Rational self-maps of K3 surfaces -- Xi Chen, November 5, 2010

It is known that some special K3 surfaces carry nontrivial rational self-maps. Such maps have been used in various ways to study arithmetical problems on K3 surfaces. On the other hand, it was conjectured that a general K3 surface does not carry such maps. A proof of this conjecture connects this to the fact that a general elliptic curve does not carry complex multiplication. I'll discuss the background of this problem and sketch its proof.