# Complex multiplication, from Abel to Zagier

## Location

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.