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

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edited Jan 31, 2014 at 7:31

Olivier

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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 K_cyc 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$K$?

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 K_cyc 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?

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 K_cyc 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 $K$?