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

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 (Mathematika (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.