代数群和类域

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