complex multiplication

2012-12-10T19:50:39Z

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?

Answer by anon for complex multiplication

2012-12-10T21:00:12Z

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.)

complex multiplication - MathOverflow

complex multiplication

Answer by anon for complex multiplication