Introduction to P-Adic Analytic Number Theory

by Murty, Maruti Ram

• ISBN13: 9780821832622
• ISBN10: 082183262X
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Amer Mathematical Society

Summary

This book is an elementary introduction to $p$-adic analysis from the number theory perspective. With over 100 exercises included, it will acquaint the non-expert to the basic ideas of the theory and encourage the novice to enter this fertile field of research. The main focus of the book is the study of $p$-adic $L$-functions and their analytic properties. It begins with a basic introduction to Bernoulli numbers and continues with establishing the Kummer congruences. Thesecongruences are then used to construct the $p$-adic analog of the Riemann zeta function and $p$-adic analogs of Dirichlet's $L$-functions. Featured is a chapter on how to apply the theory of Newton polygons to determine Galois groups of polynomials over the rational number field. As motivation for furtherstudy, the final chapter introduces Iwasawa theory. The book treats the subject informally, making the text accessible to non-experts. It would make a nice independent text for a course geared toward advanced undergraduates through beginning graduate students.

Table of Contents

Preface	ix
Historical Introduction	1	(8)
Bernoulli Numbers	9	(16)
p-adic Numbers	25	(18)
Hensel's Lemma	43	(14)
p-adic Interpolation	57	(14)
p-adic L-functions	71	(16)
p-adic Integration	87	(12)
Leopoldt's Formula for Lp(1,χ)	99	(14)
Newton Polygons	113	(22)
An Introduction to Iwasawa Theory	135	(10)
Bibliography	145	(3)
Index	148