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About this book :-
Abelian Varieties written by
J.S. Milne
These notes are an introduction to the theory of abelian varieties, including the arithmetic
of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version
of the notes was distributed during the teaching of an advanced graduate course.
Book Detail :-
Title: Abelian Varieties
Edition:
Author(s): J.S. Milne
Publisher:
Series:
Year: 2008
Pages: 172
Type: PDF
Language: English
ISBN:
Country:
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Book Contents :-
Abelian Varieties written by
J.S. Milne
cover the following topics.
Introduction
Abelian Varieties
Geometry, Definitions; Basic Properties, Abelian Varieties over the Complex Numbers, Rational Maps Into Abelian Varieties, Review of cohomology, The Theorem of the Cube, Abelian Varieties are Projective, Isogenies, The Dual Abelian Variety, The Dual Exact Sequence, Endomorphisms, Polarizations and Invertible Sheaves, The Etale Cohomology of an Abelian Variety, Weil Pairings, The Rosati Involution, Geometric Finiteness Theorems, Families of Abelian Varieties, Neron models; Semistable Reduction, Abel and Jacobi
Abelian Varieties: Arithmetic
The Zeta Function of an Abelian Variety, Abelian Varieties over Finite Fields, Abelian varieties with complex multiplication
Jacobian Varieties
Overview and definitions, The canonical maps from C to its Jacobian variety, The symmetric powers of a curve, The construction of the Jacobian variety, The canonical maps from the symmetric powers of C to its Jacobian variety, The Jacobian variety as Albanese variety; autoduality, Weil’s construction of the Jacobian variety, Generalizations, Obtaining coverings of a curve from its Jacobian, Abelian varieties are quotients of Jacobian varieties, The zeta function of a curve, Torelli’s theorem: statement and applications, Torelli’s theorem: the proof, Bibliographic notes
Finiteness Theorems
Introduction, The Tate Conjecture; Semisimplicity, Finiteness I implies Finiteness II, Finiteness II implies the Shafarevich Conjecture, Shafarevich’s Conjecture implies Mordell’s Conjecture, The Faltings Height, The Modular Height, The Completion of the Proof of Finiteness I.
Bibliography
Index
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