- Complex abelian varieties and Riemann forms
- Hodge structures and abelian varieties
- Mumford-Tate groups
- Tate modules
- Images of Galois representations

- Hodge symplectic forms
- Polarisations on Hodge structures
- Polarisable complex tori are projective
- Dual abelian varieties over the complex numbers
- Dual abelian varieties and line bundles
- Line bundles and morphisms to the dual variety
- Dual varieties over general fields
- Weil pairings: definition
- Weil pairings: the skew-symmetric pairing

- Finiteness theorems for abelian varieties
- Shafarevich and Siegel's theorems
- Siegel's theorem for curves of genus 0
- The Faltings height of an abelian variety over the rationals
- The Faltings height and normed modules

- The Masser-Wüstholz period theorem
- The Masser-Wüstholz isogeny theorem
- The matrix lemma for elliptic curves
- Main steps of the proof of the period theorem

- Vector extensions of abelian varieties
- Universal vector extensions of abelian varieties
- The Hodge filtration and universal vector extensions

- Hodge classes on abelian varieties
- Tate classes
- Deligne's Principle A and the Mumford-Tate conjecture
- Absolute Hodge classes
- Deligne's theorem on absolute Hodge classes
- Deligne's Principle B
- Tate twists in singular and de Rham cohomology
- Absolute Hodge classes in l-adic cohomology

- Periods of abelian varieties
- Period relations on abelian varieties
- Motivic Galois groups and periods