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 R_p attached to K and a prime number p. The definition of R_p 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^[update], then comes out as the statement that R_p is not zero.

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

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 ${\displaystyle r_{1}+r_{2}-1.}$

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