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An introduction to the theory of local zeta functions

Author: Jun-ichi Igusa
Publisher: Providence, R.I. : American Mathematical Society ; [Cambridge, Mass.] : International Press, ©2000.
Series: AMS/IP studies in advanced mathematics, v. 14.
Edition/Format:   Book : EnglishView all editions and formats
Database:WorldCat
Summary:

An introductory presentation to the theory of local zeta functions. As distributions, and mostly in the archimedian case, local zeta functions are called complex powers. The volume contains major  Read more...

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Document Type: Book
All Authors / Contributors: Jun-ichi Igusa
ISBN: 082182015X 9780821820155
OCLC Number: 43296790
Description: xii, 232 p. ; 26 cm.
Contents: 1 Preliminaries 1 --
1.1 Review of some basic theorems 1 --
1.2 Noetherian rings 5 --
1.3 Hilbert's theorems 8 --
2 Implicit function theorems and K-analytic manifolds 15 --
2.1 Implicit function theorem 15 --
2.2 Implicit function theorem (non-archimedean case) 21 --
2.3 Weierstrass preparation theorem 24 --
2.4 K-analytic manifolds and differential forms 28 --
2.5 Critical sets and critical values 32 --
3 Hironaka's desingularization theorem 35 --
3.1 Monoidal transformations 35 --
3.2 Hironaka's desingularization theorem (analytic form) 38 --
3.3 Desingularization of plane curves 40 --
4 Bernstein's theory 45 --
4.1 Bernstein's polynomial b[subscript f](s) 45 --
4.2 Some properties of b[subscript f](s) 47 --
4.3 Reduction of the proof 49 --
4.4 A general theorem on D-modules 52 --
4.5 Completion of the proof 55 --
5 Archimedean local zeta functions 59 --
5.1 Group [Omega](K[superscript x]) 59 --
5.2 Schwartz space S(K[superscript n]) 61 --
5.3 Local zeta function Z[subscript Phi]([omega]) 67 --
5.4 Complex power [omega](f) via desingularization 73 --
5.5 An application 77 --
6 Prehomogeneous vector spaces 83 --
6.1 Sato's b-function b(s) 83 --
6.2 [Gamma]-function (a digression) 87 --
6.3 b(s) = b[subscript f](s) and the rationality of the zeros 91 --
7 Totally disconnected spaces and p-adic manifolds 97 --
7.1 Distributions in totally disconnected spaces 97 --
7.2 Case of homogeneous spaces 101 --
7.3 Structure of eigendistributions 106 --
7.4 Integration on p-adic manifolds 108 --
7.5 Serre's theorem on compact p-adic manifolds 113 --
7.6 Integration over the fibers 114 --
8 Local zeta functions (p-adic case) 117 --
8.1 Selfduality of K and some lemmas 117 --
8.2 p-adic zeta function Z[subscript Phi]([omega]) 120 --
8.3 Weil's functions F[subscript Phi](i) and F*[subscript Phi](i*) 125 --
8.4 Relation of F[subscript Phi](i) and Z[subscript Phi]([omega]) 129 --
8.5 Poles of [omega](f) for a group invariant f 134 --
9 Some homogeneous polynomials 137 --
9.1 Quadratic forms and Witt's theorem 137 --
9.2 Quadratic forms over finite fields 141 --
9.3 Classical groups over finite fields 145 --
9.4 Composition and Jordan algebras 149 --
9.5 Norm forms and Freudenthal quartics 154 --
9.6 Gauss' identity and its corollaries 160 --
10 Computation of Z(s) 163 --
10.1 Z([omega]) in some simple cases 163 --
10.2 A p-adic stationary phase formula 167 --
10.3 A key lemma 173 --
10.4 Z(s) for a Freudenthal quartic 178 --
10.5 Z(s) for the Gramian det([superscript t]xhx) 184 --
10.6 An integration formula 188 --
10.7 Z(s) for det([superscript t]xhx) in product forms 193 --
11 Theorems of Denef and Meuser 199 --
11.1 Regular local rings 199 --
11.2 Geometric language 202 --
11.3 Hironaka's desingularization theorem (algebraic form) 205 --
11.4 Weil's zeta functions over finite fields 210 --
11.5 Degree of Z(s) 214 --
11.6 Field K[subscript e] (a digression) 217 --
11.7 Functional equation of Z(s) 221.
Series Title: AMS/IP studies in advanced mathematics, v. 14.
Responsibility: Jun-ichi Igusa.

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The book is distinguished ... self contained ... covers everything needed. Bulletin of the AMS

 
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