Class Field Theory

Class eld theoryFrom Wikipedia, the free encyclopedia

Contents

1 Abelian extension 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 AlbertBrauerHasseNoether theorem 22.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Artin L-function 43.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 The Artin conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Artin reciprocity law 74.1 Signicance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Finite extensions of global elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.2 Relation to quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.3 Cohomological interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Alternative statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Artin transfer (group theory) 115.1 Transversals of a subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Permutation representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Artin transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.3.1 Independence of the transversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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5.3.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3.3 Wreath product of H and S(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3.4 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3.5 Wreath product of S(m) and S(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3.6 Cycle decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3.7 Normal subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.4 Computational implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4.1 Abelianization of type (p,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4.2 Abelianization of type (p2,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.5 Transfer kernels and targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.6 Abelianization of type (p,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.7 Abelianization of type (p2,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.7.1 First layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7.2 Second layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7.3 Transfer kernel type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7.4 Connections between layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.8 Inheritance from quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.8.1 Passing through the abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.8.2 TTT singulets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.8.3 TKT singulets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.8.4 TTT and TKT multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.8.5 Inherited automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.9 Stabilization criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.10 Structured descendant trees (SDTs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.11 Pattern recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.11.1 Historical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.12 Commutator calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.13 Systematic library of SDTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.13.1 Coclass 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.13.2 Coclass 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.13.3 Coclass 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.14 Arithmetical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.14.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.14.2 Comparison of various primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Class eld theory 416.1 Formulation in contemporary language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3 Generalizations of class eld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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7 Class formation 457.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.2 Examples of class formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.3 The rst inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.4 The second inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.5 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.6 Tates theorem and the Artin map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.7 The Takagi existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.8 Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8 Complex multiplication 518.1 Example of the imaginary quadratic eld extension . . . . . . . . . . . . . . . . . . . . . . . . . . 518.2 Abstract theory of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.3 Kronecker and abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.4 Sample consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.5 Singular moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9 Conductor (class eld theory) 569.1 Local conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9.1.1 More general elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569.1.2 Archimedean elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9.2 Global conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2.1 Algebraic number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

10 Galois cohomology 5910.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11 Genus eld 6111.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6111.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12 GolodShafarevich theorem 6212.1 The inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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12.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13 GrunwaldWang theorem 6413.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6413.2 Wangs counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

13.2.1 An element that is an nth power almost everywhere locally but not everywhere locally . . . . 6413.2.2 An element that is an nth power everywhere locally but not globally . . . . . . . . . . . . . 65

13.3 A consequence of Wangs counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.4 Special elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.5 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6513.6 Explanation of Wangs counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6613.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

14 Hasse norm theorem 6714.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

15 Hilbert class eld 6815.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.3 Additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.4 Explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

16 Hilbert symbol 7016.1 Quadratic Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

16.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.1.2 Interpretation as an algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.1.3 Hilbert symbols over the rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.1.4 Kaplansky radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

16.2 The general Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.2.2 Hilberts reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.2.3 Power residue symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

16.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

17 Iwasawa theory 7417.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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17.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7417.3 Connections with p-adic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7417.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7517.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7517.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7517.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7617.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

18 KroneckerWeber theorem 7718.1 Field-theoretic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

19 Laorgues theorem 7919.1 Langlands conjectures for GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.2 Representations of the Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.3 Automorphic representations of GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.4 Drinfelds theorem for GL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.5 Laorgues theorem for GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

20 Langlands dual group 8220.1 Denition for separably closed elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.2 Denition for groups over more general elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8220.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

21 LanglandsDeligne local constant 8421.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.2 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8421.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8521.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

22 Local class eld theory 8622.1 Connection to Galois groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622.2 LubinTate theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8622.3 Higher local class eld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8722.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8722.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

vi CONTENTS

22.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

23 Local Fields 8823.1 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8823.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

24 Local Langlands conjectures 8924.1 Local Langlands conjectures for GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.2 Representations of the Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.3 Representations of GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8924.4 Local Langlands conjectures for GL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9024.5 Local Langlands conjectures for GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9024.6 Local Langlands conjectures for other groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9024.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9124.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

25 Non-abelian class eld theory 9325.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9325.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

26 Principalization (algebra) 9526.1 Extension of classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9526.2 Artins reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9626.3 Commutative diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9626.4 Class eld tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9726.5 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9826.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

26.6.1 Quadratic elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.6.2 Cubic elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.6.3 Sextic elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9926.6.4 Quartic elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

26.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.8 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10026.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

27 Quasi-nite eld 10227.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10227.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10227.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10327.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

28 Ray class eld 10428.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10428.2 Ray class elds using ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

CONTENTS vii

28.3 Ray class elds using ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10428.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10528.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

29 Symbol (number theory) 10629.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10729.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

30 Takagi existence theorem 10830.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.2 A well-dened correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830.3 Earlier work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10930.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10930.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10930.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

31 Tate cohomology group 11031.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11031.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11031.3 Tates theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.4 Tate-Farrell cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11131.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

32 Teichmller character 11232.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

33 Teichmller cocycle 11333.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11333.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

34 Timeline of class eld theory 11434.1 Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11434.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

35 Weil group 11635.1 Weil group of a class formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11635.2 Weil group of an archimedean local eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11635.3 Weil group of a nite eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11635.4 Weil group of a local eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11735.5 Weil group of a function eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11735.6 Weil group of a number eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11735.7 WeilDeligne group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

viii CONTENTS

35.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11735.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11735.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11835.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 119

35.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11935.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12035.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Chapter 1

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois groupis a cyclic group, we have a cyclic extension. A Galois extension is called solvable if its Galois group is solvable, i.e.if it is constructed from a series of abelian groups corresponding to intermediate extensions.Every nite extension of a nite eld is a cyclic extension. The development of class eld theory has provided detailedinformation about abelian extensions of number elds, function elds of algebraic curves over nite elds, and localelds.There are two slightly dierent concepts of cyclotomic extensions: these can mean either extensions formed byadjoining roots of unity, or subextensions of such extensions. The cyclotomic elds are examples. Any cyclotomicextension (for either denition) is abelian.If a eld K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resultingso-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n,since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th rootsof elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-directproduct. The Kummer theory gives a complete description of the abelian extension case, and the KroneckerWebertheorem tells us that if K is the eld of rational numbers, an extension is abelian if and only if it is a subeld of a eldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classies all covering spaces of a space:abelian covers are classied by its abelianisation which relates directly to the rst homology group.

1.1 References Kuz'min, L.V. (2001), cyclotomic extension, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer,

ISBN 978-1-55608-010-4

1

Chapter 2

AlbertBrauerHasseNoether theorem

In algebraic number theory, theAlbertBrauerHasseNoether theorem states that a central simple algebra over analgebraic number eld K which splits over every completion Kv is a matrix algebra over K. The theorem is an exampleof a local-global principle in algebraic number theory and leads to a complete description of nite-dimensional divisionalgebras over algebraic number elds in terms of their local invariants. It was proved independently by Helmut Hasse,Richard Brauer, and Emmy Noether and by Abraham Adrian Albert.

2.1 Statement of the theoremLet A be a central simple algebra of rank d over an algebraic number eld K. Suppose that for any valuation v, Asplits over the corresponding local eld Kv:

AK Kv 'Md(Kv):

Then A is isomorphic to the matrix algebra Md(K).

2.2 ApplicationsUsing the theory of Brauer group, one shows that two central simple algebras A and B over an algebraic number eldK are isomorphic over K if and only if their completions Av and Bv are isomorphic over the completion Kv for everyv.Together with the GrunwaldWang theorem, the AlbertBrauerHasseNoether theorem implies that every centralsimple algebra over an algebraic number eld is cyclic, i.e. can be obtained by an explicit construction from a cycliceld extension L/K .

