Complex multiplication, from Abel to Zagier
According to Hilbert, the theory of complex multiplication is the most beautiful part not only of mathematics but also of all science. It refers to a lattice in C (or an elliptic curve) which admits endomorphisms by a ring larger than Z. We begin with Kronecker's ‘Jugendtraum’ (dream of youth)—the use of complex multiplication to solve Hilbert's 12th problem. This leads us into fascinating work by Gross and Zagier on the j-invariants of elliptic curves with complex multiplication. We conclude with some recent work on the modular curve ‘at infinite level’, which is a perfectoid space, and the unexpected role that complex multiplication plays in its geometry.