Leopoldt's conjecture

<h2 id="formulation">Formulation</h2>
Let K be a <a href="/facts/Number_field/e0K0TXSh">number field</a> and for each <a href="/facts/Prime_number/L20FLkgn">prime</a> P of K above some fixed rational prime p, let UP denote the local units at P and let U1,P denote the subgroup of principal units in UP. Set

U
          
            1
          
        
        =
        
          ∏
          
            P
            
              |
            
            p
          
        
        
          U
          
            1
            ,
            P
          
        
        .
      
    
    {\displaystyle U_{1}=\prod _{P|p}U_{1,P}.}

Then let E1 denote the set of global units ε that map to U1 via the <a href="/facts/Diagonal_embedding/0BjhsmC7">diagonal embedding</a> of the global units in E.
Since 
 
 
 
 
 E
 
 1
 
 
 
 
 {\displaystyle E_{1}}
 
 is a finite-<a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> subgroup of the global units, it is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> of rank 
 
 
 
 
 r
 
 1
 
 
 +
 
 r
 
 2
 
 
 −
 1
 
 
 {\displaystyle r_{1}+r_{2}-1}
 
, where 
 
 
 
 
 r
 
 1
 
 
 
 
 {\displaystyle r_{1}}
 
 is the number of real embeddings of 
 
 
 
 K
 
 
 {\displaystyle K}
 
 and 
 
 
 
 
 r
 
 2
 
 
 
 
 {\displaystyle r_{2}}
 
 the number of pairs of complex embeddings. Leopoldt's conjecture states that the 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
-module rank of the closure of 
 
 
 
 
 E
 
 1
 
 
 
 
 {\displaystyle E_{1}}
 
 embedded diagonally in 
 
 
 
 
 U
 
 1
 
 
 
 
 {\displaystyle U_{1}}
 
 is also 
 
 
 
 
 r
 
 1
 
 
 +
 
 r
 
 2
 
 
 −
 1.
 
 
 {\displaystyle r_{1}+r_{2}-1.}

Leopoldt's conjecture is known in the special case where 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is an <a href="/facts/Abelian_extension/vIdW4Hnv">abelian extension</a> of 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 or an abelian extension of an imaginary <a href="/facts/Quadratic_field/dUnSDN4e">quadratic number field</a>: Ax (1965) reduced the abelian case to a p-adic version of <a href="/facts/Baker%2527s_theorem/GXcd8RbV">Baker's theorem</a>, which was proved shortly afterwards by Brumer (1967).
<a href="/facts/Preda_Mih%25C4%2583ilescu/yIuayGhd">Mihăilescu</a> (2009, 2011) has announced a proof of Leopoldt's conjecture for all CM-extensions of 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
.
<a href="/facts/Pierre_Colmez/lAShIbuC">Colmez</a> (1988) expressed the residue of the p-adic <a href="/facts/Dedekind_zeta_function/Nj8PwFo8">Dedekind zeta function</a> of a <a href="/facts/Totally_real_field/4rulk9kd">totally real field</a> at s = 1 in terms of the p-adic regulator. As a consequence, Leopoldt's conjecture for those fields is equivalent to their p-adic Dedekind zeta functions having a simple pole at s = 1.

<ul><li>Ax, James (1965), <a href="http://projecteuclid.org/euclid.ijm/1256059299">"On the units of an algebraic number field"</a>, Illinois Journal of Mathematics, 9 (4): 584–589, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2Fijm%2F1256059299">10.1215/ijm/1256059299</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0019-2082">0019-2082</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0181630">0181630</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0132.28303">0132.28303</a></li>
<li>Brumer, Armand (1967), "On the units of algebraic number fields", <a href="/facts/Mathematika/oKErqS6j">Mathematika</a>, 14 (2): 121–124, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2FS0025579300003703">10.1112/S0025579300003703</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5793">0025-5793</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0220694">0220694</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0171.01105">0171.01105</a></li>
<li>Colmez, Pierre (1988), "Résidu en s=1 des fonctions zêta p-adiques", <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>, 91 (2): 371–389, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1988InMat..91..371C">1988InMat..91..371C</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01389373">10.1007/BF01389373</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0922806">0922806</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:118434651">118434651</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0651.12010">0651.12010</a></li>
<li>Kolster, M. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Leopoldt's_conjecture">"Leopoldt's conjecture"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li><a href="/facts/Heinrich-Wolfgang_Leopoldt/xSbeLDXc">Leopoldt, Heinrich-Wolfgang</a> (1962), <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002179482">"Zur Arithmetik in abelschen Zahlkörpern"</a>, <a href="/facts/Journal_f%25C3%25BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">Journal für die reine und angewandte Mathematik</a>, 1962 (209): 54–71, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1962.209.54">10.1515/crll.1962.209.54</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0139602">0139602</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:117123955">117123955</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0204.07101">0204.07101</a></li>
<li><a href="/facts/Heinrich-Wolfgang_Leopoldt/xSbeLDXc">Leopoldt, H. W.</a> (1975), "Eine p-adische Theorie der Zetawerte II", <a href="/facts/Journal_f%25C3%25BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">Journal für die reine und angewandte Mathematik</a>, 1975 (274/275): 224–239, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1975.274-275.224">10.1515/crll.1975.274-275.224</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:118013793">118013793</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0309.12009">0309.12009</a>.</li>
<li>Mihăilescu, Preda (2009), The T and T* components of Λ - modules and Leopoldt's conjecture, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0905.1274">0905.1274</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2009arXiv0905.1274M">2009arXiv0905.1274M</a></li>
<li>Mihăilescu, Preda (2011), Leopoldt's Conjecture for CM fields, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1105.4544">1105.4544</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2011arXiv1105.4544M">2011arXiv1105.4544M</a></li>
<li><a href="/facts/J%25C3%25BCrgen_Neukirch/nzVc4G3F">Neukirch, Jürgen</a>; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323 (Second ed.), Berlin: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-37888-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2392026">2392026</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1136.11001">1136.11001</a></li>
<li>Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields (Second ed.), New York: Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94762-0, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0966.11047">0966.11047</a>.</li></ul>

Leopoldt's conjecture open-in-new

Leopoldt's conjecture