Explicit isomorphisms for a Herr-type complex over a metabelian extension

Let $S$ be a Banach algebra over $\mathbb{Q}_p$ whose residue fields are finite extensions of $\mathbb{Q}_p$. Given an arithmetic family $V$ of Galois representations, i.e., a finite free $S$-module $V$ with a continuous action of the absolute Galois group of a $p$-adic number field, we construct a complex associated to $V$ over false-Tate extensions and construct explicit isomorphisms between its cohomology and the Galois cohomology. This recovers earlier results by Tavares Ribeiro when $S$ is a finite extension of $\mathbb{Q}_p$.

💡 Research Summary

The paper develops an explicit cohomological description for families of $p$‑adic Galois representations over a metabelian (false‑Tate) extension of a $p$‑adic field $K$. Let $S$ be a $\mathbf Q_p$‑Banach algebra whose residue fields are finite extensions of $\mathbf Q_p$, and let $V$ be a finite free $S$‑module equipped with a continuous $G_K$‑action. Using the theory of $(\varphi,\tau)$‑modules, the authors associate to $V$ a family of étale $(\varphi,\tau)$‑modules $D^\dagger_{r,\tau,K}(V)$ defined over period rings $B^\dagger_{r,\tau,K}$ and $B^\dagger_{r,L}$ (the latter being the $H_\infty$‑fixed part of the usual $p$‑adic period ring $B^\dagger$). For sufficiently large $r$ they set \