## Abelian varieties

The theory of algebraic groups, i.e. groups that have the structure of an algebraic variety such that the multiplication and inversion operations are given by morphisms, can be reduced to the study of the following two subtopics: The theory of linear algebraic groups (=affine algebraic groups) and Abelian varieties (=complete algebraic groups).

Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. For example, they occur naturally when studying line bundles over an algebraic variety or the arithmetic of a number field. Special cases of Abelian varieties are elliptic curves and the Jacobian of an algebraic curve.

While it is often difficult to to study them directly by explicit equations, Abelian varieties can be shown to have many useful properties. The aim of this lecture is to convey the basic theory of Abelian varieties and the tools of algebraic geometry used to study them. In particular we will study

- Underlying concepts of algebraic geometry: Rigidity theorem, Seesaw principle, Theorem of the cube
- Properties of Abelian varieties: Abelian varieties are projective, smooth and commutative
- Isogenies and morphisms of Abelian varieties
- The dual Abelian variety

**Lecturer: **Dr. Paul Hamacher

**Time and venue:**

Lecture: Mon 10-12 via Zoom

Exercise class: Wed 13-14 via Zoom

**Homework:**

Week 1

Week 2

Week 3

Week 4

Week 5

Week 6

Week 7

Week 9

Week 10

Week 11

Week 12

Details and course materials can be found on the moodle site.

**Prerequisites:** The student is required to have basic knowledge in algebraic geometry.

**Literature:**

Mumford: Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5. *Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi,* 2008. xii+263 pp. ISBN: 978-81-85931-86-9; 81-85931-86-0

Milne: Abelian varieties, Lecture notes