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