2.3 See also Class eld theory

Hasse norm theorem

2.4 References Albert, A.A.; Hasse, H. (1932), A determination of all normal division algebras over an algebraic number

eld, Trans. Amer. Math. Soc. 34 (3): 722726, doi:10.1090/s0002-9947-1932-1501659-x, Zbl 0005.05003

2

2.5. NOTES 3

Hasse, H.; Brauer, R.; Noether, E. (1931), Beweis eines Hauptsatzes in der Theorie der Algebren, Journalfr Mathematik 167: 399404

Fenster, D.D.; Schwermer, J. (2005), Delicate collaboration: Adrian Albert and Helmut Hasse and the Princi-pal Theorem in Division Algebras (PDF),Archive for history of exact sciences 59 (4): 349379, doi:10.1007/s00407-004-0093-6, retrieved 2009-07-05

Pierce, Richard (1982), Associative algebras, Graduate Texts in Mathematics 88, New York-Berlin: Springer-Verlag, ISBN 0-387-90693-2, Zbl 0497.16001

Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series 28, Oxford Uni-versity Press, p. 276, ISBN 0-19-852673-3, Zbl 1024.16008

Roquette, Peter (2005), The BrauerHasseNoether theorem in historical perspective (PDF), Schriften derMathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie derWissenschaften 15, MR 2222818,Zbl 1060.01009, CiteSeerX: 10 .1 .1 .72 .4101, retrieved 2009-07-05 Revised version Roquette, Peter (2013),Contributions to the history of number theory in the 20th century, Heritage of European Mathematics, Zrich:European Mathematical Society, pp. 176, ISBN 978-3-03719-113-2, Zbl 1276.11001

Albert, Nancy E. (2005), A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8

2.5 Notes

Chapter 3

Artin L-function

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation of a Galoisgroup G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class eldtheory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to beresistant to easy proof. One of the aims of proposed non-abelian class eld theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and Langlandsphilosophy. So far, only a small part of such a theory has been put on a rm basis.

3.1 DenitionGiven , a representation of G on a nite-dimensional complex vector space V , where G is the Galois group of thenite extension L/K of number elds, the Artin L -function: L(; s) is dened by an Euler product. For each primeideal p in K 's ring of integers, there is an Euler factor, which is easiest to dene in the case where p is unramied inL (true for almost all p ). In that case, the Frobenius element Frob(p) is dened as a conjugacy class in G . Thereforethe characteristic polynomial of (Frob(p)) is well-dened. The Euler factor for p is a slight modication of thecharacteristic polynomial, equally well-dened,

charpoly((Frob(p)))1 = det [I t(Frob(p))]1 ;as rational function in t, evaluated at t = N(p)s , with s a complex variable in the usual Riemann zeta functionnotation. (Here N is the eld norm of an ideal.)When p is ramied, and I is the inertia group which is a subgroup of G, a similar construction is applied, but to thesubspace of V xed (pointwise) by I.[note 1]

The Artin L-function L(; s) is then the innite product over all prime ideals p of these factors. As Artin reciprocityshows, when G is an abelian group these L-functions have a second description (as Dirichlet L-functions when Kis the rational number eld, and as Hecke L-functions in general). Novelty comes in with non-abelian G and theirrepresentations.One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number eld that isGalois over the rational numbers. In accordance with the decomposition of the regular representation into irreduciblerepresentations, such a zeta-function splits into a product of Artin L-functions, for each irreducible representationof G. For example, the simplest case is when G is the symmetric group on three letters. Since G has an irreduciblerepresentation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation ofthe Dedekind zeta-function for such a number eld, in a product with the Riemann zeta-function (for the trivialrepresentation) and an L-function of Dirichlets type for the signature representation.

3.2 Functional equationArtin L-functions satisfy a functional equation. The function L(,s) is related in its values to L(*, 1 s), where* denotes the complex conjugate representation. More precisely L is replaced by (, s), which is L multiplied by

4

3.3. THE ARTIN CONJECTURE 5

certain gamma factors, and then there is an equation of meromorphic functions

(, s) = W()(*, 1 s)

with a certain complex number W() of absolute value 1. It is the Artin root number. It has been studied deeplywith respect to two types of properties. Firstly Langlands and Deligne established a factorisation into LanglandsDeligne local constants; this is signicant in relation to conjectural relationships to automorphic representations. Alsothe case of and * being equivalent representations is exactly the one in which the functional equation has thesame L-function on each side. It is, algebraically speaking, the case when is a real representation or quaternionicrepresentation. The Artin root number is, then, either +1 or 1. The question of which sign occurs is linked to Galoismodule theory (Perlis 2001).

3.3 The Artin conjectureThe Artin conjecture on Artin L-functions states that the Artin L-function L(,s) of a non-trivial irreducible repre-sentation is analytic in the whole complex plane.[1]

This is known for one-dimensional representations, the L-functions being then associated to Hecke characters andin particular for Dirichlet L-functions.[1] More generally Artin showed that the Artin conjecture is true for all repre-sentations induced from 1-dimensional representations. If the Galois group is supersolvable then all representationsare of this form so the Artin conjecture holds.Andr Weil proved the Artin conjecture in the case of function elds.Two dimensional representations are classied by the nature of the image subgroup: it may be cyclic, dihedral,tetrahedral, octahedral, or icosahedral. The Artin conjecture for the cyclic or dihedral case follows easily fromHeckes work. Langlands used the base change lifting to prove the tetrahedral case, and Tunnell extended his work tocover the octahedral case; Wiles used these cases in his proof of the TaniyamaShimura conjecture. Richard Taylorand others have made some progress on the (non-solvable) icosahedral case; this is an active area of research.Brauers theorem on induced characters implies that all Artin L-functions are products of positive and negative integralpowers of Hecke L-functions, and are therefore meromorphic in the whole complex plane.Langlands (1970) pointed out that the Artin conjecture follows from strong enough results from the Langlands phi-losophy, relating to the L-functions associated to automorphic representations for GL(n) for all n 1 . Moreprecisely, the Langlands conjectures associate an automorphic representation of the adelic group GL(AQ) to everyn-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois repre-sentation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphicL-function of the automorphic representation. The Artin conjecture then follows immediately from the known factthat the L-functions of cuspidal automorphic representations are holomorphic. This was one of the major motivationsfor Langlands work.

3.4 See also Equivariant L-function

3.5 Notes[1] It is arguably more correct to think instead about the coinvariants, the largest quotient space xed by I, rather than the

invariants, but the result here will be the same. Cf. HasseWeil L-function for a similar situation.

3.6 References[1] Martinet (1977) p.18

6 CHAPTER 3. ARTIN L-FUNCTION

Artin, E. (1923). "ber eine neue Art von L Reihen". Hamb. Math. Abh. 3. Reprinted in his collected works,ISBN 0-387-90686-X. English translation in Artin L-Functions: A Historical Approach by N. Snyder.

Artin, Emil (1930), Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren., Abhandlungen Ham-burg (in German) 8: 292306, doi:10.1007/BF02941010, JFM 56.0173.02

Tunnell, Jerrold (1981). Artins conjecture for representations of octahedral type. Bull. Amer. Math. Soc.N. S. 5 (2): 173175. doi:10.1090/S0273-0979-1981-14936-3.

Gelbart, Stephen (1977). Automorphic forms and Artins conjecture. Modular functions of one variable, VI(Proc. Second Internat. Conf., Univ. Bonn., Bonn, 1976). Lecture Notes in Math. 627. Berlin: Springer. pp.241276.

Langlands, Robert (1967), Letter to Prof. Weil Langlands, R. P. (1970), Problems in the theory of automorphic forms, Lectures in modern analysis and appli-cations, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 1861, doi:10.1007/BFb0079065,ISBN 978-3-540-05284-5, MR 0302614

Martinet, J. (1977), Character theory and Artin L-functions, in Frhlich, A., Algebraic Number Fields, Proc.Symp. London Math. Soc., Univ. Durham 1975, Academic Press, pp. 187, ISBN 0-12-268960-7, Zbl0359.12015

3.7 External links Perlis, R. (2001), Artin root numbers, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN

978-1-55608-010-4

Chapter 4

Artin reciprocity law

The Artin reciprocity law, established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theoremin number theory that forms a central part of global class eld theory.[1] The term "reciprocity law" refers to a longline of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and thereciprocity laws of Eisenstein and Kummer to Hilberts product formula for the norm symbol. Artins result provideda partial solution to Hilberts ninth problem.

4.1 SignicanceArtins reciprocity law implies a description of the abelianization of the absolute Galois group of a global eld Kwhich is based on the Hasse localglobal principle and the use of the Frobenius elements. Together with the Takagiexistence theorem, it is used to describe the abelian extensions of K in terms of the arithmetic of K and to understandthe behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one ofthe main theorems of global class eld theory. It can be used to prove that Artin L-functions are meromorphic andfor the proof of the Chebotarev density theorem.[2]

Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphismof I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic numberelds into the group theoretic task of determining the kernels of transfers of nite non-abelian groups.[3]

4.2 Finite extensions of global eldsThe denition of the Artin map for a nite abelian extension L/K of global elds (such as a nite abelian extensionof Q) has a concrete description in terms of prime ideals and Frobenius elements.If p is a prime of K then the decomposition groups of primes P above p are equal in Gal(L/K) since the latter groupis abelian. If p is unramied in L, then the decomposition group Dp is canonically isomorphic to the Galois group ofthe extension of residue elds OL;P/P over OK;p/p . There is therefore a canonically dened Frobenius element inGal(L/K) denoted by Frobp or

L/Kp

. If denotes the relative discriminant of L/K, the Artin symbol (or Artin

map, or (global) reciprocity map) of L/K is dened on the group of prime-to- fractional ideals, IK , by linearity:

L/K: IK ! Gal(L/K)

mYi=1

pnii 7!mYi=1

L/K

pi

ni:

The Artin reciprocity law (or global reciprocity law) states that there is a modulus c of K such that the Artin mapinduces an isomorphism

IcK/i(Kc;1)NL/K(IcL)!Gal(L/K)

7

8 CHAPTER 4. ARTIN RECIPROCITY LAW

where K, is the ray modulo c, NL/K is the norm map associated to L/K and IcL is the fractional ideals of L prime toc. Such a modulus c is called a dening modulus for L/K. The smallest dening modulus is called the conductor ofL/K and typically denoted f(L/K) .

