Abelian varieties over local and global fields

TCC course, Spring 2016

Lecture notes (If you have any comments, please email me.)

Course description: This course is a selection of topics on abelian varieties and rational points on them.

Assessment: By essay. You need to submit your essay to me by email by 1 May 2016. You are asked to write a text of about15 pages on a subject related to abelian varieties. It can either be a proof of some of the results given in my lectures without proof, or something that extends the material of the course. For example, you can write about equations defining abelian varieties in projective spaces. Alternatively, you can write about abelian varieties related to your own research, be it in geometry or in number theory.

Background: Good knowledge of algebra, algebraic geometry, algebraic number theory and elliptic curves.

Recommended books:

Over arbitrary fields: D. Mumford “Abelian varieties”, A. Polishchuk “Abelian varieties, theta functions and the Fourier transform”

Over local and global fields: J. Milne “Arithmetic duality theorems” and J. Milne “Abelian varieties”, see http://www.jmilne.org/math/CourseNotes/av.html

G. van dee Geer and B. Moonen “Abelian varieties”, see


For background in geometry, see Shafarevich “Basic algebraic geometry”. For a short introduction see lecture notes of my old course in Algebraic Geometry.