View source for Leopoldt conjecture

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:

<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>

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.

Leopoldt's conjecture is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110120/l11012021.png" />.

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.

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]]]).

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

<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>

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" />.

==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" />.

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" />.

==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

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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]].

====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>