Leopoldt's conjecture

In algebraic number theory, Leopoldt's conjecture, introduced by Template:Harvs, 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 Template:Harvs.

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: Template:Harvtxt reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by Template:Harvtxt. Template:Harvs has announced a proof of Leopoldt's conjecture for all CM-extensions of $\mathbb{Q}$.

Template:Harvs 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

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