Leopoldt's conjecture

In algebraic number theory, Leopoldt's conjecture, introduced by Heinrich-Wolfgang Leopoldt, 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.

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

$U_{1}=\prod _{P\,\mid \,p}U_{1,P}.$

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 E₁ is a finite-index subgroup of the global units, it is an abelian group of rank r₁ + r₂ – 1, where r₁ is the number of real embeddings of K and r₂ 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 U₁ is also r₁ + r₂ – 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 reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by Brumer. Mihăilescu has announced a proof of Leopoldt's conjecture for all CM-extensions of $\mathbb {Q}$.

Colmez 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. MR 0181630. Zbl 0132.28303.

• Brumer, Armand (1967). "On the units of algebraic number fields". Mathematika. 14 (2): 121–124. doi:10.1112/S0025579300003703. 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. MR 0922806. 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. 209: 54–71. doi:10.1515/crll.1962.209.54. MR 0139602. Zbl 0204.07101.

• Leopoldt, Heinrich-Wolfgang (1975). "Eine p-adische Theorie der Zetawerte. II". Journal für die reine und angewandte Mathematik. 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 [math.NT].

• Mihăilescu, Preda (2011). "Leopoldt's conjecture for CM fields". arXiv:1105.4544 [math.NT].

• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008) [2000]. Cohomology of Number Fields. A Series of Comprehensive Studies in Mathematics. Vol. 323 (second ed.). Berlin, Heidelberg: Springer. ISBN 978-3-540-37888-4. MR 2392026. Zbl 1136.11001.

• Washington, Lawrence C. (1997) [1982]. Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. Vol. 83 (second ed.). New York: Springer. ISBN 978-0-387-94762-4. Zbl 0966.11047.