Leopoldt's conjecture

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In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt (1962, 1975), 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 H.-W. Leopoldt (1962).

Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator. The conjecture, which for general K is still open as of 2009, then comes out as the statement that Rp is not zero.


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: 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 \mathbb{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.