School of Mathematics and Statistics
Number Theory Seminar 

For a (nonsingular) elliptic curve E we know that End^0(E) = (End(E) tensor Q) is either Q or an imaginary quadratic extension of Q. The aim of this talk is to find out more about the structure of End^0(X) where X is an abelian variety. In the first part I will basically show that End^0(X) is a semisimple finitedimensional Qalgebra. In particular, if X is a simple abelian variety then End^0(X) is a skew field and so its center is a field for which we have only two possibilities: it is either totally real or a CMfield. This is basically the result of section 5 of Shimura's book. At the end I would like to give the complete classification due to Albert, more or less detailed depending on time. 