In algebraic number theory, Leopoldt's conjecture, introduced by Heinrich-Wolfgang Leopoldt, 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.
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 = ∏ P ∣ p U 1 , P . {\displaystyle U_{1}=\prod _{P\,\mid \,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 E1 is a finite-index subgroup of the global units, it is an abelian group of rank r1 + r2 – 1, where r1 is the number of real embeddings of K and r2 the number of pairs of complex embeddings. Leopoldt's conjecture states that the Z p {\displaystyle \mathbb {Z} _{p}} -module rank of the closure of E 1 {\displaystyle E_{1}} embedded diagonally in U1 is also r1 + r2 – 1. Leopoldt's conjecture is known in the special case where K is an abelian extension of Q {\displaystyle \mathbb {Q} } or an abelian extension of an imaginary quadratic number field: Ax reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by Brumer. Mihăilescu has announced a proof of Leopoldt's conjecture for all CM-extensions of Q {\displaystyle \mathbb {Q} } . Colmez 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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