A Hilbert 90 Property for S-Class Groups and Applications to the Gross--Kuz'min Conjecture
Julian Feuerpfeil
TL;DR
This work extends Hilbert 90 to the S-class group setting and ties the cl^S-Hilbert 90 property to the Gross–Kuz'min conjecture within Iwasawa theory. It develops a cohomological Mackey functor framework to model S-class groups and Selmer groups, deriving a ramified criterion for cl^S-Hilbert 90 that depends only on base-field data and ramification, not on the full L-class group. A key contribution is a computable criterion involving the surjectivity of a map τ^p linked to Fermat quotients and spin symbols, yielding consequences for the Kuz'min-Tate module and the Gross–Kuz'min conjecture in cyclotomic Z_p-extensions; the authors also provide heuristic and numerical evidence suggesting finiteness of exceptional primes in the totally real case. The results offer a practical path to verify Gross–Kuz'min in many cases and illuminate connections with logarithmic class groups and Gras–Munnier governing fields, with broader applicability to ramified extensions and CM fields via transfer principles.
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 $σ\in \mathrm{Gal}(L/K)$, the kernel of the arithmetic norm map $\mathbf{cl}^S(L) \to \mathbf{cl}^S(K)$ coincides with $(1 - σ)\mathbf{cl}^S(L)$. 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.