📝 SummarXiv

Abstract

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 $\mathrm{cl}^S(L)\rightarrow \mathrm{cl}^S(K)$ coincides with $(1-\sigma)\mathrm{cl}^S(L)$. We establish an effective criterion for the $\mathbf{cl}^S$-Hilbert 90 property and investigate its connections with Iwasawa theory, in particular its relation to the Gross--Kuz'min conjecture. This yields a new method for verifying the property by explicit computation, with further links to the theory of spins of prime ideals. We conjecture that in the totally real case the property fails for at most finitely many primes, and we present numerical evidence supporting a heuristic toward this conjecture.