On the arithmetic of modified idèle class groups

Let $k$ be a number field and $S$, $T$ sets of places of $k$. For each prime $p$, we define an invariant $\mathscr{G}=\mathscr{G}_p(k_\infty/k,S,T)$ related to the Galois group of the maximal abelian extension of $k$ which is unramified outside $S$ and splits completely in $T$. In the main theorem we interpret $\mathscr{G}$ in terms of another arithmetic object $\mathscr{U}$ that involves various unit groups and uses genus theory applied to certain modules, which are technically modified from idèle groups. We show that this interpretation is functorial with respect to $S$ and $T$ and thereby provides interesting connections between $\mathscr{G}$ and $\mathscr{U}$ as $S$ and $T$ vary. The settings and methods are new, and different from the classical genus theoretic methods for idèle groups. The advantage of the new methods at the finite level not only generalizes but also strengthens certain known results involving the maximal $p$-abelian profinite Galois group of $k$ that is $S$-ramified and $T$-split in terms of the arithmetic of certain units of $k$. At the infinite level, the method relates the deep arithmetic of special units with those of profinite Galois groups. For example, for special cases of $S$ and $T$, the invariants $\mathscr{G}$ are related to the conjectures of Gross (or Kuz'min–Gross) and Leopoldt and accordingly, in these special cases, the functorial interpretation of $\mathscr{G}$ as $S$ and $T$ vary involves interesting connections between the conjectures of Gross and Leopoldt in a simpler and more concrete way. As a result, we conjecture that $\mathscr{G}$ is finite for all finite disjoint sets $S$, $T$ over the cyclotomic $\mathbb{Z}_p$-tower of $k$, which includes the conjectures of Gross and Leopoldt as special cases.