Jared Weinstein : Complex multiplication, from Abel to Zagier

Posted by James Freitag , part of the Departmental Colloquium.

At

Nov. 2, 2018, 3 p.m.

In

636 SEO

Abstract

According to Hilbert, the theory of complex multiplication is not only the most beautiful part of mathematics but also of all science. Complex multiplication refers to a lattice in the complex numbers (or an elliptic curve) which admits endomorphisms by a ring larger than the integers. We will begin with Kronecker's "Jugendtraum" -- the use of complex multiplication to solve Hilbert's twelfth problem. This will lead us into a discussion of some fascinating work by Gross and Zagier on the j-invariants of elliptic curves with complex multiplication. We will 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.