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

2019/12/10 by Yoshinosuke Hirakawa, Hirakawa, Yoshinosuke
Mathematics · #11D72 #11R18 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #primary 11D57 #secondary 11D41

paper · pdf · doi:10.48550/arxiv.1912.04620

openalex publication_date 2019/12/10 · openalex created_date 2019/12/26 · openalex updated_date 2026/07/28

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.

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