Higher Composition Laws

A bstract ........................................................................................................................ iii
Acknowledgments ....................................................................................................... iv
1 Introduction 1
2 Q uadratic com position laws 5
2.1 On 2 x 2 x 2 cubes of integers ....................................................................... 5
2.1.1 The fundamental slicings ................................................................... 6
2.1.2 Composition of binary quadratic fo rm s ......................................... -7
2.1.3 Composition of 2 x 2 x 2 c u b e s ...................................................... 9
2.1.4 Composition of binary cubic fo rm s ................................................ 10
2.1.5 Composition of pairs of binary quadratic forms ........................ 11
2.2 Relations with ideal classes in quadratic orders ......................................... 12
2.2.1 The parametrization of quadratic rin g s ........................................ 12
2.2.2 The case of binary quadratic fo rm s ............................................... 13
2.2.3 The case of 2 x 2 x 2 cubes ............................................................. 14
2.2.4 The case of binary cubic forms .......................................................... 20
2.2.5 The case of pairs of binary quadratic form s .................................. 23
Appendix: Equivalence of the Cube Law and Gauss composition . . . 25
3 Cubic com position laws 27
3.1 On 2 x 3 x 3 boxes of integers ..................................................................... 28
3.1.1 The unique T-invariant Disc(A, B) ................................................ 29
3.1.2 The parametrization of cubic rin g s ................................................ 29
3.1.3 Cubic rings and 2 x 3 x 3 integer boxes ......................................... 31
3.1.4 Cubic rings and pairs of ternary quadratic form s ........................ 36
3.2 Resulting composition laws ............................................................................ 38
3.2.1 Composition of 2 x 3 x 3 integer matrices .................................. 38
3.2.2 Composition of pairs of ternary quadratic form s ........................ 40
4 The param etrization of quartic rings 42
4.1 Resolvent rings and param etrizations ........................................................ 43
4.1.1 The Sfc-closure of a ring of rank k ................................................... 43
4.1.2 The quadratic resolvent of a cubic rin g ......................................... 45
4.1.3 Cubic resolvents of a quartic rin g .................................................. -47
4.2 Quartic rings and pairs of ternary quadratic form s ................................. 50
4.2.1 The fundamental invariant Disc(A, B) ........................................ 50
4.2.2 How much of the structure of Q is determined by (A, B )? . . . 51
4.2.3 How much of the structure of R is determined by (-4, B)? . . . 56
4.2.4 Is R the cubic resolvent of Q ? ........................................................ 58
4.2.5 The main result .................................................................................... 58
4.2.6 The content of a rin g .......................................................................... 59
4.2.7 Invariant theory of pairs of ternary quadratic forms I I ................ 62
4.2.8 Isolating Q ........................................................................................... 65
4.2.9 Local behaviour ..................................................................................... 66
4.2.10 Maximal quartic rings.......... ................................................................. 69
5 The density of discrim inants of quartic rings and fields 74
5.1 On the class numbers of pairs of ternary quadratic fo rm s .................... 77
5.1.1 Reduction th eo ry ................................................................................ 78
5.1.2 Some further n o ta tio n ...................................................................... 80
5.1.3 Preliminary estimates .......................................................................... 80
5.1.4 Estimates on reducible pairs (.4, B ) ................................................ 82
5.1.5 Cutting the cusps ................................................................................ 89
5.1.6 Proof of Lemma 5.9 ............................................................................. 93
5.1.7 Computation of the fundamental volum e ...................................... 102
5.2 Pairs of ternary quadratic forms and Theorems 5 .1 -5 .4 ....................... 105
5.2.1 Nowhere overramified quartic field s ................................................ 106
5.2.2 A uniformity e stim a te ...................................................................... 107
5.2.3 Proofs of Theorems 5.1-5.4 ...................................................... 110
Appendix: The quadratic covariant Q ......................................................... 114
6 Conclusion 116
6.1 Higher composition laws and exceptional groups .................................... 116
6.2 Modular forms on exceptional groups ........................................................ 119
6.3 Higher composition laws and prehomogeneous vector spaces ............ 120
6.4 Computational applications ........................................................................ 121
Summary of higher composition la w s ........................................................ 122