Leopoldt conjecture

Let $F$ be a totally real algebraic number field (cf. also Field; Algebraic number) and let $p$ be a prime number. Let ${\sigma _ {1} \dots \sigma _ {r _ {1} } } : F \rightarrow {\mathbf C _ {p} }$ denote the distinct embeddings of $F$ into the completion $\mathbf C _ {p}$ of the algebraic closure of $\mathbf Q _ {p}$. By the Dirichlet unit theorem (cf. also Dirichlet theorem), the unit group $U _ {F}$ of $F$ has rank $r = r _ {1} - 1$. Let $\epsilon _ {1} \dots \epsilon _ {r}$ be a $\mathbf Z$- basis of $U _ {F}$. In [a5], H.-W. Leopoldt defined the $p$- adic regulator $R _ {p} ( F )$ as the $p$- adic analogue of the Dirichlet regulator:

$$R _ {p} ( F ) = \pm { \mathop{\rm det} } \left ( { \mathop{\rm log} } _ {p} ( \sigma _ {i} ( \epsilon _ {j} ) ) _ {1 \leq i,j \leq r } \right ) ,$$

where ${ { \mathop{\rm log} } _ {p} } : {U _ {F} } \rightarrow {\mathbf C _ {p} }$ denotes the $p$- adic logarithm.

Leopoldt's conjecture is: $R _ {p} ( F ) \neq 0$.

The definition of $R _ {p} ( F )$( and therefore also the conjecture) extends to arbitrary number fields (cf. [a7]) and is nowadays considered in this generality. A. Brumer used transcendental methods developed by A. Baker to prove Leopoldt's conjecture for fields $F$ that are Abelian over $\mathbf Q$ or over an imaginary quadratic field [a2]. For specific non-Abelian fields the conjecture has also been verified (cf., e.g., [a1]), but in general it is still (1996) open.

For a totally real field $F$, Leopoldt's conjecture is equivalent to the non-vanishing of the $p$- adic $\zeta$- function $\zeta _ {F,p } ( s )$ at $s = 1$( cf. [a5], [a3]).

For a prime $v$ in $F$, let $U _ {v}$ denote the group of units of the local field $F _ {v}$. There is a canonical mapping

$${f _ {p} } : {U _ {F} \otimes \mathbf Z _ {p} } \rightarrow {\prod _ { {v \mid p } } U _ {v} }$$

and the Leopoldt defect $\delta _ {F}$ is defined as the $\mathbf Z _ {p}$- rank of ${ \mathop{\rm ker} } f _ {p}$. Class field theory yields the following equivalent formulation of the Leopoldt conjecture (cf. [a7]): Leopoldt's conjecture holds if and only if $\delta _ {F} = 0$.

Relation to Iwasawa theory.

An extension $F _ \infty /F$ of a number field $F$ is called a $\mathbf Z _ {p}$- extension if it is a Galois extension and ${ \mathop{\rm Gal} } ( F _ \infty /F ) \cong \mathbf Z _ {p}$. The number of independent $\mathbf Z _ {p}$- extensions of $F$ is related via class field theory to the $\mathbf Z _ {p}$- rank of ${ \mathop{\rm coker} } f _ {p}$ and is equal to $1 + r _ {2} ( F ) + \delta _ {F}$( cf. [a4]), where $r _ {2} ( F )$ is the number of pairs of complex-conjugate embeddings of $F$.

For $n \geq 0$, let $F _ {n}$ denote the unique subfield of $F _ \infty /F$ of degree $p ^ {n}$ over $F$ and let $\delta _ {n}$ denote the Leopoldt defect of $F _ {n}$. The $\mathbf Z _ {p}$- extension $F _ \infty /F$ satisfies the weak Leopoldt conjecture if the defects $\delta _ {n}$ are bounded independent of $n$. It is known (cf. [a4]) that the weak Leopoldt conjecture holds for the so-called cyclotomic $\mathbf Z _ {p}$- extension of $F$, i.e. for the unique $\mathbf Z _ {p}$- extension contained in $F ( \mu _ {p ^ \infty } )$.

Relation to Galois cohomology.

Let $G _ {p} ( F )$ denote the Galois group of the maximal pro- $p$- extension of $F$, which is unramified outside $p$. Leopoldt's conjecture is equivalent to the vanishing of the Galois cohomology group $H ^ {2} ( G _ {p} ( F ) , \mathbf Q _ {p} / \mathbf Z _ {p} )$[a6]. More generally, it is conjectured that

$$H ^ {2} ( G _ {p} ( F ) , \mathbf Q _ {p} / \mathbf Z _ {p} ( i ) ) = 0$$

for all $i \neq 1$[a6]. This is known to be true for $i \geq 2$ as a consequence of a profound result of A. Borel in algebraic $K$- theory.

References

[a1]	F. Bertrandias, J.-J. Payan, "$\Gamma$-extensions et invariants cyclotomiques" Ann. Sci. Ecole Norm. Sup. (4) , 5 (1972) pp. 517–543 MR0480419 MR0337882 Zbl 0246.12005 Zbl 0246.12004
[a2]	A. Brumer, "On the units of algebraic number fields" Mathematica , 14 (1967) pp. 121–124 MR0220694 Zbl 0171.01105
[a3]	P. Colmez, "Résidu en $s=1$ des fonctions zêta $p$-adiques" Invent. Math. , 91 (1988) pp. 371–389
[a4]	K. Iwasawa, "On $\ZZ_\ell$-extensions of algebraic number fields" Ann. of Math. , 98 (1973) pp. 246–326 MR349627
[a5]	H.-W. Leopoldt, "Zur Arithmetik in abelschen Zahlkörpern" J. Reine Angew. Math. , 209 (1962) pp. 54–71 MR0139602 Zbl 0204.07101
[a6]	P. Schneider, "Über gewisse Galoiscohomologiegruppen" Math. Z. , 168 (1979) pp. 181–205 MR0544704 Zbl 0421.12024
[a7]	L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982) MR0718674 Zbl 0484.12001