Leopoldt's conjecture

In algebraic number theory, Leopoldt's conjecture, introduced by Heinrich-Wolfgang Leopoldt,Template:SfnpTemplate:Sfnp 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,Template:Sfnp which was proved shortly afterwards by Brumer.Template:Sfnp Mihăilescu has announced a proof of Leopoldt's conjecture for all CM-extensions of $\mathbb{Q}$.Template:SfnpTemplate:Sfnp

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.Template:Sfnp 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

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