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Abelian Varieties by J.S. Milne

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Abelian Varieties written by J.S. Milne . This is an other book of mathematics 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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