Leopoldt conjecture

Let be a totally real algebraic number field (cf. also Field; Algebraic number) and let be a prime number. Let denote the distinct embeddings of into the completion of the algebraic closure of . By the Dirichlet unit theorem (cf. also Dirichlet theorem), the unit group of has rank . Let be a -basis of . In [a5], H.-W. Leopoldt defined the -adic regulator as the -adic analogue of the Dirichlet regulator:

where denotes the -adic logarithm.

Leopoldt's conjecture is: .

The definition of (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 that are Abelian over 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 , Leopoldt's conjecture is equivalent to the non-vanishing of the -adic -function at (cf. [a5], [a3]).

For a prime in , let denote the group of units of the local field . There is a canonical mapping

and the Leopoldt defect is defined as the -rank of . Class field theory yields the following equivalent formulation of the Leopoldt conjecture (cf. [a7]): Leopoldt's conjecture holds if and only if .

Relation to Iwasawa theory.

An extension of a number field is called a -extension if it is a Galois extension and . The number of independent -extensions of is related via class field theory to the -rank of and is equal to (cf. [a4]), where is the number of pairs of complex-conjugate embeddings of .

For , let denote the unique subfield of of degree over and let denote the Leopoldt defect of . The -extension satisfies the weak Leopoldt conjecture if the defects are bounded independent of . It is known (cf. [a4]) that the weak Leopoldt conjecture holds for the so-called cyclotomic -extension of , i.e. for the unique -extension contained in .

Relation to Galois cohomology.

Let denote the Galois group of the maximal pro--extension of , which is unramified outside . Leopoldt's conjecture is equivalent to the vanishing of the Galois cohomology group [a6]. More generally, it is conjectured that

for all [a6]. This is known to be true for as a consequence of a profound result of A. Borel in algebraic -theory.

References

[a1]	F. Bertrandias, J.-J. Payan, "-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 des fonctions zêta -adiques" Invent. Math. , 91 (1988) pp. 371–389
[a4]	K. Iwasawa, "On -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