Counterexamples to the Hasse principle

TL;DR

This paper constructs counterexamples to the Hasse principle using accessible undergraduate techniques, extending classical examples and highlighting their significance in elliptic curve theory and the Tate-Shafarevich group.

Contribution

It introduces new counterexamples to the Hasse principle with minimal prerequisites, making the topic accessible to nonspecialists and connecting to elliptic curve theory.

Findings

Extended classical counterexamples of Lind and Reichardt

Counterexamples are relevant to elliptic curves and Tate-Shafarevich group

Accessible approach using undergraduate number theory techniques

Abstract

In this article we develop counterexamples to the Hasse principle using only techniques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this interesting area of number theory. The counterexamples considered here extend the classical counterexample of Lind and Reichardt. As discussed in an appendix, this type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in the Tate--Shafarevich group.