Leopoldt's conjecture

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

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 diagonal embedding of the global units in E.

Since E 1 {\displaystyle E_{1}} is a finite-index subgroup of the global units, it is an abelian group 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 abelian extension of Q {\displaystyle \mathbb {Q} } 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 Q {\displaystyle \mathbb {Q} } .

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.