Reciprocity Laws II. From Kummer to Hilbert

12. The Failure of Unique Factorization

12.1 The Prehistory: Before Gauss
12.2 From Gauss to Dedekind
12.3 Discovering Capitulation
12.4 Connections with Reciprocity

13. Eisenstein's Conjecture

13.1 Quadratic Reciprocity
13.2 Eisenstein's Function
13.3 Eisenstein's Hypothesis
13.4 Eisenstein's Formulation of the Reciprocity Law

14. Kummer's Reciprocity Law

14.1 The Quest for the Reciprocity Law
14.2 Kummer Characters
14.3 Supplementary Laws and Special Cases
14.4 Kummer's Genus Theory
14.5 The p-adic interpretation

15. Hilbert's Reciprocity Law

15.1 Hilbert's Theory of Genus Characters
15.2 Hilbert's Quadratic Reciprocity Law
15.3 Hilbert's Proof of Kummer's Reciprocity Law

16. Theta Functions and Reciprocity

16.1 Classical Theta Functions
16.2 Hecke's Method
16.3 Weil and Kubota
16.4 Theta and Zeta
16.5 Theta Functions and Abelian Varieties
16.6 Hecke's Construction of Hilbert 2-Class Fields

17. Hilbert Class Field Theory

17.1 Hilbert's Conjectures
17.2 The ambiguous class number formula
17.3 Furtwängler's proofs
17.4 Rédei-Reichardt-Scholz

18. Quaternion Algebras

18.1 Quaternion Algebras
18.2 Tensor Products
18.3 Quaternion Algebras over Local Fields
18.4 Quaternion Algebras over Global Fields
18.5 Cyclic Algebras
18.6 Clifford Algebras

19. Quadratic Function Fields

19.1 Arithmetic in Rational Function Fields
19.2 Quadratic Function Fields
19.3 The Reciprocity Law
19.4 Genus Theory and Gauss sums

20. Quadratic Function Fields

20.1 Formal Properties
20.2 Eisenstein's Reciprocity Law in Quadratic Fields

References

Reciprocity Laws III. From Takagi to Artin

21. Classical Class Field Theory

21.1 Weber's Number Groups
21.2 Takagi's Class Field Theory
21.3 The Reciprocity Law
21.4 Genus Theory

22. Power Residue Character of Units

22.1 The Quadratic Character of Quadratic Units
22.2 The Quartic and Octic Character of Quadratic Units
22.3 The Cubic Character of Quadratic Units
22.4 Scholz's General Reciprocity Law
22.5 The l-th Power Character of Cyclotomic Units

23. Artin's Reciprocity Law

23.1 Density of primes
23.2 L-series
23.3 The Reciprocity Law
23.4 Applications

24. Class Field Theory after Artin

24.1 Local Class Field Theory
24.2 The Idèlic Version
24.3 The Cohomological Version
24.4 The Knots of Scholz and Jehne

25. Explicit Reciprocity Laws

25.1 Explicit Formulas
25.2 The Cubic Character of Quadratic Units Revisited
25.3 Applications of Artin's Reciprocity Law

26. Algebras

26.1 Brauer Groups
26.2 Crossed Products
26.3 The Hasse-Brauer-Noether Theory

27. Function Fields

27.1 Arithmetic in Function Fields
27.2 Riemann-Roch
27.3 Class Field Theory

28. Elliptic Curves

28.1 The Group Law
28.2 Mordell-Weil, Selmer and Tate-Shafarevich groups
28.3 The Birch - Swinnerton-Dyer Conjecture
28.4 Vélu's example

29. The Non-Abelian Horizon

29.1 An Example of Jacobi and Smith
29.2 Shimura's Example
29.3 The Serre Conjectures
29.4 Fermat's Last Theorem
29.5 The Langlands Program

References