Leopoldt's conjecture

In algebraic number theory, Leopoldt's conjecture, introduced by, 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.

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


 * $$ U_1 = \prod_{P|p} U_{1,P}. $$

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 $$E_1$$ 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 $$\mathbb{Z}_p$$-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 $$\mathbb{Q}$$ or an abelian extension of an imaginary quadratic number field: reduced the abelian case to a p-adic version of Baker's theorem, which was  proved shortly afterwards by. has announced a proof of Leopoldt's conjecture for all CM-extensions of $$\mathbb{Q}$$.

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.