The p-adic L-function

Abstract

In this chapter we return to p-adic integration theory, and give Mazur’s formulation of the p-adic L-function as Mellin transform. It turns out to be more convenient as a basic definition, than Iwasawa’s previous formulation in terms of power series. The connection is made via Example 2 of §1. We derive further analytic properties, which allow us to make explicit its value at s = 1, thereby obtaining Leopoldt’s formula in the p-adic case, analogous to that of the complex case. We also give Leopoldt’s version of the p-adic class number formula and regulator.