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内容简介
CHAPTER ISummary of Main Results1. Generalized Jacobians2. Abelian coverings3. Other resultsBibliographic noteCHAPTER IIAlgebraic Curves
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目录
目 录 CHAPTER I Summary of Main Results 1. Generalized Jacobians 2. Abelian coverings 3. Other results Bibliographic note CHAPTER II Algebraic Curves 1. Algebraic curves 2. Local rings 3. Divisors, linear equivalence, linear series 4. The Riemann-Roch theorem first form 5. Classes of repartitions 6. Dual of the space of classes of repartitions 7. Differentials, residues 8. Duality theorem 9. The Riemann-Roch theorem definitive form 10. Remarks on the duality theorem 11. Proof of the invariance of the residue 12. Proof of the residue formula 13. Proof of lemma 5 Bibliographic note CHAPTER III Maps From a Curve to a Commutative Group 1. Local symbols 1. Definitions 2. First properties of local symbols 3. Example of a local symbol: additive group case 4. Example of a local symbol: multiplicative group case 2. Proof of theorem 1 5. First reduction 6. Proof in characteristic 0 7. Proof in characteristic p > 0: reduction of the problem 8. Proof in characteristic p > 0: case a 9. Proof in characteristic p > 0: reduction of case b to the unipotent case 10. End of the proof: case where G is a unipotent group 3. Auxiliary results 11. Invariant differential forms on an algebraic group 12. Quotient of a variety by a finite group of automorphisms 13. Some formulas related to coverings 14. Symmetric products 15. Symmetric products and coverings Bibliographic note CHAPTER IV Singular Algebraic Curves 1. Structure of a singular curve 1. Normalization of an algebraic variety 2. Case of an algebraic curve 3. Construction of a singular curve from its normalization 4. Singular curve defined by a modulus 2. Riemann-Roch theorems 5. Notations 6. The Pdemann-Roch theorem first form 7. Application to the computation of the genus of an alge- braic curve 8. Genus of a curve on a surface 3. Differentials on a singular curve 9. Regular differentials on X1 10. Duality theorem 11. The equality nQ = 2Q 12. Complements Bibliographic note CHAPTER V Generalized Jacobians 1. Construction of generalized Jacobians 1. Divisors rational over a field 2. Equivalence relation defined by a modulus 3. Preliminary lemmas 4. Composition law on the symmetric product X 5. Passage from a birational group to an algebraic group 6. Construction of the Jacobian Jm 2. Universal character of generalized Jacobians 7. A homomorphism from the group of divisors of X to Jm 8. The canonical map from X to Jm 9. The universal property of the Jacobians Jm 10. Invariant differential forms on Jm 3. Structure of the Jacobians Jm 11. The usual Jacobian 12. Relations between Jacobians Jm 13. Relation between Jm and J 14. Algebraic structure on the local groups U/U n 15. Structure of the group V n in characteristic zero 16. Structure of the group V n in characteristic p > 0 17. Relation between Jm and J: determination of the alge- braic structure of the group Lm 18. Local symbols 19. Complex case 4. Construction of generalized Jacobians: case of an arbitrary base field 20. Descent of the base field 21. Principal homogeneous spaces 22. Construction of the Jacobian Jm over a perfect field 23. Case of an arbitrary base field Bibliographic note CHAPTER VI Class Field Theory 1. The isogeny x →xq→z 1. Algebraic varieties defined over a finite field 2. Extension and descent of the base field 3. Tori over a finite field 5. Quadratic forms over a finite field 6. The isogeny x→xq→x: commutative case 2. Coverings and isogenies 7. Review of definitions about isogenies 8. Construction of coverings as pull-backs of isogenies 9. Special cases 10. Case of an unramified covering 11. Case of curves 12. Case of curves: conductor 3. Projective system attached to a variety 13. Maximal maps 14. Some properties of maximal maps 15. Maximal maps defined over k 4. Class field theory 16. Statement of the theorem 17. Construction of the extensions Ea 18. End of the proof of theorem 1: first method 19. End of the proof of theorem 1: second method 20. Absolute class fields 21. Complement: the trace map 5. The reciprocity map 22. The Frobenius substitution 23. Geometric interpretation of the Frobenius substitution 24. Determination of the Frobenius substitution in an exten- sion of type a 25. The reciprocity map: statement of results 26. Proof of theorems 3, 3'''', and 3 starting from the case of curves 27. Kernel of the reciprocity map 6. Case of curves 28. Comparison of the divisor class group and generalized Jacobians 29. The idele class group 30. Explicit reciprocity laws 7. Cohomology 31. A criterion for class formations 32. Some properties of the cohomology class uF/E 33. Proof of theorem 5 34. Map to the cycle class group Bibliographic note CHAPTER VII Group Extension and Cohomology 1. Extensions of groups 1. The groups Ext A, B 2. The first exact sequence of Ext 3. Other exact sequences 4. Factor systems 5. The principal fiber space defined by an extension 6. The case of linear groups 2. Structure of commutative connected unipotent groups 7. The group Ext Ga, Ga 8. Witt groups 9. Lemmas 10. Isogenies with a product of Witt groups 11. Structure of connected unipotent groups: particular cases 12. Other results 13. Comparison with generalized Jacobians 3. Extensions of Abelian varieties 14. Primitive cohomology classes 15. Comparison between Ext A, B and H1 A, BA 16. The case B = Gm 17. The case B = Ga 18. Case where B is unipotent 4. Cohomology of Abelian varieties 19. Cohomology of Jacobians 20. Polar part of the maps m 21. Cohomology of Abelian varieties 22. Absence of homological torsion on Ahelian varieties 23. Application to the functor Ext A, B Bibliographic note Bibliography Supplementary Bibliography Index
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