For any positive integer , we define to be the smallest number such that every diagonal form in variables with integer coefficients must have a nontrivial zero in every -adic field . An old conjecture of Norton is that we should have for all . For many years, 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.
"Infinitely Many Counterexamples to a Conjecture of Norton." Michigan Math. J. 69 (3) 533 - 543, August 2020. https://doi.org/10.1307/mmj/1596700817