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

Anand Chitrao, Aditya Karnataki, Jishnu Ray

Abstract

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 $\mathbb{Q}_p$.

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

Abstract

Let be a Banach algebra over whose residue fields are finite extensions of . Given an arithmetic family of Galois representations, i.e., a finite free -module with a continuous action of the absolute Galois group of a -adic number field, we construct a complex associated to over false-Tate extensions and construct explicit isomorphisms between its cohomology and the Galois cohomology. This recovers earlier results by Tavares Ribeiro when is a finite extension of .
Paper Structure (17 sections, 21 theorems, 77 equations)

This paper contains 17 sections, 21 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Notations:
  3. $(\varphi,\Gamma)$-modules
  4. $(\varphi,\tau)$-modules
  5. Preliminaries
  6. Period rings
  7. More on the theory of $(\varphi, \tau)$-modules
  8. Families of $(\varphi, \tau)$-modules
  9. Cohomology of $G_{\infty}$-modules and $H_{\infty}$-modules
  10. Cohomology of $G_{\infty}$-modules
  11. Cohomology of $H_{\infty}$-modules
  12. The inflation-restriction sequence for $H_{\infty}$ and $G_{\infty}$
  13. A four-term complex for ${\mathcal{G}}_K$-cohomology
  14. Explicit maps from group cohomology to cohomology of $(\varphi, \tau)$-modules
  15. Computations for $H^1({\mathcal{G}}_K, V)$
  16. ...and 2 more sections

Key Result

Theorem 1

Let $V$ be a family of representations of ${\mathcal{G}}_K$ and ${\bf D}^{\dagger, r}_{\tau, K}(V)$ the associated $(\varphi, \tau)$-module as in section Families of phi tau modules (Theorem theo ana families). For $r \geq r_0$, let ${\bf D}^{r}_L = S\widehat{\otimes}{\widetilde{\bf{B}}}^{\dagger, r is isomorphic to the Galois cohomology of $V$, where $\delta = \frac{\tau^{\chi(\gamma)} - 1}{\tau

Theorems & Definitions (42)

  • Theorem 1: see Theorem \ref{['Overconvergent theorem']}
  • Remark 2
  • Proposition 3
  • Remark 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • Definition 8
  • Theorem 9
  • Definition 10
  • ...and 32 more