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 U_P denote the local units at P and let U_1,P denote the subgroup of principal units in U_P. Set

[math]\displaystyle{ U_1 = \prod_{P|p} U_{1,P}. }[/math]

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 [math]\displaystyle{ E_1 }[/math] is a finite-index subgroup of the global units, it is an abelian group of rank [math]\displaystyle{ r_1 + r_2 - 1 }[/math], where [math]\displaystyle{ r_1 }[/math] is the number of real embeddings of [math]\displaystyle{ K }[/math] and [math]\displaystyle{ r_2 }[/math] the number of pairs of complex embeddings. Leopoldt's conjecture states that the [math]\displaystyle{ \mathbb{Z}_p }[/math]-module rank of the closure of [math]\displaystyle{ E_1 }[/math] embedded diagonally in [math]\displaystyle{ U_1 }[/math] is also [math]\displaystyle{ r_1 + r_2 - 1. }[/math]

Leopoldt's conjecture is known in the special case where [math]\displaystyle{ K }[/math] is an abelian extension of [math]\displaystyle{ \mathbb{Q} }[/math] 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 [math]\displaystyle{ \mathbb{Q} }[/math].

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, http://projecteuclid.org/euclid.ijm/1256059299 
• Brumer, Armand (1967), "On the units of algebraic number fields", Mathematika 14 (2): 121–124, doi:10.1112/S0025579300003703, ISSN 0025-5793 
• 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, Bibcode: 1988InMat..91..371C 
• Hazewinkel, Michiel, ed. (2001), "l/l110120", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=l/l110120 
• 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, 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, Bibcode: 2009arXiv0905.1274M 
• Mihăilescu, Preda (2011), Leopoldt's Conjecture for CM fields, Bibcode: 2011arXiv1105.4544M 
• 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 
• Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields (Second ed.), New York: Springer, ISBN 0-387-94762-0 .