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?

containsan imaginary quadratic field (even infinitely many). – user29720 Dec 11 '12 at 6:43