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An Introduction to the Theory of Local Zeta Functions
Jun-ichi Igusa, Johns Hopkins University, Baltimore, MD
A co-publication of the AMS and International Press of Boston, Inc..
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2000; 232 pp; softcover
Volume: 14
ISBN-10: 0-8218-2907-6
ISBN-13: 978-0-8218-2907-3
List Price: US$52 Member Price: US$41.60
Order Code: AMSIP/14.S

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 co-published with International Press of Boston, Inc., Cambridge, MA.

Reviews

"This nice book ... is the first one providing an introduction to this theory; it is essentially self-contained 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 algebro-geometric 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 ... self-contained ... covers everything needed to begin investigating local zeta functions ... has broad scope and meticulous self-sufficiency ... full of concrete examples ... thorough and well-organized ... 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

• 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