# Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields?

More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, unramified outside a finite set S of primes of K, containing the cyclotomic $\mathbb{Z}_p$ - extension Kcyc of K and whose Galois group G is abelian with an open subgroup isomorphic to ${\mathbb{Z}_p}^d$ for some $d$, then does there exist an element of the Iwasawa algebra $\mathbb{Z}_p[[G]]$ (or some localisation of it) whose values at finite order characters χ of $G$ are the special values of the Hecke L-functions $L_S(\chi, 1)$ (with the Euler factors at primes in S removed)?

When K is totally real Leopoldt's conjecture forces $d = 1$; but I'm interested in other cases. I know there is a 2-variable Iwasawa main conjecture for imaginary quadratic fields, which I understand has been proved by Rubin, but I'm just asking about the existence of the L-function (not about any connection to annihilators of class groups). What is known in this direction for more general $K$?

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Hey David. I don't know so much about this stuff, but I always found the introduction to Katz' "p-adic L-functions for CM fields" (Inventiones 49, 1978) very helpful. In particular he points out that in many cases (in this generality) the p-adic L-function should be the zero function for "trivial reasons". –  Kevin Buzzard Mar 9 '10 at 12:37
Thanks, that paper is really useful and seems to answer all of my questions -- I will study it carefully. –  David Loeffler Mar 9 '10 at 13:00
"I will study it carefully" is perhaps the compulsory response when your question is answered by your PhD supervisor ;-) –  Kevin Buzzard Mar 9 '10 at 13:38
Hi David, I just saw this question. I guess you worked everything out, buy now, but just in case... Hida and Tilouine, for instance, discuss $p$-adic $L$ functions for arbitrary CM fields, which have many variables (something like half of the degree plus one plus Leopoldt defect). They describe this nicely in the introduction to their Inventiones, 1994. –  Filippo Alberto Edoardo Sep 29 '12 at 7:55