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 ""Asterisque"" 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.
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-Poincare characteristic. Application to the functional equation Bibliography Index.