Martin Orr's Blog

Abelian varieties posts

Complex abelian varieties and the Mumford-Tate conjecture

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

Polarisations, dual abelian varieties and the Weil pairing

• 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 and the Faltings height

• 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 isogeny theorem

• 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

Endomorphisms of abelian varieties

• Rosati involutions
• Endomorphisms of simple abelian varieties

Universal vector extensions of abelian varieties

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

Absolute Hodge classes

• 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

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

The Chowla-Selberg formula

• The Chowla-Selberg formula
• Gross's proof of the Chowla-Selberg formula