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).

Leopoldt proposed a definition of a p-adic regulator R_p attached to K and a prime number p. The definition of R_p uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator. The conjecture, which for general K is still open as of 2009, then comes out as the statement that R_p is not zero.

Formulation

Let K be a number field and for each prime P of K above some fixed rational prime p, let U_P denote the local units at P and let U_1,P denote the subgroup of principal units in U_P. Set

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 $E_{1}$ is a finite-index subgroup of the global units, it is an abelian group of rank $r_1 + r_2 - 1$ , where $r_{1}$ is the number of real embeddings of $K$ and $r_{2}$ the number of pairs of complex embeddings. Leopoldt's conjecture states that the $\mathbb {Z} _{p}$ -module rank of the closure of $E_{1}$ embedded diagonally in $U_{1}$ is also $r_{1}+r_{2}-1.$

Leopoldt's conjecture is known in the special case where $K$ is an abelian extension of $\mathbb {Q}$ 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 $\mathbb {Q}$ .

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: 584–589, ISSN 0019-2082, MR 0181630, Zbl 0132.28303
Brumer, Armand (1967), "On the units of algebraic number fields", Mathematika. A Journal of Pure and Applied Mathematics, 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, doi:10.1007/BF01389373, ISSN 0020-9910, MR 922806, Zbl 0651.12010
Kolster, M. (2001), "l/l110120", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Leopoldt, Heinrich-Wolfgang (1962), "Zur Arithmetik in abelschen Zahlkörpern", Journal für die reine und angewandte Mathematik, 209: 54–71, ISSN 0075-4102, MR 0139602, 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, Zbl 0309.12009 .
Mihăilescu, Preda (2009), The T and T* components of Λ - modules and Leopoldt's conjecture, arXiv:0905.1274
Mihăilescu, Preda (2011), Leopoldt's Conjecture for CM fields, arXiv:1105.4544
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, Zbl 1136.11001, MR 2392026
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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