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 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, 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 -module rank of the closure of E_1 embedded diagonally in U_1 is also 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.