In this setting, a “symmetry” of the group occurs wherever it’s possible to rearrange elements of the group in a way that preserves the addition structure of the group. For Group 2, there are two such symmetries: the “identity” symmetry (in which you don’t change the places of any elements), and the symmetry that swaps x with z. (Because x + x = y and z + z = y, x and z are interchangeable.)

Group 1 has more symmetries. The elements a, b, and c are all interchangeable, since a + a = 0, b + b = 0, and c + c = 0. Given that, every way of rearranging these three elements is a symmetry (or “automorphism”) of the group. If you work through all the combinations you see there are six symmetries in all. Putting this together, Group 1 has three times as many symmetries as Group 2. You’d therefore expect to find Group 2 three times as often as you would Group 1, in keeping with the rule that arrangements occur in inverse proportion to their number of symmetries. This law is as true for groups with four simple elements like Group 1 and Group 2 as it is for other, more complicated, groups of ideals.

When mathematicians are confronted with a class number, they want to know the structure of the underlying group it represents. If they can establish the structure of the underlying group, and establish how frequently groups of a given structure arise, they can bring that information back to the surface and use it to understand how often a given class number should occur.

If you start to examine the group structure and its symmetries, then “suddenly it gives you what the distribution of class numbers should be on the nose,” said Bhargava.

A New Way to Test Structure

The two groups above are (relatively) easy to parse. Groups of ideals are much harder to pin down; it’s not easy to sketch out their addition tables. Instead, mathematicians have ways of probing the groups, testing their structure, even when they can’t see the whole thing completely. In particular, they test how far each element in the group is from zero.

Recall that every group has a zero element that, when added to any other number, leaves that number unchanged. To investigate the structure of class groups, mathematicians try to get a feel for the number of elements in a given class group that have what they call “n-torsion,” which means that when you add n copies of the element, you wind up at the zero of the group. An element is 2-torsion, for example, if x + x = 0, 3-torsion if x + x + x = 0, 4-torsion if x + x + x + x = 0 and so on.

One way to make clear the difference between the two groups above is to consider how many of their elements are 2-torsion. In Group 1, all four elements are 2-torsion, which is evident by the line of zeroes on the diagonal: 0 + 0 = 0, a + a = 0, b + b = 0, c + c = 0. In Group 2, only 0 and y are 2-torsion. The amount of different types of torsion in the group is an exact reflection of the group’s overall structure.

“If the number of n-torsion elements in two groups is the same for all n then they’re the same group. Investigating how many n-torsion elements there are is a simple strategy that probes the group and is enough to recover the group if you understand everything about torsion,” said Bhargava.

A lot of the work on the Cohen-Lenstra heuristics today has to do with establishing how many elements in a class group have different types of torsion. The Cohen-Lenstra predictions with respect to torsion are quite easy to state. For example, if you’re adjoining the square roots of negative numbers, how many ideals in their class group should have 3-torsion? Cohen-Lenstra predict that there should be on average two 3-torsion elements per number ring. How many should have 5-torsion? 7-torsion? 11-torsion? The answer again, for each prime, is two.

This constancy is striking because from a naïve perspective, you’d expect the number of elements with a given torsion to grow as the size of the class group grows. Yet even as the sizes of the class groups vary, the Cohen-Lenstra heuristics predict that the number of elements with, say, 3-torsion, will on average remain constant.

“It’s interesting that this prediction is independent of the prime number,” said Bhargava. “It’s an amazing prediction.”

It’s an amazing prediction that’s been borne out statistically in countless computer runs, yet remains hard to prove.

Lowering the Bound

The Cohen-Lenstra heuristics, further extended by Cohen and Jacques Martinet in 1987, have been around for more than 40 years. Yet you could summarize progress on them on a Post-it. Only two cases have ever been proved: one in 1971, by Harold Davenport and Hans Heilbronn, and another in 2005 by Bhargava. Otherwise, “almost nothing has been proven,” said Bhargava.

With proofs of the heuristics being hard to come by, mathematicians have adopted more modest goals. They’d like to prove that the average number of n-torsion elements for a given prime is as expected, but short of that, they’ll settle for at least putting a ceiling on the number. This is called establishing an upper bound, and mathematicians have been making gradual progress in this regard.

When you’re adjoining the square root of a negative number to your number system, the class number grows in proportion to the size of the square root. If you’re adjoining the square root of –13, you can expect the class group to be, at most, about square root of 13 elements in size. Another way of writing the square root for any number n is n0.5, and that number — the 0.5 in the exponent — is the place mathematicians start when trying to fix an upper bound. If the whole class group contains n0.5 elements, then you know from the start that there can’t be more than n0.5 elements with say, 3-torsion, because that would be every element. For that reason, n0.5 is considered the trivial bound on n-torsion in the class group.

Mathematicians typically use one of several general approaches to lowering these bounds. One is an approach called a “sieve,” which you can analogize as “panning” for n-torsion elements the way a prospector pans for gold. The two other methods involve complicated transformations through which elements with n-torsion can be counted as lattice points in a region or on a curve.

One of the first to break the trivial bound was Lillian Pierce, a mathematician at Duke University, when, in 2006, she proved that the number of 3-torsion elements in a particular number ring is at most n0.49. It was a small improvement over the trivial bound, but it started a trail that other mathematicians followed. Independently and around the same time, Venkatesh and Harald Helfgott of the University of Göttingen lowered the bound to n0.44, and the next year Venkatesh and Jordan Ellenberg of the University of Wisconsin-Madison brought the bound down even further, to n0.33. These are not expected to be the optimal bounds, but they do move the field forward. “From my point of view it’s much more important to prove anything at all in the first place,” said Venkatesh.

The most recent result in this area comes from Bhargava and five coauthors, Arul Shankar, Takashi Taniguchi, Frank Thorne, Jacob Tsimerman, and Yongqiang Zhao. In January, they posted a paper to the scientific preprint site arxiv.org that lowered the bound for 2-torsion in cubic and quartic number rings to n0.28. In that same paper they also proved that they can break the trivial bound for 2-torsion for number rings in any degree.

“It is just a small savings, but it’s chipped away at the trivial bound for the first time in infinitely many cases,” said Pierce.

Even that small savings has already paid mathematical dividends. The methods Bhargava and his collaborators used have proved useful for bounding the number of solutions to a specific class of polynomial equation called elliptic curves, which is consistent with the way that class numbers seem to be situated at the intersection of many different mathematical fields. And, while there’s a long way to go before this happens, progress on class numbers could end up redeeming the original purpose of the number rings they describe.

“A proof of Fermat’s Last Theorem has never been obtained just by studying these class numbers,” said Bhargava. “If we fully understood how class groups behave in general, it seems conceivable a proof of that kind could work for FLT and for many other equations. It’s hard to say because we still have a long way to go.”

Correction: On March 6 this article was updated to clarify that the integers, not the whole numbers, are in a sense the original group.

This article was reprinted on ScientificAmerican.com.