In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt (1962, 1975), states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by H.-W. Leopoldt (1962).
Formulation
Let K be a number field and for each prime 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
Then let E1 denote the set of global units ε that map to U1 via the diagonal embedding of the global units in E.
Since is a finite-index subgroup of the global units, it is an abelian group of rank , where is the number of real embeddings of and the number of pairs of complex embeddings. Leopoldt's conjecture states that the -module rank of the closure of embedded diagonally in is also
Leopoldt's conjecture is known in the special case where is an abelian extension of or an abelian extension of an imaginary quadratic number field: Ax (1965) reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by Brumer (1967). Mihăilescu (2009, 2011) has announced a proof of Leopoldt's conjecture for all CM-extensions of .
Colmez (1988) expressed the residue of the p-adic Dedekind zeta function of a totally real field 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.
References
- Ax, James (1965), "On the units of an algebraic number field", Illinois Journal of Mathematics, 9 (4): 584–589, doi:10.1215/ijm/1256059299, ISSN 0019-2082, MR 0181630, Zbl 0132.28303
- Brumer, Armand (1967), "On the units of algebraic number fields", Mathematika, 14 (2): 121–124, doi:10.1112/S0025579300003703, ISSN 0025-5793, MR 0220694, Zbl 0171.01105
- Colmez, Pierre (1988), "Résidu en s=1 des fonctions zêta p-adiques", Inventiones Mathematicae, 91 (2): 371–389, Bibcode:1988InMat..91..371C, doi:10.1007/BF01389373, ISSN 0020-9910, MR 0922806, S2CID 118434651, Zbl 0651.12010
- Kolster, M. (2001) [1994], "Leopoldt's conjecture", Encyclopedia of Mathematics, EMS Press
- Leopoldt, Heinrich-Wolfgang (1962), "Zur Arithmetik in abelschen Zahlkörpern", Journal für die reine und angewandte Mathematik, 1962 (209): 54–71, doi:10.1515/crll.1962.209.54, ISSN 0075-4102, MR 0139602, S2CID 117123955, Zbl 0204.07101
- Leopoldt, H. W. (1975), "Eine p-adische Theorie der Zetawerte II", Journal für die reine und angewandte Mathematik, 1975 (274/275): 224–239, doi:10.1515/crll.1975.274-275.224, S2CID 118013793, Zbl 0309.12009.
- Mihăilescu, Preda (2009), The T and T* components of Λ - modules and Leopoldt's conjecture, arXiv:0905.1274, Bibcode:2009arXiv0905.1274M
- Mihăilescu, Preda (2011), Leopoldt's Conjecture for CM fields, arXiv:1105.4544, Bibcode:2011arXiv1105.4544M
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001
- Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields (Second ed.), New York: Springer, ISBN 0-387-94762-0, Zbl 0966.11047.
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