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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101201.png" /> be a totally real algebraic number field (cf. also [[Field|Field]]; [[Algebraic number|Algebraic number]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101202.png" /> be a prime number. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101203.png" /> denote the distinct embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101204.png" /> into the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101205.png" /> of the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101206.png" />. By the Dirichlet unit theorem (cf. also [[Dirichlet theorem|Dirichlet theorem]]), the unit group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101207.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101208.png" /> has rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l1101209.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012010.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012011.png" />-basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012012.png" />. In [[#References|[a5]]], H.-W. Leopoldt defined the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012014.png" />-adic regulator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012015.png" /> as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012016.png" />-adic analogue of the Dirichlet regulator:
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012017.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012018.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012020.png" />-adic logarithm.
+
Let  $  F $
 +
be a totally real algebraic number field (cf. also [[Field|Field]]; [[Algebraic number|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|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 [[#References|[a5]]], H.-W. Leopoldt defined the  $  p $-
 +
adic regulator  $  R _ {p} ( F ) $
 +
as the  $  p $-
 +
adic analogue of the Dirichlet regulator:
  
Leopoldt's conjecture is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012021.png" />.
+
$$
 +
R _ {p} ( F ) = \pm  { \mathop{\rm det} } \left ( { \mathop{\rm log} } _ {p} ( \sigma _ {i} ( \epsilon _ {j} ) ) _ {1 \leq  i,j \leq  r }  \right ) ,
 +
$$
  
The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012022.png" /> (and therefore also the conjecture) extends to arbitrary number fields (cf. [[#References|[a7]]]) and is nowadays considered in this generality. A. Brumer used transcendental methods developed by A. Baker to prove Leopoldt's conjecture for fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012023.png" /> that are Abelian over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012024.png" /> or over an imaginary quadratic field [[#References|[a2]]]. For specific non-Abelian fields the conjecture has also been verified (cf., e.g., [[#References|[a1]]]), but in general it is still (1996) open.
+
where  $  { { \mathop{\rm log} } _ {p} } : {U _ {F} } \rightarrow {\mathbf C _ {p} } $
 +
denotes the $  p $-
 +
adic logarithm.
  
For a totally real field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012025.png" />, Leopoldt's conjecture is equivalent to the non-vanishing of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012028.png" />-adic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012029.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012030.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012031.png" /> (cf. [[#References|[a5]]], [[#References|[a3]]]).
+
Leopoldt's conjecture is: $  R _ {p} ( F ) \neq 0 $.
  
For a prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012033.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012034.png" /> denote the group of units of the local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012035.png" />. There is a canonical mapping
+
The definition of  $  R _ {p} ( F ) $(
 +
and therefore also the conjecture) extends to arbitrary number fields (cf. [[#References|[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 [[#References|[a2]]]. For specific non-Abelian fields the conjecture has also been verified (cf., e.g., [[#References|[a1]]]), but in general it is still (1996) open.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012036.png" /></td> </tr></table>
+
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. [[#References|[a5]]], [[#References|[a3]]]).
  
and the Leopoldt defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012037.png" /> is defined as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012038.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012039.png" />. [[Class field theory|Class field theory]] yields the following equivalent formulation of the Leopoldt conjecture (cf. [[#References|[a7]]]): Leopoldt's conjecture holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012040.png" />.
+
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|Class field theory]] yields the following equivalent formulation of the Leopoldt conjecture (cf. [[#References|[a7]]]): Leopoldt's conjecture holds if and only if $  \delta _ {F} = 0 $.
  
 
==Relation to Iwasawa theory.==
 
==Relation to Iwasawa theory.==
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012041.png" /> of a number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012042.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012044.png" />-extension if it is a [[Galois extension|Galois extension]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012045.png" />. The number of independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012046.png" />-extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012047.png" /> is related via class field theory to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012048.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012049.png" /> and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012050.png" /> (cf. [[#References|[a4]]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012051.png" /> is the number of pairs of complex-conjugate embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012052.png" />.
+
An extension $  F _  \infty  /F $
 +
of a number field $  F $
 +
is called a $  \mathbf Z _ {p} $-
 +
extension if it is a [[Galois extension|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. [[#References|[a4]]]), where $  r _ {2} ( F ) $
 +
is the number of pairs of complex-conjugate embeddings of $  F $.
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012053.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012054.png" /> denote the unique subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012055.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012056.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012057.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012058.png" /> denote the Leopoldt defect of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012059.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012060.png" />-extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012061.png" /> satisfies the weak Leopoldt conjecture if the defects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012062.png" /> are bounded independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012063.png" />. It is known (cf. [[#References|[a4]]]) that the weak Leopoldt conjecture holds for the so-called cyclotomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012065.png" />-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012066.png" />, i.e. for the unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012067.png" />-extension contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012068.png" />.
+
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. [[#References|[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.==
 
