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Berg M.C. — The Fourier-Analytic Proof of Quadratic Reciprocity
Berg M.C. — The Fourier-Analytic Proof of Quadratic Reciprocity



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Название: The Fourier-Analytic Proof of Quadratic Reciprocity

Автор: Berg M.C.

Аннотация:

A unique synthesis of the three existing Fourier-analytic treatments of quadratic reciprocity.

The relative quadratic case was first settled by Hecke in 1923, then recast by Weil in 1964 into the language of unitary group representations. The analytic proof of the general n-th order case is still an open problem today, going back to the end of Hecke's famous treatise of 1923. The Fourier-Analytic Proof of Quadratic Reciprocity provides number theorists interested in analytic methods applied to reciprocity laws with a unique opportunity to explore the works of Hecke, Weil, and Kubota.

This work brings together for the first time in a single volume the three existing formulations of the Fourier-analytic proof of quadratic reciprocity. It shows how Weil's groundbreaking representation-theoretic treatment is in fact equivalent to Hecke's classical approach, then goes a step further, presenting Kubota's algebraic reformulation of the Hecke-Weil proof. Extensive commutative diagrams for comparing the Weil and Kubota architectures are also featured.

The author clearly demonstrates the value of the analytic approach, incorporating some of the most powerful tools of modern number theory, including adèles, metaplectric groups, and representations. Finally, he points out that the critical common factor among the three proofs is Poisson summation, whose generalization may ultimately provide the resolution for Hecke's open problem.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 115

Добавлена в каталог: 09.04.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute quadratic reciprocity      xviii
Absolute quadratic reciprocity law      17
Adelic character      71 93
Adelic cocycles      92 ff
Adelic splitting over the rational points      33 84
Adelization      71 ff 92
Analytic proof of higher reciprocity      103 ff
Analytic proof of quadratic reciprocity      1 &c.
Anisotropic forms      68
Artin root number      107
Artin’s reciprocity law      17
Automorphic form of half-integral weight      xiv 2
Automorphic functions      xvi 14 54 80 112
Automorphic representation      xv
Automorphy      xiv 52
Bilinear form      25 ff 66 80
Borge Jessen’s Theorem      72
Cauchy, Augustin-Louis      vii xiv 2
Central extension      xv 21 27 42 78
Cocycle      xvi 14 21
Cocycle, Kubota’s      86 ff
Cover, double      xv 27 43 77 91 101
Covers, metaplectic      xvi ff 64 99
Covers, symplectic      106
Deligne, Pierre      108
Double cover      xv 27 43 77 91 101
Dual pairing      32
Eisenstein, Gotthold      2
Ergaenzungssaetze      1 19
Euler, Leonhard      xiii xviii 1
Even more gruesome diagram      xx 102
Fourier transform, generalized      36 104
Gauss sums      xiv ff 2
Gauss — Euler (quadratic) reciprocity      xvii ff 1 103
Gauss — Hecke sums      6 &c
Gauss, Carl Friedrich      xii 1
Gaussian kernel (or density)      104 107
Gelbart, Steve      28 57
Genera      xiii 2
Generalized Fourier transform      36 104
Generalized theta functions      xv 27 93
Grothendieck group      66 ff
Group extensions      xx 31 83
Gruesome diagram      xx 99—102
Hahn — Banach theorem      53
Hamlet      102
Harmonic analysis      viii 79 110
Hecke, Erich      vii ff 1
Heisenberg group      xix ff 28
Heisenberg group, linearized      29 ff
Higher Kubota formalism      106
Higher metaplectic groups      106
Higher Weil formalism      106
Hilbert reciprocity      xiv ff 16 17 103
Hilbert symbol, quadratic      16 17
Hilbert, David      xiv ff
Howe, Roger      28
Hyperbolic spaces (planes)      67
Isometry (of quadratic spaces)      65 ff
Isotropy      xix 66 79 104
Kronecker product      67
Kubota formalism, higher      106
Kubota, Tomio      xvi &c
Lam, T.Y.      xix 66
Laumon, Gerard      xvii 108
Legendre symbol      xii xviii 1 3 16 62 105
Legendre symbol, generalized      14
Linearized Heisenberg group      29 ff
Locally compact abelian (=LCA) groups      xix
Mackey, George      xix 26 75
Metaplectic covers      xvi 64 99
Metaplectic groups      31 73 98 111
Metaplectic groups, higher      105
Moore, С.С.      xvi 27 30 83
Multiplier representation      30 71 76
normalizer      33
Plancherel’s theorem      5 36 53
Poisson summation      xv ff 27 80 105
Pontryagin dual      21 28 32
Pontryagin duality      48 ff 57
Primary algebraic integer      11 ff
Projective representation      15 30 49 76
Quadratic forms      xiv 3
Quadratic forms, analytic theory of      xv
Quadratic Hilbert symbol      xv xviii 65 69 105
Quadratic reciprocity      i &c 4
Quadratic reciprocity (absolute)      xviii
Quadratic reciprocity (Gauss — Euler)      xvii ff 101 108
Quadratic reciprocity (Hilbert)      xvi ff 16 17
Quadratic reciprocity law      xv 2
Quadratic reciprocity law (absolute)      17
Quadratic reciprocity law (relative)      xv 2 9 14
Quadratic spaces      66 ff
Quaternions over a local field      63 ff
Reciprocity law, Artin’s      17
Reciprocity law, higher      vii xvi 104
Reciprocity, quadratic      see “Quadratic reciprocity (lots of entries)”
Remak — Klein — Fricke Theorem      99
Representation, unitary      32 ff
Representation, Weil [projective]      xv &c
Schroedinger model      50
Schur’s lemma      30 76
Schwartz — Bruhat functions      29 ff 105
Second-degree characters      44 48 58
Segal, I.E.      28
Semi-direct product      27 35 50 102
Shimura, Goro      108
Siegel, Carl Ludwig      xv
Skew-symmetric bilinear form      25 ff
Splitting of primes      xiii 2
Stone — Von Neumann theorem      xviii ff 20 73
Symplectic covers      106
Theta constants      3 ff 109
Theta functions, generalized      xv 27 93
Theta functions, Hecke      3 ff
Theta series      52 111
Theta-functional (Weil)      80 92
Totally imaginary number field      xvi
Totally isotropic forms      68
Transfer of Hilbert space structure      35 37
Unitary groups      xv 110
Unitary representations      32 ff
Varadarajan, V.S.      xix 20 107
Vector spaces over local fields      57 ff
Weil formalism, higher      107
Weil index      61 63
Weil theta functional      80 92
Weil [projective] representation      xv &c
Weil — Kubota formalism      103 ff
Weil, Andre      vii &c
Weyl group (physics)      28
Weyl, Hermann      28
Witt group      66 67 70
Witt — Grothendieck group      66 67
Witt — Grothendieck ring      66 67
Zahlbericht      xiv 16
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