4.2.1 ExamplesQuadratic elds

If d 6= 1 is a squarefree integer, K = Q, and L=Q(pd) , then the Galois group Gal(L/Q) can be identied with {1}.The discriminant of L over Q is d or 4d depending on whether d 1 (mod 4) or not. The Artin map is then denedon primes p that do not divide by

p 7!

p

where

p

is the Kronecker symbol.[4] More specically, the conductor of L/Q is the principal ideal () or ()

according to whether is positive or negative,[5] and the Artin map on a prime-to- ideal (n) is given by the Kroneckersymbol

n

: This shows that a prime p is split or inert in L according to whether

p

is 1 or 1.

Cyclotomic elds

Let m>1 be either an odd integer or a multiple of 4, let m be a primitive mth root of unity, and let L = Q(m) bethe mth cyclotomic eld. The Galois group Gal(L/Q) can be identied with (Z/mZ) by sending to a given by therule

(m) = am :

The conductor of L/Q is (m),[6] and the Artin map on a prime-to-m ideal (n) is simply n (mod m) in (Z/mZ).[7]

4.2.2 Relation to quadratic reciprocityLet p and be distinct odd primes. For convenience, let * = (1)(1)/2 (which is always 1 (mod 4)). Then, quadraticreciprocity states that

`

p

=p`

:

The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic eld F=Q(p`) andthe cyclotomic eld L=Q(`) as follows.[4] First, F is a subeld of L, so if H = Gal(L/F) and G = Gal(L/Q), thenGal(F/Q) = G/H. Since the latter has order 2, the subgroup H must be the group of squares in (Z/Z). A basicproperty of the Artin symbol says that for every prime-to- ideal (n)

F/Q(n)

=

L/Q(n)

(mod H):

When n = p, this shows that`p

= 1 if, and only if, p (mod ) is in H, i.e. if, and only if, p is a square modulo .

4.3 Cohomological interpretationLet LvKv be a Galois extension of local elds with Galois group G. The local reciprocity law describes a canonicalisomorphism

4.4. ALTERNATIVE STATEMENT 9

v : Kv /NLv/Kv (L

v )! Gab;

called the local Artin symbol, the local reciprocity map or the norm residue symbol.[8][9]

Let LK be a Galois extension of global elds and CL stand for the idle class group of L. The maps v for dierentplaces v of K can be assembled into a single global symbol map by multiplying the local components of an idleclass. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism[10][11]

: CK/NL/K(CL)! Gal(L/K)ab:

A cohomological proof of the global reciprocity law can be achieved by rst establishing that

(Gal(Ksep/K); lim!CL)

constitutes a class formation in the sense of Artin and Tate.[12] Then one proves that

H^0(Gal(L/K); CL) ' H^2(Gal(L/K);Z);

where H^i denote the Tate cohomology groups. Working out the cohomology groups establishes that is an isomor-phism.

4.4 Alternative statementAn alternative version of the reciprocity law, leading to the Langlands program, connects Artin L-functions associatedto abelian extensions of a number eld with Hecke L-functions associated to characters of the idle class group.[13]

A Hecke character (or Grencharakter) of a number eld K is dened to be a quasicharacter of the idle classgroup of K. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic groupGL(1) over the ring of adeles of K.[14]

Let E K be an abelian Galois extension with Galois group G. Then for any character : G C (i.e. one-dimensionalcomplex representation of the group G), there exists a Hecke character of K such that

LArtinE/K(; s) = LHeckeK (; s)

where the left hand side is the Artin L-function associated to the extension with character and the right hand sideis the Hecke L-function associated with , Section 7.D of.[14]

The formulation of the Artin reciprocity law as an equality of L-functions allows formulation of a generalisation ton-dimensional representations, though a direct correspondence is still lacking.

4.5 Notes[1] Helmut Hasse, History of Class Field Theory, in Algebraic Number Theory, edited by Cassels and Frlich, Academic Press,

1967, pp. 266279

[2] Jrgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, Chapter VII

[3] Artin, Emil (December 1929), Idealklassen in oberkrpern und allgemeines reziprozittsgesetz, Abhandlungen aus demMathematischen Seminar der Universitt Hamburg 7 (1): 4651, doi:10.1007/BF02941159.

[4] Lemmermeyer 2000, 3.2

[5] Milne 2008, example 3.11

10 CHAPTER 4. ARTIN RECIPROCITY LAW



[8] Serre (1967) p.140

[9] Serre (1979) p.197

[10] Neukirch (1999) p.391

[11] Jrgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, p. 408. In fact, a more precise version of the reciprocity lawkeeps track of the ramication.

[12] Serre (1979) p.164

[13] James Milne, Class Field Theory

[14] Gelbart, Stephen S. (1975), Automorphic forms on adle groups, Annals of Mathematics Studies 83, Princeton, N.J.:Princeton University Press, MR 0379375.

4.6 References Emil Artin, ber eine neue Art von L-Reihen, Abh. Math. Semin. Univ. Hamburg, 3 (1924), 89108;

Collected Papers, Addison Wesley, 1965, 105124

Emil Artin, Beweis des allgemeinen Reziprozittsgesetzes, Abh. Math. Semin. Univ. Hamburg, 5 (1927),353363; Collected Papers, 131141

Emil Artin, Idealklassen in Oberkrpern und allgemeines Reziprozittsgesetzes, Abh. Math. Semin. Univ.Hamburg, 7 (1930), 4651; Collected Papers, 159164

Frei, Gnther (2004), On the history of the Artin reciprocity law in abelian extensions of algebraic numberelds: how Artin was led to his reciprocity law, in Olav Arnnn Laudal; Ragni Piene, The legacy of NielsHenrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3-8, 2002,Berlin: Springer-Verlag, pp. 267294, ISBN 978-3-540-43826-7, MR 2077576, Zbl 1065.11001

Janusz, Gerald (1973), Algebraic Number Fields, Pure and Applied Mathematics 55, Academic Press, ISBN0-12-380250-4

Lang, Serge (1994),Algebraic number theory, Graduate Texts in Mathematics 110 (2 ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR 1282723

Lemmermeyer, Franz (2000), Reciprocity laws: From Euler to Eisenstein, Springer Monographs in Mathemat-ics, Berlin: Springer-Verlag, ISBN 978-3-540-66957-9, MR 1761696, Zbl 0949.11002

Milne, James (2008), Class eld theory (v4.0 ed.), retrieved 2010-02-22 Neukirch, Jrgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322,

Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl0956.11021

Serre, Jean-Pierre (1979), Local elds, Graduate Texts in Mathematics 67, Translated from the French by Mar-vin Jay Greenberg, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 3-540-90424-7, Zbl 0423.12016

Serre, Jean-Pierre (1967), VI. Local class eld theory, in Cassels, J.W.S.; Frhlich, A., Algebraic numbertheory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATOAdvanced Study Institute) with the support of the International Mathematical Union, London: Academic Press,pp. 128161, Zbl 0153.07403

Tate, John (1967), VII. Global class eld theory, in Cassels, J.W.S.; Frhlich, A., Algebraic number theory.Proceedings of an instructional conference organized by the London Mathematical Society (a NATO AdvancedStudy Institute) with the support of the International Mathematical Union, London: Academic Press, pp. 162203, Zbl 0153.07403

Chapter 5

Artin transfer (group theory)

In the mathematical eld of group theory, an Artin transfer is a certain homomorphism from a group to thecommutator quotient group of a subgroup of nite index. Originally, such mappings arose as group theoretic coun-terparts of class extension homomorphisms of abelian extensions of algebraic number elds by applying Artins reci-procity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups.However, independently of number theoretic applications, the kernels and targets of Artin transfers have recentlyturned out to be compatible with parent-descendant relations between nite p-groups, which can be visualized indescendant trees. Therefore, Artin transfers provide a valuable tool for the classication of nite p-groups and forsearching particular groups in descendant trees by looking for patterns dened by the kernels and targets of Artintransfers. These methods of pattern recognition are useful in purely group theoretic context, as well as for applica-tions in algebraic number theory concerning higher p-class groups and Hilbert p-class eld towers.

