An Introduction to the Theory of Local Zeta Functions ( AMS/IP Studies in Advanced Mathematics )

Publication series ：AMS/IP Studies in Advanced Mathematics

Author： Jun-ichi Igusa  

Publisher： American Mathematical Society‎

Publication year： 2017

E-ISBN: 9781470438050

P-ISBN(Paperback): 9780821829073

Subject： O156.4 Analytic number theory

Keyword： 暂无分类

Language： ENG

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An Introduction to the Theory of Local Zeta Functions

Description

This book is an introductory presentation to the theory of local zeta functions. Viewed as distributions, and mostly in the archimedean case, local zeta functions are also called complex powers. The volume contains major results on analytic and algebraic properties of complex powers by Atiyah, Bernstein, I. M. Gelfand, S. I. Gelfand, and Sato. Chapters devoted to $p$-adic local zeta functions present Serre's structure theorem, a rationality theorem, and many examples found by the author. The presentation concludes with theorems by Denef and Meuser.

Chapter

Cover

Title page

Contents

Introduction

Preliminaries

Implicit function theorems and 𝐾-analytic manifolds

Hironaka’s desingularization theorem

Bernstein’s theory

Archimedean local zeta functions

Prehomogeneous vector spaces

Totally disconnected spaces and 𝑝-adic manifolds

Local zeta functions (𝑝-adic case)

Some homogeneous polynomials

Computation of 𝑍(𝑠)

Theorems of Denef and Meuser

Bibliography

Index

Back Cover

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