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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 the first chapter these above theorems in Lang's book require the abelian variety being over a characteristic 0 field?
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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 the first chapter in Lang's book require the abelian variety being over a characteristic 0 field(looks like not this case)?
show/hide this revision's text 2 added 27 characters in body

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 the first chapter in Lang's book require the abelian variety being over a characteristic 0 field (looks like not this case)?
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