The Heisenberg coboundary equation: appendix to Explicit Chabauty-Kim theory

TL;DR

This paper discusses the challenge of explicitly lifting cocycles in Galois cohomology related to the Heisenberg coboundary equation, contributing to the explicit Chabauty-Kim theory in number theory.

Contribution

It provides an analysis and makes available an appendix to the explicit Chabauty-Kim theory, focusing on the Heisenberg coboundary equation and its cohomological lifting problem.

Findings

Clarifies the cohomological lifting problem for the Heisenberg group

Provides an analysis following Romyar Sharifi's note

Makes the appendix to the theory available online

Abstract

Let p be a regular prime number, let Gp denote the Galois group of the maximal unramified away from p extension of Q, and let H_et denote the Heisenberg group over Qp with Gp-action given by H_et = Qp(1)^2 \oplus Qp(2). Although Soul\'e vanishing guarantees that the map H^1(Gp, H_et) ---> H^1(Gp, Qp(1)^2) is bijective, the problem of constructing an explicit lifting of an arbitrary cocycle in H^1(Gp, Qp(1)^2) proves to be a challenge. We explain how we believe this problem should be analyzed, following an unpublished note by Romyar Sharifi, hereby making the original appendix to Explicit Chabauty-Kim theory available online in an arXiv-only note.