5.1 Transversals of a subgroupLet G be a group and H < G be a subgroup of nite index n = (G : H) .Denitions. [1]

A left transversal of H in G is an ordered system (g1; : : : ; gn) of representatives for the leftcosets of H in G such that G = _Sni=1 giH is a disjoint union.

Similarly, a right transversal of H in G is an ordered system (d1; : : : ; dn) of representatives forthe right cosets of H in G such that G = _Sni=1Hdi is a disjoint union.

Remarks. [2]

For any transversal of H in G , there exists a unique subscript 1 i0 n such that gi0 2 H ,resp. di0 2 H . Of course, this element may be, but need not be, replaced by the neutral element1 .

If G is non-abelian and H is not a normal subgroup of G , then we can only say that the inverseelements (g11 ; : : : ; g1n ) of a left transversal (g1; : : : ; gn) form a right transversal of H in G ,since G = _Sni=1 giH implies G = G1 = _Sni=1 (giH)1 = _Sni=1H1g1i = _Sni=1Hg1i .

However, if H / G is a normal subgroup of G , then any left transversal is also a right transversalof H in G , since xH = Hx for each x 2 G .

5.2 Permutation representationSuppose (g1; : : : ; gn) is a left transversal of a subgroup H < G of nite index n = (G : H) in a group G . A xedelement x 2 G gives rise to a unique permutation x 2 Sn of the left cosets of H in G such that xgiH = gx(i)H ,resp. xgi 2 gx(i)H , resp. hx(i) := g1x(i)xgi 2 H , for each 1 i n .

11

12 CHAPTER 5. ARTIN TRANSFER (GROUP THEORY)

Similarly, if (d1; : : : ; dn) is a right transversal of H in G , then a xed element x 2 G gives rise to a uniquepermutation x 2 Sn of the right cosets of H in G such that Hdix = Hdx(i) , resp. dix 2 Hdx(i) , resp.x(i) := dixd

1x(i)

2 H , for each 1 i n .Denition. [1]

The mapping G! Sn; x 7! x , resp. x 7! x , is called the permutation representation of G in Sn with respectto (g1; : : : ; gn) , resp. (d1; : : : ; dn) .The mapping G ! Hn Sn; x 7! (hx(1); : : : ; hx(n);x) , resp. x 7! (x(1); : : : ; x(n); x) , is called themonomial representation of G in Hn Sn with respect to (g1; : : : ; gn) , resp. (d1; : : : ; dn) .Remark.For the special right transversal (g11 ; : : : ; g1n ) associated to the left transversal (g1; : : : ; gn) we have x(i) =g1i xgx(i) but on the other hand hx(i)1 = (g1x(i)xgi)

1 = g1i x1gx(i) = g

1i x

1gx1 (i) = x1(i), for each 1 i n . This relation simultaneously shows that, for any x 2 G , the associated permutationrepresentations are connected by x1 = x , and the associated monomial representations are connected additionallyby x1(i) = hx(i)1 , for each 1 i n .

5.3 Artin transferLet G be a group and H < G be a subgroup of nite index n = (G : H) . Assume that (g1; : : : ; gn) , resp.(d1; : : : ; dn) , is a left, resp. right, transversal of H in G .Denition. [2] [3]

Then the Artin transfer TG;H : G ! H/H 0 from G to the abelianization of H with respect to (g1; : : : ; gn) ,resp. (d1; : : : ; dn) , is dened by T (g)G;H(x) :=

Qni=1 g

1x(i)

xgi H 0 or briey TG;H(x) =Qn

i=1 hx(i) H 0 , resp.T(d)G;H(x) :=

Qni=1 dixd

1x(i)

H 0 or briey TG;H(x) =Qn

i=1 x(i) H 0 , for x 2 G .

5.3.1 Independence of the transversal

Assume that (1; : : : ; n) is another left transversal of H in G such that G = _[ni=1 iH . Then there exists a uniquepermutation 2 Sn such that giH = (i)H , for all 1 i n . Consequently, hi := g1i (i) 2 H , resp.

(i) = gihi with hi 2 H , for all 1 i n . For a xed element x 2 G , there exists a unique permutationx 2 Sn such that we have x((i))H = x(i)H = xgihiH = xgiH = gx(i)H = gx(i)hx(i)H = (x(i))H, for all 1 i n . Therefore, the permutation representation of G with respect to (1; : : : ; n) is given byx = x , resp. x = x 1 2 Sn , for x 2 G . Furthermore, for the connection betweenthe elements kx(i) := 1x(i)xi 2 H and hx(i) := g

1x(i)

xgi 2 H , we obtain kx((i)) = 1x((i))x(i) =

1(x(i))xgihi = (gx(i)hx(i))

1xgihi = h1x(i)g1x(i)

xgihi = h1x(i)

hx(i)hi , for all 1 i n . Finally,due to the commutativity of the quotient group H/H 0 and the fact that ; x are permutations, the Artin transferturns out to be independent of the left transversal: T ()G;H(x) =

Qni=1 kx((i)) H 0 =

Qni=1 h

1x(i)

hx(i)hi H 0=Qn

i=1 hx(i)Qn

i=1 h1x(i)

Qni=1 hi H 0 =

Qni=1 hx(i) 1 H 0 =

Qni=1 hx(i) H 0 = T (g)G;H(x) , as dened above.

It is clear that a similar proof shows that the Artin transfer is independent of the choice between two dierent righttransversals. It remains to show that the Artin transfer with respect to a right transversal coincides with the Artintransfer with respect to a left transversal. For this purpose, we select the special right transversal (g11 ; : : : ; g1n )associated to the left transversal (g1; : : : ; gn) . Using the commutativity of H/H 0 and the remark in the previoussection, we consider the expression T (g

1)G;H (x) =

Qni=1 g

1i xgx(i) H 0 =

Qni=1 x(i) H 0 =

Qni=1 hx1(i)

1 H 0 = (

Qni=1 hx1(i) H 0)1 = (T (g)G;H(x1))1 = T (g)G;H(x) . The last step is justied by the fact that the Artin

transfer is a homomorphism. This will be shown in the following section.

5.3.2 Homomorphisms

Letx; y 2 G be two elements with transfer imagesTG;H(x) =Qn

i=1 g1x(i)

xgiH 0 andTG;H(y) =Qn

j=1 g1y(j)

ygj H 0 . Since H/H 0 is abelian and y is a permutation, we can change the order of the factors in the following product:

5.3. ARTIN TRANSFER 13

TG;H(x)TG;H(y) =Qn

i=1 g1x(i)

xgiH0Qnj=1 g1y(j)ygj H 0 =Qnj=1 g1x(y(j))xgy(j)H 0Qnj=1 g1y(j)ygj H 0

=Qn

j=1 g1x(y(j))

xgy(j)g1y(j)

ygj H 0 =Qn

j=1 g1(xy)(j))xygj H 0 = TG;H(xy) . This relation simultane-

ously shows that the Artin transfer TG;H and the permutation representation G! Sn; x 7! x are homomorphisms,since xy = x y .It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial representation.The images of the factors x; y are given by TG;H(x) =

Qni=1 hx(i) H 0 and TG;H(y) =

Qnj=1 hy(j) H 0 . The

image of the product xy turned out to be TG;H(xy) =Qn

j=1 hx(y(j)) hy(j) H 0 , which is a very peculiar lawof composition discussed in more detail in the following section.The law reminds of the crossed homomorphisms x 7! hx in the rst cohomology group H1(G;M) of a G -moduleM , which have the property hxy = hyx hy .

5.3.3 Wreath product of H and S(n)The peculiar structures which arose in the previous section can also be interpreted by endowing the cartesian productHn Sn with a special law of composition known as the wreath product H o Sn of the groups H and Sn withrespect to the set f1; : : : ; ng . For x; y 2 G , it is given by (hx(1); : : : ; hx(n);x) (hy(1); : : : ; hy(n);y) :=(hx(y(1)) hy(1); : : : ; hx(y(n)) hy(n);x y) = (hxy(1); : : : ; hxy(n);xy) , which causes the monomialrepresentation G ! H o Sn; x 7! (hx(1); : : : ; hx(n);x) also to be a homomorphism. In fact, it is an injectivehomomorphism, also called a monomorphism or embedding, in contrast to the permutation representation.

