Infinitely Many Counterexamples to a Conjecture of Norton

Abstract

For any positive integer $k$, we define ${\Gamma }^{\ast }\left(k\right)$ to be the smallest number $s$ such that every diagonal form $a_1{x}_1^k+a_2{x}_2^k+\cdots +a_s{x}_s^k$ in $s$ variables with integer coefficients must have a nontrivial zero in every $p$-adic field ${\mathbb{Q}}_p$. An old conjecture of Norton is that we should have ${\Gamma }^{\ast }\left(k\right)\equiv 1\left(\text{mod}k\right)$ for all $k$. For many years, ${\Gamma }^{\ast }\left(8\right)=39$ 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.