Abelian varieties with complex multiplication

Abstract: Milne, Complex multiplication, Sections 2 and 3

Motivated by the complex multiplication theory of elliptic curves, we now consider the complex multiplication theory of abelian varieties, which is a higher-dimensional generalization of elliptic curves. We explain how to construct abelian varieties whose endomorphism ring is an order in an arbitrary CM field. We also define and classify abelian varieties with complex multiplication.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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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.

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