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Lang S. — Cyclotomic Fields II (Graduate Texts in Mathematics)
Lang S. — Cyclotomic Fields II (Graduate Texts in Mathematics)

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Название: Cyclotomic Fields II (Graduate Texts in Mathematics)

Автор: Lang S.


This second volume incorporates a number of results which were discovered and/or systematized since the first volume was being written. Again, I limit myself to the cyclotomic fields proper without introducing modular functions. As in the first volume, the main concern is with class number formulas, Gauss sums, and the like. We begin with the Ferrero-Washington theorems, proving Iwasawa's conjecture that the p-primary part of the ideal class group in the cyclotomic Zp-extension of a cyclotomic field grows linearly rather than exponentially. This is first done for the minus part (the minus referring, as usual, to the eigenspace for complex conjugation), and then it follows for the plus part because of results bounding the plus part in terms of the minus part. Kummer had already proved such results (e.g. if pxh- then pxhp+). These are now formulated in ways applicable to the Iwasawa invariants, following Iwasawa himself. After that we do what amounts to "Dwork theory," to derive the Gross-Koblitz formula expressing Gauss sums in terms of the p-adic gamma function. This lifts Stickelberger's theorem p-adically. Half of the proof relies on a course of Katz, who had first obtained Gauss sums as limits of certain factorials, and thought of using Washnitzer-Monsky cohomology to prove the Gross-Koblitz formula. Finally, we apply these latter results to the Ferrero-Greenberg theorem, showing that Lp'(O, X) not =0 under the appropriate conditions. We take this opportunity to introduce a technique of Washington, who defined the p-adic analogues of the Hurwitz partial zeta functions, in a way making it possible to parallel the treatment from the complex case to the p-adic case, but in a much more efficient way.
Read more at http://ebookee.org/Cyclotomic-Fields-II-Graduate-Texts-in-Mathematics_1269288.html#MVSZejTdR2VXeDxz.99

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1980

Количество страниц: 175

Добавлена в каталог: 22.08.2014

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Предметный указатель
$B_{k, \chi}$      9
Analytic      43
Artin — Hasse power series      76
Artin — Schreier curve      127
Associated analytic function      44
Associated rational function      45
Banach basis      106
Banach space      105
Barsky      82
Bernoulli distribution      142
Bernoulli numbers      9 15
Class numbers      15
completely continuous      108
Davenport — Hasse distribution      143
Diamond function      150
Dieudonne — Dwork lemma      76
Differential operator      89
Distribution relation of gamma function      74
Distribution relations      139
Dwork power series      79
Dwork Trace Formula      98
Dwork — Robba      119
Equidistribution      29
Exponential invariant      6 52
Ferrero — Greenberg      144
Ferrero — Washington      12
Formal multiplicative group      44
Frobenius      93 135
Gamma function      72 101
Gauss sums      99 103 143
Gross — Koblitz      103
Growth conditions      118
Hurwitz — Washington function      148
Iwasawa algebra      38
Iwasawa coefficients      6
Iwasawa congruences      12 27
Iwasawa invariants      5 52
Kummer lemma      69
L-function      9 146
Lifting      122
Linear invariants      6 52
Measures      2 37
Normal family      30
p-adic gamma function      72
p-adic L-function      9
p-rank      55
Partial zeta function      148
Probabilities      18
Pure group      143
Rank      55
Rational function of a measure      45
Stickelberger distribution      140
Stickelberger theorem      104
Twist      7
Unitization operator      46
Washington theorem      22
Washnitzer — Monsky cohomology      131
Washnitzer — Monsky ring      118
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