A Hilbert 90 Property for S-Class Groups and Applications to the Gross--Kuz'min Conjecture

Let $L/K$ be a cyclic extension of number fields, and let $S$ be a finite set of places of $K$ containing the ramified and Archimedean ones. We say that $L/K$ has the $\mathbf{cl}^S$-Hilbert 90 property if, for any generator $\sigma \in \mathrm{Gal}(L/K)$, the kernel of the arithmetic norm map $\mathbf{cl}^S(L) \to \mathbf{cl}^S(K)$ coincides with $(1 - \sigma)\mathbf{cl}^S(L)$. In this article, we first provide a method to verify the $\mathbf{cl}^S$-Hilbert 90 property, which does not require any knowledge of the class group of $L$. Then we investigate a connection between the $\mathbf{cl}^S$-Hilbert 90 property and the Gross-Kuz'min conjecture from Iwasawa theory. In doing so, we derive a new criterion for the Gross-Kuz'min conjecture, related to Fermat quotients and spin symbols of prime ideals, which can easily be checked by explicit computation. We conjecture that, in the totally real case, the condition holds for all but finitely many primes. Finally, we present numerical evidence supporting a heuristic in favor of this conjecture.