Leopoldt's conjecture

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

: $U\_1\; =\; \backslash prod\_\{P|p\}\; U\_\{1,P\}.$

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 $\backslash mathbb\{Z\}\_p$-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 $\backslash mathbb\{Q\}$ or an abelian extension of an imaginary quadratic number field: {{harvtxt|Ax|1965}} reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by {{harvtxt|Brumer|1967}}.

{{harvs|txt|authorlink=Preda Mihăilescu|last=Mihăilescu|year1=2009|year2=2011}} has announced a proof of Leopoldt's conjecture for all CM-extensions of $\backslash mathbb\{Q\}$.

{{harvs|txt|authorlink=Pierre Colmez|last=Colmez|year=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.

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Category:Algebraic number theory

Category:Conjectures

Category:Unsolved problems in number theory