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

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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.