Infinitely Many Counterexamples to a Conjecture of Norton

Abstract

For any positive integer k, we define Γ∗(k) to be the smallest number s such that every diagonal form a1x1k+a2x2k+⋯+asxsk in s variables with integer coefficients must have a nontrivial zero in every p-adic field Qp. An old conjecture of Norton is that we should have Γ∗(k)≡1(modk) for all k. For many years, Γ∗(8)=39 was the only known counterexample to this conjecture, and in recent years two more counterexamples have been found. In this article, we produce infinitely many counterexamples to Norton’s conjecture.