Counterexamples to the local-global principle associated with Swinnerton-Dyer's cubic form

TL;DR

This paper constructs new explicit families of counterexamples to the local-global principle for certain odd-degree cubic forms in multiple variables, extending Swinnerton-Dyer's classical work without requiring explicit local solubility calculations.

Contribution

It introduces a novel method to generate infinite families of counterexamples for cubic forms of various degrees and variables, avoiding explicit local solubility computations.

Findings

New explicit counterexamples for cubic forms in multiple variables.

Counterexamples exist for forms of degree 2n+1 in 4, 5, 6, ..., 2n+2 variables.

Method avoids detailed local solubility calculations.

Abstract

In this paper, we imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematica (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of $4, 5, 6, ..., 2n+2$ variables of degree $2n+1$ for infinitely many integers $n$. It is contrastive to Swinnerton-Dyer's original construction that we do not need any concrete calculation in the proof of local solubility.