Leopoldt’s conjecture for the number field F

Prove that the Leopoldt defect δ(F) vanishes for the number field F, equivalently that the compositum of all Z_p-extensions of F has Galois group of rank r_2(F)+1 over F.

Background

The paper uses the Leopoldt defect δ(F) to describe the rank of the Galois group of the compositum of all Z_p-extensions of a number field F. Vanishing of this defect is the content of Leopoldt’s conjecture.

The conjecture is mentioned as a standing condition under which a particular S-ramified Z_p-extension is unique and as the assertion that the rank contribution δ(F) is zero. It is not proved in the paper.

References

It follows from Theorem 2.1 that the $\ZZ_p$-extension $F_{\infty}/F$ unramified outside $S$ exists, and it is unique if we assume that Leopoldt's conjecture holds for $F$.

— Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions  (2608.25309 - Qi et al., 26 Aug 2026) in Section 1, subsection “Preliminaries” and Section 1, subsection “Main results and tools”

(Leopoldt's conjecture predicts that $d_0=d-1$.)

— Completed cohomology of Hilbert modular varieties below middle degree  (2609.27310 - Pan, 23 Sep 2026) in Section 3, subsection “Dimension Theory,” paragraph introducing the subgroup H and the parameter d_0

One main reason for introducing this very unnatural definition is that we don't know the value of $e$ in general. It was predicted by Leopoldt's conjecture that $e=1$.

— Plectic Lie Algebra Action on Hilbert modular varieties  (2609.34465 - Jiang et al., 28 Sep 2026) in Remark following Definition \ref{dfn-rhokVk}, Section 4.2