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

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 $\mathrm{cl}^S(L) \to \mathrm{cl}^S(K)$ coincides with $(1 - \sigma)\mathrm{cl}^S(L)$. We establish a criterion for the $\mathbf{cl}^S$-Hilbert 90 property that depends only on arithmetic data of the base field and does not require any knowledge of the class group of $L$. We then show that, for $\mathbb{Z}_p$-extensions, the $\mathbf{cl}^{S_p}$-Hilbert 90 property at finite layers already implies the finiteness of the coinvariants of the associated Kuz'min-Tate module, providing a new finite-level criterion for an Iwasawa-theoretic property related to the Gross--Kuz'min conjecture. This criterion can be expressed in terms of a local map closely related to Fermat quotients, making the criterion amenable to explicit computation. Motivated by extensive numerical experiments, we formulate a conjecture predicting that this criterion is satisfied for all but finitely many primes in the totally real case and present a random matrix heuristic supporting this prediction.