Ancient Equations Offer New Look at Number Groups | Quanta Magazine

Ancient Equations Offer New Look at Number Groups | Quanta Magazine

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 $latex \sqrt{d}$ 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.

And when Koymans and Pagano tried to apply his techniques to just the class groups they cared about, the methods broke down immediately. The pair would need to make significant changes to get them to work. Moreover, they weren’t just characterizing one class group, but rather the discrepancy that might exist between two different class groups (doing so would be a major part of their proof of Stevenhagen’s conjecture) — which would also require some different tools.

So Koymans and Pagano started combing more carefully through Smith’s paper in hopes of pinpointing exactly where things started to go off the rails. It was difficult, painstaking work, not just because the material was so complicated, but because Smith was still refining his preprint at the time, making needed corrections and clarifications. (He posted the new version of his paper online last month.)

For a whole year, Koymans and Pagano learned the proof together, line by line. They met every day, discussing a given section over lunch before spending a few hours at a blackboard, helping each other work through the relevant ideas. If one of them made progress on his own, he texted the other to update him. Shusterman recalls sometimes seeing them working long into the night. In spite of (or perhaps because of) the challenges it entailed, “that was very fun to do together,” Koymans said.

They ultimately identified where they’d need to try a fresh approach. At first, they were only able to make modest improvements. Together with the mathematicians Stephanie Chan and Djordjo Milovic, they figured out how to get a handle on some additional elements in the class group, which allowed them to get better bounds than Fouvry and Klüners had. But significant pieces of the class group’s structure still eluded them.

One major problem they had to tackle — something for which Smith’s method no longer worked in this new context — was ensuring that they were truly analyzing “average” behavior for class groups as the values of d got larger and larger. To establish the proper degree of randomness, Koymans and Pagano proved a complicated set of rules, called reciprocity laws. In the end, that allowed them to gain the control they needed over the difference between the two class groups.

That advance, coupled with others, allowed them to finally complete the proof of Stevenhagen’s conjecture earlier this year. “It’s amazing that they were able to solve it completely,” Chan said. “Previously, we had all these issues.”

What they did “surprised me,” Smith said. “Koymans and Pagano have sort of kept my old language and just used it to push further and further in a direction that I barely understand anymore.”

The Sharpest Tool

From the time he introduced it five years ago, Smith’s proof of one part of the Cohen-Lenstra heuristics was seen as a way to open doors to a host of other problems, including questions about elliptic curves and other structures of interest. (In their paper, Koymans and Pagano list about a dozen conjectures they hope to use their methods on. Many have nothing to do with the negative Pell equation or even class groups.)

“A lot of objects have structures that are not dissimilar to these sorts of algebraic groups,” Granville said. But many of the same roadblocks that Koymans and Pagano had to confront are also present in these other contexts. The new work on the negative Pell equation has helped dismantle these roadblocks. “Alexander Smith has told us how to build these saws and hammers, but now we have to make them as sharp as possible and as hard-hitting as possible and as adaptable as possible to different situations,” Bartel said. “One of the things this paper does is go a great deal in that direction.”

All of this work, meanwhile, has refined mathematicians’ understanding of just one facet of class groups. The rest of the Cohen-Lenstra conjectures remain out of reach, at least for the moment. But Koymans and Pagano’s paper “is an indication that the techniques we have for attacking problems in Cohen-Lenstra are kind of growing up,” Smith said.

Lenstra himself was similarly optimistic. It is “absolutely spectacular,” he wrote in an email. “It really opens up a new chapter in a branch of number theory that is just as old as number theory itself.”


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