Complex Multiplication, Shimura-Taniyama formula

Abstract: Motivated by the desire to construct canonical models of Shimura curves (in the final talk), we introduce the theory of complex multiplication (CM) of abelian varieties. After briefly discussing the connection between CM and canonical models, we will cover the basic properties of CM abelian varieties, state the Shimura-Taniyama formula (without proof), and then give the main theorem of complex multiplication. Our main reference for all of this will be chapters 10 and 11 of Milne.

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 2025 topic: Weil conjectures.

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.