Leopoldt's conjecture

In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt (1962, 1975), states that 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^[update], 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 $r_1 %2B r_2 - 1$ , 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 $r_1 %2B r_2 - 1.$

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 number fields.

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, http://projecteuclid.org/euclid.ijm/1256059299
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
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
Kolster, M. (2001), "Leopoldt's conjecture", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=l/l110120
Leopoldt, Heinrich-Wolfgang (1962), "Zur Arithmetik in abelschen Zahlkörpern", Journal für die reine und angewandte Mathematik 209: 54–71, ISSN 0075-4102, MR 0139602, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002179482
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 .
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.
Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields (Second ed.), New York: Springer, ISBN 0387947620 .