complex multiplication - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:25:14Z http://mathoverflow.net/feeds/question/116010 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116010/complex-multiplication complex multiplication unknown (google) 2012-12-10T19:50:39Z 2012-12-10T21:00:12Z <p>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.)</p> <p>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.)</p> <p>But I also saw a paragraph on another book:</p> <p><a href="http://postimage.org/image/tuoj3u709/" rel="nofollow">http://postimage.org/image/tuoj3u709/</a></p> <p>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?</p> http://mathoverflow.net/questions/116010/complex-multiplication/116018#116018 Answer by anon for complex multiplication anon 2012-12-10T21:00:12Z 2012-12-10T21:00:12Z <p>There are a number of definitions of complex multiplication in the literature.</p> <p>(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.</p> <p>(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).</p> <p>(c) Classical algebraic geometers say an abelian variety has complex multiplication if it is acted on by an order in a CM field.</p> <p>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.)</p>