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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).

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|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 {\displaystyle E_{1}} is a finite-index subgroup of the global units, it is an abelian group of rank r 1 + r 2 − 1 {\displaystyle r_{1}+r_{2}-1} , where r 1 {\displaystyle r_{1}} is the number of real embeddings of K {\displaystyle K} and r 2 {\displaystyle r_{2}} 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 U 1 {\displaystyle U_{1}} is also r 1 + r 2 − 1. {\displaystyle r_{1}+r_{2}-1.}

Leopoldt's conjecture is known in the special case where K {\displaystyle K} is an abelian extension of Q {\displaystyle \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 Q {\displaystyle \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.

References

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