Leopoldt's Conjecture - Formulation

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

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 is a finite-index subgroup of the global units, it is an abelian group of rank, where is the number of real embeddings of and the number of pairs of complex embeddings. Leopoldt's conjecture states that the -module rank of the closure of embedded diagonally in is also

Leopoldt's conjecture is known in the special case where is an abelian extension of 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 .

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.