5.3.4 CompositionLet G be a group with nested subgroups K H G such that the index (G : K) = (G : H) (H : K) = n mis nite. Then the Artin transfer TG;K is the compositum of the induced transfer ~TH;K : H/H 0 ! K/K 0 and theArtin transfer TG;H , that is, TG;K = ~TH;K TG;H . This can be seen in the following manner.If (g1; : : : ; gn) is a left transversal ofH inG and (h1; : : : ; hm) is a left transversal ofK inH , that isG = _[ni=1 giHand H = _[mj=1 hjK , then G = _[ni=1 _[mj=1 gihjK is a disjoint left coset decomposition of G with respect to K. Given two elements x 2 G and y 2 H , there exist unique permutations x 2 Sn , and y 2 Sm , such thathx(i) := g

1x(i)

xgi 2 H , for each 1 i n , and ky(j) := h1y(j)yhj 2 K , for each 1 j m . ThenTG;H(x) =

Qni=1 hx(i) H 0 , and ~TH;K(y H 0) = TH;K(y) =

Qmj=1 ky(j) K 0 . For each pair of subscripts

1 i n and 1 j m , we have xgihj = gx(i)g1x(i)xgihj = gx(i)hx(i)hj = gx(i)hyi (j)kyi(j) , resp.h1yi (j)g

1x(i)

xgihj = kyi(j) , where yi := hx(i) . Therefore, the image of x under the Artin transfer TG;K is givenby TG;K(x) =

Qni=1

Qmj=1 kyi(j) K 0 =

Qni=1

Qmj=1 h

1yi (j)

g1x(i)xgihj K 0 =Qn

i=1

Qmj=1 h

1yi (j)

hx(i)hj K 0 =

Qni=1

Qmj=1 h

1yi (j)

yihj K 0 =Qn

i=1~TH;K(yi H 0) = ~TH;K(

Qni=1 yi H 0) = ~TH;K(

Qni=1 hx(i) H 0) =

~TH;K(TG;H(x)) .Finally, we want to emphasize the structural peculiarity of the corresponding monomial representationG! KnmSnm , x 7! (`x(1; 1); : : : ; `x(n;m); x) , dening `x(i; j) := ((gh)x(i;j))1x(gh)(i;j) 2 K for a permutation

x 2 Snm , and using the symbolic notation (gh)(i;j) := gihj for all pairs of subscripts 1 i n , 1 j m .The preceding proof has shown that `x(i; j) = h1yi (j)g

1x(i)

xgihj . Therefore, the action of the permutation x onthe set [1; n] [1;m] is given by x(i; j) = (x(i); hx(i)(j)) . The action on the second component depends onthe rst component (via the permutation hx(i) 2 Sm to be selected) whereas the action on the rst component isindependent of the second component. Therefore, the permutation x 2 Snm can be identied with the multiplet

(x;hx(1); : : : ; hx(n)) 2 Sn Snm;which will be written in twisted form in the next section.

5.3.5 Wreath product of S(m) and S(n)The permutations x , which arose as second components of the monomial representation G ! Knm Snm ,x 7! (`x(1; 1); : : : ; `x(n;m); x) , in the previous section, are of a very special kind. They belong to the stabilizer


of the natural equipartition of the set [1; n] [1;m] into the n rows of the corresponding matrix (rectangular array).Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer isisomorphic to the wreath product Sm o Sn of the groups Sm and Sn with respect to the set f1; : : : ; ng , whoseunderlying set Snm Sn is endowed with the following law of composition x z = (hx(1); : : : ; hx(n);x) (hz(1); : : : ; hz(n);z) = (hx(z(1)) hz(1); : : : ; hx(z(n)) hz(n);x z) = (hxz(1); : : : ; hxz(n);xz)= xz for all x; z 2 G .This law reminds of the chain ruleD(g f)(x) = D(g)(f(x)) D(f)(x) for the Frchet derivative in x 2 E of thecompositum of dierentiable functions f : E ! F and g : F ! G between normed spaces.The above considerations establish a third representation, the stabilizer representation,G! SmoSn; x 7! (hx(1); : : : ; hx(n);x)of the group G in the wreath product Sm o Sn , similar to the permutation representation and the monomial repre-sentation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, if G isinnite.

5.3.6 Cycle decomposition

Let (g1; : : : ; gn) be a left transversal of a subgroup H < G of nite index n = (G : H) in a group G . Suppose theelement x 2 G gives rise to the permutation x 2 Sn of the left cosets of H in G such that xgiH = gx(i)H , resp.g1x(i)xgi =: hx(i) 2 H , for each 1 i n .If x has the decomposition x =

Qtj=1 j into pairwise disjoint cycles j 2 Sn of lengths fj 1 , which is unique

up to the ordering of the cycles, more explicitly, if (gjH; gj(j)H; g2j (j)H; : : : ; gfj1j (j)H) = (gjH;xgjH;x2gjH; : : : ; x

fj1gjH)

, for 1 j t , and Ptj=1 fj = n , then the image of x under the Artin transfer TG;H is given by TG;H(x) =Qtj=1 g

1j x

fjgj H 0 .The reason for this fact is that we obtain another left transversal ofH inG by putting j;k := xkgj for 0 k fj1and 1 j t , since G = _[tj=1 _[fj1k=0 xkgjH . Let us x a value of 1 j t . For 0 k fj 2 , wehave xj;k = xxkgj = xk+1gj = j;k+1 = j;k+1 1 , resp. hx(j; k) = 1 . However, for k = fj 1 ,we obtain xj;fj1 = xxfj1gj = xfjgj 2 gjH , resp. g1j xfjgj = hx(j; fj 1) 2 H . Consequently,TG;H(x) =

Qtj=1

Qfj1k=0 hx(j; k) H 0 =

Qtj=1 (

Qfj2k=0 1) hx(j; fj 1) H 0 =

Qtj=1 g

1j x

fjgj H 0 . Thecycle decomposition corresponds to a double coset decomposition G = _[tj=1 hxigjH of the group G modulo thecyclic group hxi and the subgroup H . It was actually this cycle decomposition form of the transfer homomorphismwhich was given by E. Artin in his original 1929 paper.[3]

5.3.7 Normal subgroup

Let H / G be a normal subgroup of nite index n = (G : H) in a group G . Then we have xH = Hx , for allx 2 G , and there exists the quotient group G/H of order n . For an element x 2 G , we let f := ord(xH) denotethe order of the coset xH in G/H . Then, hxHi is a cyclic subgroup of order f of G/H , and a (left) transversal(g1; : : : ; gt) of the subgroup hx;Hi in G , where t = n/f and G = _[tj=1 gjhx;Hi , can be extended to a (left)transversalG = _[tj=1 _[f1k=0 gjxkH ofH inG . Hence, the formula for the image of x under the Artin transfer TG;Hin the previous section takes the particular shape TG;H(x) =

Qtj=1 g

1j x

fgj H 0 with exponent f independent ofj .In particular, the inner transfer of an element x 2 H of order f = 1 is given as a symbolic power TG;H(x) =Qt

j=1 g1j xgj H 0 =

Qtj=1 x

gj H 0 = xPt

j=1 gj H 0 with the trace element TrG(H) =Pt

j=1 gj 2 Z[G] of Hin G as symbolic exponent. The other extreme is the outer transfer of an element x 2 G nH which generates Gmodulo H , that is G = hx;Hi and f = n . It is simply an n th power TG;H(x) =

Q1j=1 1

1 xn 1 H 0 = xn H 0.Explicit implementations of Artin transfers in the simplest situations are presented in the following section.

5.4 Computational implementation

5.5. TRANSFER KERNELS AND TARGETS 15

5.4.1 Abelianization of type (p,p)

LetG be a p-group with abelianizationG/G0 of elementary abelian type (p; p) . ThenG has p+1 maximal subgroupsHi < G (1 i p + 1) of index (G : Hi) = p . For each 1 i p + 1 , let Ti : G ! Hi/H 0i be the Artintransfer homomorphism fromG to the abelianization ofHi . According to Burnsides basis theorem, the groupG hasgenerator rank d(G) = 2 and can therefore be generated as G = hx; yi by two elements x; y such that xp; yp 2 G0. For each of the normal subgroups Hi / G , a generator hi with respect to G0 , and a generator ti of a transversalmust be given such that Hi = hhi; G0i and G = hti;Hii . A convenient selection is given by h1 = y , t1 = x ,and hi = xyi2 , ti = y , for all 2 i p + 1 . Then, for each 1 i p + 1 , it is sucient to dene theinner transfer by Ti(hi) = hTrG(Hi)i H 0i = h1+ti+t

2i++tp1i

i H 0i , which can also be expressed as a productof two pth powers hi t1i hiti t2i hit2i tp+1i hitp1i H 0i = (hit1i )ptpi H 0i , and the outer transfer as acomplete pth power by Ti(ti) = tpi H 0i . The reason is that f = ord(hiHi) = 1 and f = ord(tiHi) = p . Itshould be pointed out that the complete specication of the Artin transfers also requires explicit knowledge of thederived subgroups H 0i . Since G0 is a normal subgroup of index p in Hi , a certain general reduction is possible byH 0i = [Hi;Hi] = [G

0;Hi] = (G0)hi1 , [4] but a presentation of G must be known for determining generators ofG0 = hs1; : : : ; sni , whence H 0i = h[s1; hi]; : : : ; [sn; hi]i .

