AMS/IP Studies in Advanced Mathematics 2000; 232 pp; softcover Volume: 14 ISBN10: 0821829076 ISBN13: 9780821829073 List Price: US$49 Member Price: US$39.20 Order Code: AMSIP/14.S
Temporarily out of stock. Expected date of availability is June 12, 2014.
 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. Titles in this series are copublished with International Press, Cambridge, MA. Readership Advanced undergraduates, graduate students and research mathematicians interested in number theory. Reviews "This nice book ... is the first one providing an introduction to this theory; it is essentially selfcontained and very well written, and fills the serious background gap that everybody starting to work in this field was confronted with ... "There is an excellent chapter on Bernstein polynomials ... a remarkable short introduction to algebraic geometry and an interesting explanation of the notion of `good reduction modulo almost all primes \(p\)' ... nice algebrogeometric preparations ... a very good and extremely useful introduction to this theory."  Mathematical Reviews "A substantial addition to the literature ... The book is distinguished not only by the depth and breadth of the material that it introduces but also by the care and thoroughness with which it does so ... selfcontained ... covers everything needed to begin investigating local zeta functions ... has broad scope and meticulous selfsufficiency ... full of concrete examples ... thorough and wellorganized ... clear pedagogical goals ... many intriguing problems ... [contains] all the necessary tools from a wide variety of areas ... lays out Professor Igusa's personal view of the subject that he has done so much to create and contains insights and reminiscences that could only come from him."  Bulletin of the AMS Table of Contents  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