==Relation to Galois cohomology.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012069.png" /> denote the [[Galois group|Galois group]] of the maximal pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012070.png" />-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012071.png" />, which is unramified outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012072.png" />. Leopoldt's conjecture is equivalent to the vanishing of the [[Galois cohomology|Galois cohomology]] group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012073.png" /> [[#References|[a6]]]. More generally, it is conjectured that
+
Let $  G _ {p} ( F ) $
 +
denote the [[Galois group|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|Galois cohomology]] group $  H  ^ {2} ( G _ {p} ( F ) , \mathbf Q _ {p} / \mathbf Z _ {p} ) $[[#References|[a6]]]. More generally, it is conjectured that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012074.png" /></td> </tr></table>
+
$$
 +
H  ^ {2} ( G _ {p} ( F ) , \mathbf Q _ {p} / \mathbf Z _ {p} ( i ) ) = 0
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012075.png" /> [[#References|[a6]]]. This is known to be true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012076.png" /> as a consequence of a profound result of A. Borel in [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012077.png" />-theory]].
+
for all $  i \neq 1 $[[#References|[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|algebraic $  K $-
 +
theory]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Bertrandias, J.-J. Payan, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012078.png" />-extensions et invariants cyclotomiques" ''Ann. Sci. Ecole Norm. Sup. (4)'' , '''5''' (1972) pp. 517–543 {{MR|0480419}} {{MR|0337882}} {{ZBL|0246.12005}} {{ZBL|0246.12004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Brumer, "On the units of algebraic number fields" ''Mathematica'' , '''14''' (1967) pp. 121–124 {{MR|0220694}} {{ZBL|0171.01105}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Colmez, "Résidu en <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012079.png" /> des fonctions zêta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012080.png" />-adiques" ''Invent. Math.'' , '''91''' (1988) pp. 371–389</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Iwasawa, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012081.png" />-extensions of algebraic number fields" ''Ann. of Math.'' , '''98''' (1973) pp. 246–326 {{MR|349627}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H.-W. Leopoldt, "Zur Arithmetik in abelschen Zahlkörpern" ''J. Reine Angew. Math.'' , '''209''' (1962) pp. 54–71 {{MR|0139602}} {{ZBL|0204.07101}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Schneider, "Über gewisse Galoiscohomologiegruppen" ''Math. Z.'' , '''168''' (1979) pp. 181–205 {{MR|0544704}} {{ZBL|0421.12024}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982) {{MR|0718674}} {{ZBL|0484.12001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Bertrandias, J.-J. Payan, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012078.png" />-extensions et invariants cyclotomiques" ''Ann. Sci. Ecole Norm. Sup. (4)'' , '''5''' (1972) pp. 517–543 {{MR|0480419}} {{MR|0337882}} {{ZBL|0246.12005}} {{ZBL|0246.12004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Brumer, "On the units of algebraic number fields" ''Mathematica'' , '''14''' (1967) pp. 121–124 {{MR|0220694}} {{ZBL|0171.01105}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Colmez, "Résidu en <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012079.png" /> des fonctions zêta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012080.png" />-adiques" ''Invent. Math.'' , '''91''' (1988) pp. 371–389</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Iwasawa, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012081.png" />-extensions of algebraic number fields" ''Ann. of Math.'' , '''98''' (1973) pp. 246–326 {{MR|349627}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H.-W. Leopoldt, "Zur Arithmetik in abelschen Zahlkörpern" ''J. Reine Angew. Math.'' , '''209''' (1962) pp. 54–71 {{MR|0139602}} {{ZBL|0204.07101}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Schneider, "Über gewisse Galoiscohomologiegruppen" ''Math. Z.'' , '''168''' (1979) pp. 181–205 {{MR|0544704}} {{ZBL|0421.12024}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982) {{MR|0718674}} {{ZBL|0484.12001}} </TD></TR></table>

Revision as of 22:16, 5 June 2020


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, "-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
How to Cite This Entry:
Leopoldt conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leopoldt_conjecture&oldid=47614
This article was adapted from an original article by M. Kolster (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article