Lectures On P-adic L-functions

LECTURES ON p-ADIC L-FUNCTIONS BY KENKICHI IWASA WA PRINCETON UNIVERSITY PRESS AND UNNERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1972 Annals of Mathematics Studies Number 74 Copyright © 1972 �y Princeton University Press All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. LC Card: 78-39058 ISBN: 0-691-08112-3 AMS 1971: 10.14, 10.65, 12.50 Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press Printed in the United States of America PREFACE These are notes of lectures given at Princeton University during the fall semester of 1969. The notes present an introduction to p-adic L­ functions originated in Kubota-Leopoldt [10] as p-adic analogues of classi­ cal L-functions of Dirichlet. An outline of the contents is as follows. In §1, classical results on Dirichlet's L-functions are briefly reviewed. For some of these, a sketch of a proof is provided in the Appendix. In §2, we define generalized Ber­ noulli numbers following Leopoldt [121 and discuss some of the fundamen­ tal p roperties of these numbers. In §3, we introduce p-adic L-functions and prove the existence and the uniqueness of such functions; our method [10]. §4 consists of preliminary remarks p-adic regulators. In §S, we prove a formula of is slightly different from that in on p-adic logarithms and Leopo1dt for the values of p-adic L-functions at s announced in [10], = 1. The formula was but the proof has not yet been published. With his per­ mission, we describe here Leopoldt's original p roof of the formula (see [1], [7] for a1 temate approach). In §6, we explain another method to de­ fine p-adic L-functions . Here we follow an idea in [9] motivated by the study of cyclotomic fields. In §7, we discuss some applications of the results obtained in the preceding sections, indicating deep relations which exist between p-adic L-functions and cyclotomic fields. Conclud­ in g remarks on problems and future investigations in this area are also mentioned briefly at the end of §7. Throughout the notes, it is assumed that the reader has basic knowl­ edge of al gebraic number theory as presented, for examp le, in Borevich­ Shafarevich [2] or Lang [11]. However, except in few places where cer­ tain facts on L-functions and class numbers ar� referred to, no deeper v understanding of that theory may be required to follow the el ementary arguments in most of these notes. As for the notations, some of the symbols used throughout the notes are as follows: Z, Q, R, and C denote the ring of (rational) integers, the field of rational numbers, the field of real numbers , and the field of complex numbers, respectively. Z p and Qp will denote the ring of p­ adic integers and the field of p-adic numbers, respectively, p being, of course, a prime number. In general, i f R is a commutative ring with a unit, RX denotes the multiplicative group of all invertible elements in R, and R[[x]] the ring