Complex tori and abelian varieties

Abstract: We will define abelian varieties and discuss polarization. We then discuss Riemann's criterion for when a period matrix gives rise to an abelian variety, and if we have time, will see how the Siegel upper-half space parametrizes abelian varieties.

Reference: Section 1.1 (and maybe 1.2) of Genestier and Ngo, Lectures on Shimura varieties.

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.

Spring 2026 topic: The descent obstruction.

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.