Counter-examples to the Hasse principle among the twists of the Klein quartic

TL;DR

This paper investigates the twists of the Klein quartic, providing families that are conjecturally counterexamples to the Hasse principle by analyzing local and global points using geometric and modular methods.

Contribution

It introduces new families of twists of the Klein quartic that serve as counterexamples to the Hasse principle, combining geometric and modular approaches.

Findings

Identified families of twists with local points everywhere

Constructed global counterexamples to the Hasse principle

Utilized modular interpretation to analyze twists

Abstract

In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that at counter-examples to the Hasse principle.