Appears in collection : Bourbaki - Mars 2026

In 1983, Henri Cohen and Hendrik Lenstra proposed a conjecture about the distribution of the $N$-torsion of the class group of a random quadratic field, supported by what was at the time a large amount of computational evidence. The Cohen–Lenstra heuristics, which are still almost entirely unproven, have become one of the central foundational problems in arithmetic statistics. Recent years have seen a rapidly accelerated pace of development in Cohen-Lenstra problems. I will give a tour of these developments, including the work of Wood and her collaborators developing a fully fleshed out roster of generalized Cohen–Lenstra conjectures, with support from topology; Smith’s theorems proving the Cohen–Lenstra conjectures for the $2$-primary part of the class group, as part of more general theorems about Selmer groups in quadratic twists, leading to a resolution of the minimalist conjecture for elliptic curves; and recent work by Koymans and Pagano in the $2$-primary case, expanding on Smith’s work and proving Stevenhagen’s conjecture on the negative Pell equation.