Traditionally, $p$-adic $L$-functions have been constructed from complex $L$-functions via special values and Iwasawa theory. In this volume, Perrin-Riou presents a theory of $p$-adic $L$-functions coming directly from $p$-adic Galois representations (or, more generally, from motives). This theory encompasses, in particular, a construction of the module of $p$-adic $L$-functions via the arithmetic theory and a conjectural definition of the $p$-adic $L$-function via its special values.

Since the original publication of this book in French (see Astérisque 229, 1995), the field has undergone significant progress. These advances are noted in this English edition. Also, some minor improvements have been made to the text.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students and research mathematicians interested in number theory.

Reviews

"Written in a concise but readable style and can be recommended to readers interested in this rapidly growing subject."

-- European Mathematical Society Newsletter

Table of Contents

• Construction of the module of $p$-adic $L$-functions without factors at infinity
• Modules of $p$-adic $L$-functions of $V$
• Values of the module of $p$-adic $L$-functions
• The $p$-adic $L$-function of a motive
• Results in Galois cohomology
• The weak Leopoldt conjecture
• Local Tamagawa numbers and Euler-Poincaré characteristic. Application to the functional equation
• Bibliography
• Index