5.4.2 Abelianization of type (p2,p)

Let G be a p-group with abelianization G/G0 of non-elementary abelian type (p2; p) . Then G has p + 1 maximalsubgroups Hi < G (1 i p + 1) of index (G : Hi) = p and p + 1 subgroups Ui < G (1 i p + 1) ofindex (G : Ui) = p2 . For each 1 i p + 1 , let T1;i : G ! Hi/H 0i , resp. T2;i : G ! Ui/U 0i , be theArtin transfer homomorphism from G to the abelianization of Hi , resp. Ui . Burnsides basis theorem asserts thatthe group G has generator rank d(G) = 2 and can therefore be generated as G = hx; yi by two elements x; ysuch that xp2 ; yp 2 G0 . We begin by considering the rst layer of subgroups. For each of the normal subgroupsHi / G (1 i p) , we select a generator hi = xyi1 such that Hi = hhi; G0i . These are the cases where thefactor group Hi/G0 is cyclic of order p2 . However, for the distinguished maximal subgroup Hp+1 , for whichthe factor group Hp+1/G0 is bicyclic of type (p; p) , we need two generators hp+1 = y and h0 = xp such thatHp+1 = hhp+1; h0; G0i . Further, a generator ti of a transversal must be given such that G = hti;Hii , for each1 i p + 1 . It is convenient to dene ti = y , for 1 i p , and tp+1 = x . Then, for each 1 i p + 1, we have the inner transfer T1;i(hi) = hTrG(Hi)i H 0i = h1+ti+t

2i+:::+t

p1i

i H 0i , which equals (hit1i )ptpi H 0i ,and the outer transfer T1;i(ti) = tpi H 0i , since f = ord(hiHi) = 1 and f = ord(tiHi) = p . Now we continueby considering the second layer of subgroups. For each of the normal subgroups Ui / G (1 i p + 1) , weselect a generator u1 = y , ui = xpyi1 for 2 i p , and up+1 = xp , such that Ui = hui; G0i . Amongthese subgroups, the Frattini subgroup Up+1 = hxp; G0i = Gp G0 is particularly distinguished. A uniform way ofdening generators ti; wi of a transversal such that G = hti; wi; Uii , is to set ti = x;wi = xp , for 1 i p , andtp+1 = x;wp+1 = y . Since f = ord(uiUi) = 1 , but on the other hand f = ord(tiUi) = p2 and f = ord(wiUi) =p , for 1 i p+ 1 , with the single exception that f = ord(tp+1Up+1) = p , we obtain the following expressionsfor the inner transfer T2;i(ui) = uTrG(Ui)i U 0i = u

Pp1j=0

Pp1k=0 w

ji tki

i U 0i =Qp1

j=0

Qp1k=0 (w

ji tki )1uiw

ji tki U 0i ,

and for the outer transfer T2;i(ti) = tp2

i U 0i , exceptionally T2;p+1(tp+1) = (tpp+1)1+wp+1+w2p+1+:::+w

p1p+1 U 0p+1

, and T2;i(wi) = (wpi )1+ti+t2i+:::+t

p1i U 0i , for 1 i p+ 1 . Again, it should be emphasized that the structure

of the derived subgroups H 0i and U 0i must be known to specify the action of the Artin transfers completely.

5.5 Transfer kernels and targetsLet G be a group with nite abelianization G/G0 . Suppose that (Hi)i2I denotes the family of all subgroups Hi /Gwhich contain the commutator subgroup G0 and are therefore necessarily normal, enumerated by means of the niteindex set I . For each i 2 I , let Ti := TG;Hi be the Artin transfer from G to the abelianization Hi/H 0i .Denition. [5]

The family of normal subgroups {H(G) = (ker(Ti))i2I is called the transfer kernel type (TKT) ofGwith respectto (Hi)i2I , and the family of abelianizations (resp. their abelian type invariants) H(G) = (Hi/H 0i)i2I is calledthe transfer target type (TTT) of G with respect to (Hi)i2I . Both families are also called multiplets whereas asingle component will be referred to as a singulet.


Important examples for these concepts are provided in the following two sections.

5.6 Abelianization of type (p,p)LetG be a p-group with abelianizationG/G0 of elementary abelian type (p; p) . ThenG has p+1 maximal subgroupsHi < G (1 i p + 1) of index (G : Hi) = p . For each 1 i p + 1 , let Ti : G ! Hi/H 0i be the Artintransfer homomorphism from G to the abelianization of Hi .Denition.The family of normal subgroups {H(G) = (ker(Ti))1ip+1 is called the transfer kernel type (TKT) of G withrespect to H1; : : : ; Hp+1 .Remarks.

For brevity, the TKT is identied with the multiplet ({(i))1ip+1 , whose integer componentsare given by {(i) =

(0 if ker(Ti) = G;j if ker(Ti) = Hj some for 1 j p+ 1:

Here, we take into consid-

eration that each transfer kernel ker(Ti) must contain the commutator subgroup G0 of G , sincethe transfer target Hi/H 0i is abelian. However, the minimal case ker(Ti) = G0 cannot occur.

A renumeration of the maximal subgroups Ki = H(i) and of the transfers Vi = T(i) by meansof a permutation 2 Sp+1 gives rise to a new TKT K(G) = (ker(Vi))1ip+1 with respect toK1; : : : ;Kp+1 , identied with ((i))1ip+1 , where(i) =

(0 if ker(Vi) = G;j if ker(Vi) = Kj some for 1 j p+ 1:

It is adequate to view the TKTs K(G) {H(G) as equivalent. Since we have K(i) =ker(Vi) = ker(T(i)) = H{((i)) = K~1({((i))) , the relation between and { is givenby = ~1 { . Therefore, is another representative of the orbit {Sp+1 of { under theoperation (; ) 7! ~1 of the symmetric group Sp+1 on the set of all mappings fromf1; : : : ; p + 1g to f0; : : : ; p + 1g , where the extension ~ 2 Sp+2 of the permutation 2 Sp+1is dened by ~(0) = 0 , and formally H0 = G , K0 = G .

Denition.The orbit {(G) = {Sp+1 of any representative { is an invariant of the p-group G and is called its transfer kerneltype, briey TKT.Remark.Let #H0(G) := #f1 i p + 1 j {(i) = 0g denote the counter of total transfer kernels ker(Ti) = G , whichis an invariant of the group G . In 1980, S. M. Chang and R. Foote [6] proved that, for any odd prime p and forany integer 0 n p+ 1 , there exist metabelian p-groups G having abelianization G/G0 of type (p; p) such that#H0(G) = n . However, for p = 2 , there do not exist non-abelian 2 -groups G with G/G0 ' (2; 2) , which mustbe metabelian of maximal class, such that #H0(G) 2 . Only the elementary abelian 2 -group G = C2 C2 has#H0(G) = 3 . See Figure 5.In the following concrete examples for the counters #H0(G) , and also in the remainder of this article, we useidentiers of nite p-groups in the SmallGroups Library by H. U. Besche, B. Eick and E. A. O'Brien .[7] [8]

For p = 3 , we have

#H0(G) = 0 for the extra special groupG = h27; 4i of exponent 9 with TKT{ = (1111) (Figure6),

#H0(G) = 1 for the two groups G 2 fh243; 6i; h243; 8ig with TKTs { 2 f(0122); (2034)g(Figures 8 and 9),

#H0(G) = 2 for the group G = h243; 3i with TKT { = (0043) (Figure 4 in the article ondescendant trees),

#H0(G) = 3 for the group G = h81; 7i with TKT { = (2000) (Figure 6), #H0(G) = 4 for the extra special groupG = h27; 3i of exponent 3 with TKT{ = (0000) (Figure

6).

5.7. ABELIANIZATION OF TYPE (P2,P) 17

5.7 Abelianization of type (p2,p)Let G be a p-group with abelianization G/G0 of non-elementary abelian type (p2; p) . Then G possesses p + 1maximal subgroups Hi < G (1 i p+1) of index (G : Hi) = p , and p+1 subgroups Ui < G (1 i p+1)of index (G : Ui) = p2 .Assumption.Suppose that Hp+1 =

Qp+1j=1 Uj is the distinguished maximal subgroup which is the product of all subgroups of

index p2 , and Up+1 = \p+1j=1 Hj is the distinguished subgroup of index p2 which is the intersection of all maximalsubgroups, that is the Frattini subgroup (G) of G .

