Math Conjectures

Leopoldt’s Conjecture

OPENMajorConjectureProposed 1962 · Full conjecture

Canonical statement

Let KK be a number field and pp a rational prime. For each place v∣pv\mid p, let KvK_v be the completion and let log⁡v:OKv×→Kv\log_v:\mathcal O_{K_v}^{\times}\to K_v be the local pp-adic logarithm: if mv\mathfrak m_v is the maximal ideal of OKv\mathcal O_{K_v}, its restriction to 1+mv1+\mathfrak m_v is
log⁡v(1+x)=∑m=1∞(−1)m+1xmm(x∈mv), \log_v(1+x)=\sum_{m=1}^{\infty} \frac{(-1)^{m+1}x^m}{m}\qquad(x\in\mathfrak m_v),
and it vanishes on torsion units. Then the map
OK×⊗ZQp⟶∏v∣pKv,u⟼(log⁡vu)v∣p, \mathcal O_K^\times\otimes_{\mathbb Z}\mathbb Q_p \longrightarrow \prod_{v\mid p}K_v,\qquad u\longmapsto(\log_v u)_{v\mid p},
is injective.
View source LaTeX
Let \(K\) be a number field and \(p\) a rational prime. For each place \(v\mid p\), let \(K_v\) be the completion and let \(\log_v:\mathcal O_{K_v}^{\times}\to K_v\) be the local \(p\)-adic logarithm: if \(\mathfrak m_v\) is the maximal ideal of \(\mathcal O_{K_v}\), its restriction to \(1+\mathfrak m_v\) is
\[
  \log_v(1+x)=\sum_{m=1}^{\infty}
    \frac{(-1)^{m+1}x^m}{m}\qquad(x\in\mathfrak m_v),
\] and it vanishes on torsion units. Then the map
\[
  \mathcal O_K^\times\otimes_{\mathbb Z}\mathbb Q_p
     \longrightarrow \prod_{v\mid p}K_v,\qquad
  u\longmapsto(\log_v u)_{v\mid p},
\] is injective.
NT-IWAS-023:[Leopoldt1962][Brumer1967Leopoldt][FerriJohnston2026Leopoldt]
Number theoryAlgebra and representation theory

Notes

Leopoldt's conjecture, formulated by Heinrich-Wolfgang Leopoldt in 1962 in his study of the arithmetic of abelian number fields [Leopoldt1962], concerns the pp-adic behaviour of the units of a number field KK. Mapping a unit to the vector of its local pp-adic logarithms at the places above a prime pp, the conjecture predicts that the induced map on OK×⊗ZQp\mathcal O_K^\times\otimes_{\mathbb Z}\mathbb Q_p is injective: the global units acquire no unexpected relations when embedded pp-adically, so the pp-adic regulator does not degenerate.

The abelian case over Q\mathbb Q was settled by Brumer [Brumer1967Leopoldt], using pp-adic transcendence methods for logarithms of algebraic numbers, and the conjecture is likewise known for various further classes of fields. Beyond these cases it is open for a general number field and a general prime. Recent work brings representation-theoretic tools and explicit unit computations to bear on new families of extensions [FerriJohnston2026Leopoldt]. A full resolution would require controlling the pp-adic linear independence of unit logarithms in complete generality, which remains out of reach.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.