Complex multiplication

Abstract: Previously, we showed that every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic extension. Motivated by this theorem, we begin this talk with the ideal-theoretic formulation of global class field theory for $K$ an arbitrary number field. One can then ask if the story for $K = \mathbb{Q}$ can be replicated in other settings. We will see that this answer is in the affirmative for $K$ a quadratic imaginary field; this is realized beautifully in an analogy between cyclotomic theory and the theory of elliptic curves. Along the way, we will develop some powerful tools that will allow us to traverse between the realms of algebraic geometry and transcendental complex analysis.

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2026 topic: Explicit class field theory.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.