5.7.1 First layer

For each 1 i p+ 1 , let T1;i : G! Hi/H 0i be the Artin transfer homomorphism from G to the abelianizationof Hi .Denition.The family {1;H;U (G) = (ker(T1;i))1ip+1 is called the rst layer transfer kernel type of G with respect to

H1; : : : ; Hp+1 andU1; : : : ; Up+1 , and is identied with ({1(i))1ip+1 , where{1(i) =(0 if ker(T1;i) = Hp+1;j if ker(T1;i) = Uj some for 1 j p+ 1:

Remark.Here, we observe that each rst layer transfer kernel is of exponent p with respect to G0 and consequently cannotcoincide with Hj for any 1 j p , since Hj/G0 is cyclic of order p2 , whereas Hp+1/G0 is bicyclic of type (p; p).

5.7.2 Second layer

For each 1 i p+ 1 , let T2;i : G ! Ui/U 0i be the Artin transfer homomorphism from G to the abelianizationof Ui .Denition.The family {2;U;H(G) = (ker(T2;i))1ip+1 is called the second layer transfer kernel type of G with respect to

U1; : : : ; Up+1 andH1; : : : ;Hp+1 , and is identied with ({2(i))1ip+1 , where{2(i) =(0 if ker(T2;i) = G;j if ker(T2;i) = Hj some for 1 j p+ 1:

5.7.3 Transfer kernel type

Combining the information on the two layers, we obtain the (complete) transfer kernel type{H;U (G) = ({1;H;U (G);{2;U;H(G))of the p-group G with respect to H1; : : : ;Hp+1 and U1; : : : ; Up+1 .Remark.The distinguished subgroups Hp+1 and Up+1 = (G) are unique invariants of G and should not be renumer-ated. However, independent renumerations of the remaining maximal subgroups Ki = H(i) (1 i p)and the transfers V1;i = T1;(i) by means of a permutation 2 Sp , and of the remaining subgroups Wi =U(i) (1 i p) of index p2 and the transfers V2;i = T2;(i) by means of a permutation 2 Sp , giverise to new TKTs 1;K;W (G) = (ker(V1;i))1ip+1 with respect to K1; : : : ;Kp+1 and W1; : : : ;Wp+1 , identi-

ed with (1(i))1ip+1 , where 1(i) =(0 if ker(V1;i) = Kp+1;j if ker(V1;i) = Wj some for 1 j p+ 1;

and 2;W;K(G) =

(ker(V2;i))1ip+1 with respect to W1; : : : ;Wp+1 and K1; : : : ;Kp+1 , identied with (2(i))1ip+1 , where

2(i) =

(0 if ker(V2;i) = G;j if ker(V2;i) = Kj some for 1 j p+ 1:

It is adequate to view the TKTs1;K;W (G) {1;H;U (G)and 2;W;K(G) {2;U;H(G) as equivalent. Since we have W1(i) = ker(V1;i) = ker(T1;^(i)) = U{1(^(i)) =


W~1({1(^(i))) , resp. K2(i) = ker(V2;i) = ker(T2;^(i)) = H{2(^(i)) = K~1({2(^(i))) , the relations be-tween 1 and {1 , resp. 2 and {2 , are given by 1 = ~1 {1 ^ , resp. 2 = ~1 {2 ^ . Therefore, = (1; 2) is another representative of the orbit {SpSp of{ = ({1;{2) under the operation ((; ); (1; 2)) 7!(~1 1 ^ ; ~1 2 ^) of the product of two symmetric groups Sp Sp on the set of all pairs of mappingsfrom f1; : : : ; p + 1g to f0; : : : ; p + 1g , where the extensions ^ 2 Sp+1 and ~ 2 Sp+2 of a permutation 2 Spare dened by ^(p + 1) = ~(p + 1) = p + 1 and ~(0) = 0 , and formally H0 = K0 = G , Kp+1 = Hp+1 ,U0 = W0 = Hp+1 , and Wp+1 = Up+1 = (G) .Denition.The orbit {(G) = {SpSp of any representative { = ({1;{2) is an invariant of the p-group G and is called itstransfer kernel type, briey TKT.

5.7.4 Connections between layersThe Artin transfer T2;i : G ! Ui/U 0i from G to a subgroup Ui of index (G : Ui) = p2 ( 1 i p + 1 ) is thecompositum T2;i = ~THj ;Ui T1;j of the induced transfer ~THj ;Ui : Hj/H 0j ! Ui/U 0i from Hj to Ui and the Artintransfer T1;j : G! Hj/H 0j from G to Hj , for any intermediate subgroup Ui < Hj < G of index (G : Hj) = p (1 j p+ 1 ). There occur two situations:

For the subgroups U1; : : : ; Up only the distinguished maximal subgroup Hp+1 is an intermediatesubgroup.

For the Frattini subgroup Up+1 = (G) all maximal subgroups H1; : : : ;Hp+1 are intermediatesubgroups.

This causes restrictions for the transfer kernel type{2(G) of the second layer, since ker(T2;i) = ker( ~THj ;UiT1;j) ker(T1;j) , and thus

ker(T2;i) ker(T1;p+1) , for all 1 i p , but even ker(T2;p+1) h[p+1j=1 ker(T1;j)i .

Furthermore, when G = hx; yi with xp /2 G0 and yp 2 G0 , an element xyk1 ( 1 k p ) which is of order p2with respect to G0 , can belong to the transfer kernel ker(T2;i) only if its p th power xp is contained in ker(T1;j) , forall intermediate subgroups Ui < Hj < G , and thus:

xyk1 2 ker(T2;i) , for certain 1 i; k p , enforces the rst layer TKT singulet {1(p+ 1) =p+ 1 ,

but xyk1 2 ker(T2;p+1) , for some 1 k p , even species the complete rst layer TKTmultiplet {1 = ((p+ 1)p+1) , that is {1(j) = p+ 1 , for all 1 j p+ 1 .

5.8 Inheritance from quotientsThe common feature of all parent-descendant relations between nite p-groups is that the parent (G) is a quotientG/N of the descendant G by a suitable normal subgroup N / G . Thus, an equivalent denition can be given byselecting an epimorphism ' fromG onto a group ~Gwhose kernel ker(') plays the role of the normal subgroupN /G. In the following sections, this point of view will be taken, generally for arbitrary groups.

5.8.1 Passing through the abelianizationIf ' : G! A is a homomorphism from a group G to an abelian group A , then there exists a unique homomorphism~' : G/G0 ! A such that ' = ~' ! , where ! : G! G/G0 denotes the canonical projection. The kernel of ~' isgiven by ker( ~') = ker(')/G0 . The situation is visualized in Figure 1.The uniqueness of ~' is a consequence of the condition ' = ~' ! , which implies that ~' must be dened by~'(xG0) = ~'(!(x)) = ( ~' !)(x) = '(x) , for any x 2 G . The relation ~'(xG0 yG0) = ~'((xy)G0) = '(xy) =

5.8. INHERITANCE FROM QUOTIENTS 19

Figure 1: Factoring through the abelianization.

'(x) '(y) = ~'(xG0) ~'(xG0) , for x; y 2 G , shows that ~' is a homomorphism. For the commutator of x; y 2 G, we have '([x; y]) = ['(x); '(y)] = 1 , since A is abelian. Thus, the commutator subgroup G0 of G is contained inthe kernel ker(') , and this nally shows that the denition of ~' is independent of the coset representative, xG0 = yG0) x1y 2 G0 ker(')) ~'(xG0)1 ~'(yG0) = ~'(x1yG0) = '(x1y) = 1) ~'(xG0) = ~'(yG0) .

5.8.2 TTT singulets

Let G and ~G be groups such that ~G = '(G) is the image of G under an epimorphism ' : G! ~G and ~H = '(H)is the image of a subgroup H G .


Figure 2: Epimorphisms and derived quotients.

The commutator subgroup of ~H is the image of the commutator subgroup ofH , that is ~H 0 = '(H 0) . If ker(') H, then ~H ' H/ ker(') , ' induces a unique epimorphism ~' : H/H 0 ! ~H/ ~H 0 , and thus ~H/ ~H 0 is epimorphicimage of H/H 0 , that is a quotient of H/H 0 . Moreover, if even ker(') H 0 , then ~H 0 ' H 0/ ker(') , the map ~'is an isomorphism, and ~H/ ~H 0 ' H/H 0 . See Figure 2 for a visualization of this scenario.

