Mathematicians Crack a Simple but Stubborn Class of Equations


Smith had been trying to understand properties of solutions to equations called elliptic curves. In doing so, he worked out a specific part of the Cohen-Lenstra heuristics. Not only was it the first major step in cementing those broader conjectures as mathematical fact, but it involved precisely the piece of the class group that Koymans and Pagano needed to understand in their work on Stevenhagen’s conjecture. (This piece included the elements that Fouvry and Klüners had studied in their partial result, but it also went far beyond them.)

However, Koymans and Pagano couldn’t simply use Smith’s methods right away. (If that had been possible, Smith himself would probably have done so.) Smith’s proof was about class groups associated to the right number rings (ones in which d−−√ gets adjoined to the integers) — but he considered all integer values of d. Koymans and Pagano, on the other hand, were only thinking about a tiny subset of those values of d. As a result, they needed to assess the average behavior among a much smaller fraction of class groups.

Those class groups essentially constituted 0% of Smith’s class groups — meaning that Smith could throw them away when he was writing his proof. They didn’t contribute to the average behavior that he was studying at all.