A modern answer to an old pattern

Carl Friedrich Gauss’s 1801 work on quadratic forms revealed that these algebraic expressions could be composed in a way that produces a finite cycle. Repeatedly combining a form with itself eventually returns to the starting point. The length and structure of these cycles are encoded by objects now called class groups, which remain central to algebraic number theory.

Gauss did not have a general rule for how these structures are distributed as one ranges over many quadratic forms or quadratic fields. That question later evolved into a statistical one: rather than predicting the precise behaviour of a single class group, can mathematics predict the average behaviour of an enormous family of them?

A pair of recent preprints by Aaron Landesman and Ishan Levy makes a substantial advance on that programme. Their work computes previously inaccessible averages, known as moments, for class groups in a function-field setting. It also provides a new method for calculating stable homology of certain geometric spaces. The result is a significant step in arithmetic statistics, although it is not a complete proof of the original Cohen–Lenstra heuristics for quadratic number fields.

From Gauss’s forms to Cohen–Lenstra heuristics

Quadratic forms can be written in the shape ax² + bxy + cy². Forms with a common discriminant can be grouped into equivalence classes, and Gauss defined a composition law that lets these classes be combined. Under this operation they form a finite abelian group. The number of elements in that group is the class number, while the internal arrangement of the group carries richer arithmetic information.

In 1983, Henri Cohen and Hendrik Lenstra proposed influential heuristics for the distribution of the odd-order parts of class groups of quadratic number fields. Their proposal assigns a probability to each finite abelian group, with groups having fewer automorphisms expected to appear more frequently. It is a probabilistic prediction about a deterministic collection of arithmetic objects.

One way to test such a prediction is through moments. For a fixed finite abelian group H, researchers average the number of surjective homomorphisms from a varying class group onto H. If all the appropriate moments can be established under suitable conditions, they can determine the predicted distribution.

This approach is powerful because it turns a difficult question about every individual class group into a question about averages across a large family. But computing these averages has remained exceptionally hard. In the number-field setting, only limited cases have been proved.

Why function fields matter

Landesman and Levy work with quadratic extensions of Fq(t), the field of rational functions over a finite field Fq. These are called function fields. They are not the same as quadratic extensions of the rational numbers, but they provide a closely related setting in which algebraic, geometric and topological techniques can interact more directly.

The researchers consider families associated with hyperelliptic curves, which can be described by equations such as y² = f(x). As the degree of the square-free polynomial f grows, the corresponding quadratic function fields form a family large enough for meaningful averages to emerge.

Their principal result calculates the average number of surjections from these class groups onto any fixed finite abelian group of odd order, subject to conditions including that the finite field be sufficiently large relative to H. In the simplest case, where the relevant roots-of-unity obstruction is absent, the predicted average is 1 for one parity of the degree and 1 divided by the size of H for the other.

Those formulas supply new moments predicted by Cohen–Lenstra-type heuristics. The authors describe them as the first known such moments for these class groups beyond the previously understood special case involving a cyclic group of order three.

The geometric obstacle

The breakthrough depends on Hurwitz spaces, geometric spaces that organise branched covers of a line or curve. They translate questions about field extensions and class groups into questions about the geometry and topology of moduli spaces.

Earlier work by Jordan Ellenberg, Akshay Venkatesh and Craig Westerland showed that certain homology groups of these Hurwitz spaces stabilise as the number of branch points grows. Stabilisation means that, beyond a certain range, a feature of the spaces stops changing. That earlier result supplied an important route to weaker function-field versions of Cohen–Lenstra predictions.

Yet knowing that homology stabilises is not the same as knowing the stable value. The missing calculation limited the arithmetic conclusions that could be drawn. A 2012 attempt to obtain a stronger proof contained a flaw, which its authors later acknowledged.

Landesman and Levy’s work addresses that gap. They calculate stable rational homology for a class of Hurwitz spaces associated with non-splitting conjugacy data. Their method combines algebraic topology, higher algebra and arithmetic geometry. Once that stable homology is known, point-counting techniques connect the geometry of the spaces to the desired class-group averages over finite fields.

The work therefore does more than supply individual numerical averages. It identifies a structural mechanism through which topological stabilisation can produce arithmetic statistics.

What has—and has not—been proved

The new results warrant the description of a breakthrough, but their scope matters. They concern quadratic function fields over sufficiently large finite fields, not the original family of quadratic number fields over the integers. They also calculate moments for a fixed finite group H when q is sufficiently large in relation to H.

That distinction prevents the result from being a full proof of the Cohen–Lenstra heuristics, even in the function-field setting. For a fixed finite field, a proof of the entire distribution would require control over all relevant groups H, as well as a fuller understanding of unstable homology and the action of Frobenius on the geometric objects involved.

The papers are available as arXiv preprints and were revised in October 2025. Their claims should therefore be understood as research results presented in the preprint literature rather than as a final resolution of every question connected to Gauss’s original work.

Still, the advance is notable for both its mathematical content and its method. A problem rooted in Gauss’s composition of quadratic forms has led, more than two centuries later, to a solution built from the interaction of number theory, geometry and homotopy theory. The result narrows a long-standing gap between probabilistic predictions about class groups and rigorous theorems capable of explaining them.

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