5.8. INHERITANCE FROM QUOTIENTS 21

The statements can be seen in the following manner. The image of the commutator subgroup is'(H 0) = '([H;H]) ='(h[u; v] j u; v 2 Hi) = h['(u); '(v)] j u; v 2 Hi = ['(H); '(H)] = '(H)0 = ~H 0 . If ker(') H, then ' can be restricted to an epimorphism 'jH : H ! ~H , whence ~H = '(H) ' H/ ker(') . Accord-ing to the previous section, the composite epimorphism (! ~H 'jH) : H ! ~H/ ~H 0 from H onto the abeliangroup ~H/ ~H 0 factors through H/H 0 by means of a uniquely determined epimorphism ~' : H/H 0 ! ~H/ ~H 0 suchthat ~' !H = ! ~H 'jH . Consequently, we have ~H/ ~H 0 ' (H/H 0)/ ker( ~') . Furthermore, the kernel of~' is given explicitly by ker( ~') = ker(! ~H 'jH)/H 0 = (H 0 ker('))/H 0 . Finally, if ker(') H 0 , then~H 0 = '(H 0) ' H 0/ ker(') and ~' is an isomorphism, since ker( ~') = H 0/H 0 = 1 .Denition. [9]

Due to the results in the present section, it makes sense to dene a partial order on the set of abelian type invariantsby putting ~H/ ~H 0 H/H 0 , when ~H/ ~H 0 ' (H/H 0)/ ker( ~') , and ~H/ ~H 0 = H/H 0 , when ~H/ ~H 0 ' H/H 0 .

5.8.3 TKT singulets

Suppose thatG and ~G are groups, ~G = '(G) is the image ofG under an epimorphism ' : G! ~G , and ~H = '(H)is the image of a subgroup H G of nite index n = (G : H) . Let TG;H be the Artin transfer from G to H/H 0and T ~G; ~H be the Artin transfer from ~G to ~H/ ~H 0 .If ker(') H , then the image ('(g1); : : : ; '(gn)) of a left transversal (g1; : : : ; gn) of H in G is a left transversalof ~H in ~G , and the inclusion '(ker(TG;H)) ker(T ~G; ~H) holds. Moreover, if even ker(') H 0 , then the equation'(ker(TG;H)) = ker(T ~G; ~H) holds. See Figure 3 for a visualization of this scenario.The truth of these statements can be justied in the following way. Let (g1; : : : ; gn) be a left transversal of H in G .Then G = _[ni=1 giH is a disjoint union but '(G) = _[ni=1 '(gi)'(H) is not necessarily disjoint. For 1 j; k n ,we have '(gj)'(H) = '(gk)'(H), '(H) = '(gj)1'(gk)'(H) = '(g1j gk)'(H), '(g1j gk) = '(h) forsome element h 2 H , '(h1g1j gk) = 1, h1g1j gk =: k 2 ker(') . However, if the condition ker(') His satised, then we are able to conclude that g1j gk = hk 2 H , and thus j = k .Let ~' : H/H 0 ! ~H/ ~H 0 be the epimorphism obtained in the manner indicated in the previous section. For the imageofx 2 G under the Artin transfer, we have ~'(TG;H(x)) = ~'(

Qni=1 g

1x(i)

xgiH 0) =Qn

i=1 '(gx(i))1'(x)'(gi))

'(H 0) = . Since '(H 0) = '(H)0 = ~H 0 , the right hand side equals T ~G; ~H('(x)) , provided that ('(g1); : : : ; '(gn))is a left transversal of ~H in ~G , which is correct, when ker(') H . This shows that the diagram in Figure 3 iscommutative, that is ~' TG;H = T ~G; ~H ' . Consequently, we obtain the inclusion '(ker(TG;H)) ker(T ~G; ~H) ,if ker(') H . Finally, if ker(') H 0 , then the previous section has shown that ~' is an isomorphism. Using theinverse isomorphism, we get TG;H = ~'1 T ~G; ~H ' , which proves the equation '(ker(TG;H)) = ker(T ~G; ~H) .Denition. [9]

In view of the results in the present section, we are able to dene a partial order of transfer kernels by settingker(TG;H) ker(T ~G; ~H) , when'(ker(TG;H)) ker(T ~G; ~H) , and ker(TG;H) = ker(T ~G; ~H) , when'(ker(TG;H)) =ker(T ~G; ~H) .

5.8.4 TTT and TKT multiplets

Suppose G and ~G are groups, ~G = '(G) is the image of G under an epimorphism ' : G ! ~G , and both groupshave isomorphic nite abelianizations G/G0 ' ~G/ ~G0 . Let (Hi)i2I denote the family of all subgroups Hi /G whichcontain the commutator subgroup G0 (and thus are necessarily normal), enumerated by means of the nite index setI , and let ~Hi = '(Hi) be the image of Hi under ' , for each i 2 I . Assume that, for each i 2 I , Ti := TG;Hidenotes the Artin transfer from G to the abelianization Hi/H 0i , and ~Ti := T ~G; ~Hi denotes the Artin transfer from ~Gto the abelianization ~Hi/ ~H 0i . Finally, let J I be any non-empty subset of I .Then it is convenient to dene {H(G) = (ker(Tj))j2J , called the (partial) transfer kernel type (TKT) of Gwith respect to (Hj)j2J , and H(G) = (Hj/H 0j)j2J , called the (partial) transfer target type (TTT) of G withrespect to (Hj)j2J .Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey thefollowing fundamental inheritance laws:


Figure 3: Epimorphisms and Artin transfers.

1. If ker(') \j2J Hj , then ~H( ~G) H(G) , in the sense that ~Hj/ ~H 0j Hj/H 0j , for each

5.9. STABILIZATION CRITERIA 23

j 2 J , and {H(G) { ~H( ~G) , in the sense that ker(Tj) ker( ~Tj) , for each j 2 J .2. If ker(') \j2J H 0j , then ~H( ~G) = H(G) , in the sense that ~Hj/ ~H 0j = Hj/H 0j , for each

j 2 J , and {H(G) = { ~H( ~G) , in the sense that ker(Tj) = ker( ~Tj) , for each j 2 J .

5.8.5 Inherited automorphisms

A further inheritance property does not immediately concern Artin transfers but will prove to be useful in applicationsto descendant trees.Let G and ~G be groups such that ~G = '(G) ' G/ ker(') is the image of G under an epimorphism ' : G ! ~G .Suppose that 2 Aut(G) is an automorphism of G .If (ker(')) ker(') , then there exists a unique epimorphism ~ : ~G ! ~G such that ' = ~ ' . If(ker(')) = ker(') , then ~ 2 Aut( ~G) is also an automorphism.The justication for these facts is based on the isomorphic representation ~G = '(G) ' G/ ker(') , which permitsto identify ~(g ker('))=^~('(g)) = '((g))=^(g) ker(') for all g 2 G and proves the uniqueness of ~ . If(ker(')) ker(') , then the consistency follows from g ker(') = h ker(')) h1g 2 ker(')) (h1g) 2ker(') ) (g) ker(') = (h) ker(') . And if (ker(')) = ker(') , then injectivity of ~ is a consequence of~(g ker('))=^(g) ker(') = ker(')) (g) 2 ker(')) g = 1((g)) 2 ker(') , since 1(ker(')) ker(').Now, let us denote the canonical projection from G to its abelianization G/G0 by ! : G ! G/G0 . There exists aunique induced automorphism 2 Aut(G/G0) such that ! = ! , that is, (gG0) = (!(g)) = !((g)) =(g)G0 , for all g 2 G . The reason for the injectivity of is that (g)G0 = (gG0) = G0 ) (g) 2 G0 )g = 1((g)) 2 G0 , since G0 is a characteristic subgroup of G .Denition.G is called a -group, if there exists an automorphism 2 Aut(G) such that the induced automorphism acts likethe inversion on G/G0 , that is, (g)G0 = (gG0) = g1G0 , resp. (g) g1 (mod G0) , for all g 2 G .The supplementary inheritance property asserts that, if G is a -group and (ker(')) = ker(') , then ~G is also a -group, the required automorphism being ~ .This can be seen by applying the epimorphism ' to the equation (g)G0 = (gG0) = g1G0 , for g 2 G , whichyields ~(x) ~G0 = ~('(g)) ~G0 = '((g))'(G0) = '(g1)'(G0) = '(g)1 ~G0 = x1 ~G0 , for all x = '(g) 2'(G) = ~G .

5.9 Stabilization criteriaIn this section, the results concerning the inheritance of TTTs and TKTs from quotients in the previous section areapplied to the simplest case, which is characterized by the followingAssumption.The parent (G) of a group G is the quotient (G) = G/N of G by the last non-trivial term N = c(G) / G ofthe lower central series of G , where c denotes the nilpotency class of G . The corresponding epimorphism fromG onto (G) = G/c(G) is the canonical projection, whose kernel is given by ker() = c(G) .Under this assumption, the kernels and targets of Artin transfers turn out to be compatible with parent-descendantrelations between nite p-groups.Compatibility criterion.Let p be a prime number. Suppose that G is a non-abelian nite p-group of nilpotency class c = cl(G) 2 .Then the TTT and the TKT of G and of its parent (G) are comparable in the sense that ((G)) (G) and{(G) {((G)) .The simple reason for this fact is that, for any subgroupG0 H G , we have ker() = c(G) 2(G) = G0 H, since c 2 .For the remaining part of this section, the investigated groups are supposed to be nite metabelian p-groups G withelementary abelianization G/G0 of rank 2 , that is of type (p; p) .

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