Geodesic
The vanishing of Iwasawa's $\mu$-invariant implies the weak Leopoldt conjecture
Documenta mathematica, Tome 27 (2022), pp. 2275-2299.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $K$ denote a number field containing a primitive $p$-th root of unity; if $p=2$, then we assume $K$ to be totally imaginary. If $K_\infty/K$ is a $\mathbb{Z}_p$-extension such that no prime above $p$ splits completely in $K_\infty/K$, then the vanishing of Iwasawa's invariant $\mu(K_\infty/K)$ implies that the weak Leopoldt Conjecture holds for $K_\infty/K$. This is actually known due to a result of Ueda, which appears to have been forgotten. We present an elementary proof which is based on a reflection formula from class field theory. \par In the second part of the article, we prove a generalisation in the context of non-commutative Iwasawa theory: we consider admissible $p$-adic Lie extensions of number fields, and we derive a variant for fine Selmer groups of Galois representations over admissible $p$-adic Lie extensions.
Classification : 11R23
Keywords: class field theory, reflection formula, weak Leopoldt conjecture, Iwasawa \(\mu\)-invariant, uniform \(p\)-adic Lie extension, \(p\)-adic Galois representation 
@article{DOCMA_2022__27__a10,
     author = {Kleine, S\"oren},
     title = {The vanishing of {Iwasawa's} \(\mu\)-invariant implies the weak {Leopoldt} conjecture},
     journal = {Documenta mathematica},
     pages = {2275--2299},
     publisher = {mathdoc},
     volume = {27},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a10/}
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Kleine, Sören. The vanishing of Iwasawa's \(\mu\)-invariant implies the weak Leopoldt conjecture. Documenta mathematica, Tome 27 (2022), pp. 2275-2299. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a10/