P-adic L-function

The Dirichlet L-function is given by the analytic continuation of :L(s,\chi) = \sum_n\frac{\chi(n)}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-\chi(p)p^{-s}} The Dirichlet L-function at negative integers is given by :L(1-n, \chi) = -\frac{B_{n,\chi}}{n} where Bn,χ is a generalized Bernoulli number defined by : \displaystyle \sum_{n=0}^\infty B_{n,\chi}\frac{t^n}{n!} = \sum_{a=1}^f\frac{\chi(a)te^{at}}{e^{ft}-1} for χ a Dirichlet character with conductor f. ==Definition using interpolation==

The Kubota–Leopoldt p-adic L-function Lp(s, χ) interpolates the Dirichlet L-function with the Euler factor at p removed. More precisely, Lp(s, χ) is the unique continuous function of the p-adic number s such that : L_p(1-n, \chi) = \left(1-\chi(p)p^{n-1}\right) L(1-n, \chi) for positive integers n divisible by p − 1. The right hand side is just the usual Dirichlet L-function, except that the Euler factor at p is removed, otherwise it would not be p-adically continuous. The continuity of the right hand side is closely related to the Kummer congruences. When n is not divisible by p − 1 this does not usually hold; instead : L_p(1-n, \chi) = \left(1-\chi\omega^{-n}(p)p^{n-1}\right) L\left(1-n, \chi\omega^{-n}\right) for positive integers n. Here χ is twisted by a power of the Teichmüller character ω. ==Viewed as a p-adic measure==

p-adic L-functions can also be thought of as p-adic measures (or p-adic distributions) on p-profinite Galois groups. The translation between this point of view and the original point of view of Kubota–Leopoldt (as Qp-valued functions on Zp) is via the Mazur–Mellin transform (and class field theory). ==Totally real fields==

, building upon previous work of , constructed analytic p-adic L-functions for totally real fields. Independently, and did the same, but their approaches followed Takuro Shintani's approach to the study of the L-values. ==References==