Leopoldt's conjecture

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 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 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 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 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.