature.

(a) Shimura says that an abelian variety of dimension g has complex multiplication if its endomorphism algebra $End(A)\otimes Q$ contains a field of degree 2g.

(b) Deligne et al. say an abelian variety has complex multiplication if it is a product of abelian varieties with complex multiplication in the sense of Shimura (equivalently, but better, if its Mumford-Tate group is a torus).

(c) Classical algebraic geometers say an abelian variety has complex multiplication if it is acted on by an order in a CM field.

With definition (a), the field is automatically a CM field in characteristic zero, but not otherwise. (A CM field is a quadratic totally imaginary extension of a totally real field.)

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Thanks. It seems that those theorems I listed above require characteristic zero. –  user565739 Dec 10 '12 at 21:14

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