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).

Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp 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, then comes out as the statement that Rp is not zero.

Formulation

Let K be a number field and for each prime P of K above some fixed rational prime p, let UP denote the local units at P and let U1,P denote the subgroup of principal units in UP. Set

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

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 ${\displaystyle E_{1}}$ is a finite-index subgroup of the global units, it is an abelian group of rank ${\displaystyle r_{1}+r_{2}-1}$, where ${\displaystyle r_{1}}$ is the number of real embeddings of ${\displaystyle K}$ and ${\displaystyle r_{2}}$ the number of pairs of complex embeddings. Leopoldt's conjecture states that the ${\displaystyle \mathbb {Z} _{p}}$-module rank of the closure of ${\displaystyle E_{1}}$ embedded diagonally in ${\displaystyle U_{1}}$ is also ${\displaystyle r_{1}+r_{2}-1.}$

Leopoldt's conjecture is known in the special case where ${\displaystyle K}$ is an abelian extension of ${\displaystyle \mathbb {Q} }$ 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 CM-extensions of ${\displaystyle \mathbb {Q} }$.

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.