Cohen-Lenstra heuristics and local conditions

Abstract

We prove function field theorems supporting the Cohen-Lenstra heuristics for real quadratic fields, and natural strengthenings of these analogs from the affine class group to the Picard group of the associated curve. Our function field theorems also support a conjecture of Bhargava on how local conditions on the quadratic field do not affect the distribution of class groups. Our results lead us to make further conjectures refining the Cohen-Lenstra heuristics, including on the distribution of certain elements in class groups. We prove instances of these conjectures in the number field case. Our function field theorems use a homological stability result of Ellenberg, Venkatesh, and Westerland.

Nonabelian Cohen–Lenstra moments

“…When G is abelian and of odd order, then Theorem 1.2 is roughly the same as a result of Achter [Ach06]. Also, when |G| is odd, Ellenberg, Venkatesh, and Westerland [EVW16] (for abelian G, in the imaginary quadratic case), the author [Woo17b] (for abelian G, in the real quadratic case) and Boston and the author [BW17] (for any G, in the imaginary quadratic case) prove a result like Theorem 1.2, but with a limit in n, before a limit in q, making it closer to the analog of Conjecture 1.1.…”

“…The other lines are all exact values forẼ ≤X (A, A −1 C 2 ), (or more precisely its analog for quadratic fields of discriminant congruent to 5 mod 8) where A is an abelian group. These other three lines are all conjectured, by the Cohen-Lenstra Heuristics [CL84] (see also [Woo17b] and [BV16, Corollary 4]), to go to 1. The new insight of our paper leads to predicting that, unlike averages of unramified abelian extensions,Ẽ ≤X (A 4 , G ) should instead go to 2.…”

“…This is a local condition at 2 on the quadratic fields. It is predicted that such local conditions do not affect the Cohen-Lenstra heuristics and in particular the conjectured averages in the case that G is abelian (see [BV15,BV16,Woo17b]).…”

Random Partitions and Cohen–Lenstra Heuristics

“…Setting d = ∞, u = 1 and noting that this inequality holds for all p ≥ 3 gives Lemma 5.1. • Similarly, setting d = ∞, u = 1/p w (with p prime and w a positive integer) gives Proposition 2.3 of [26].…”

“…Friedman and Washington do not discuss this explicitly, but using the same methods as in [12] one can show that taking the limit as d → ∞ of the probability that a randomly chosen d × (d + w) matrix over Z p has cokernel isomorphic to a finite abelian p-group of type λ is given by P ∞,1/p w (λ). See the discussion above Proposition 2.3 of [26]. Similarly, Tse considers rectangular matrices with more rows than columns and shows that P ∞,1/p w (λ) is equal to the d → ∞ probability that a randomly chosen (d + w) × d matrix over Z p has cokernel isomorphic to Z w p ⊕ G, where G is a finite abelian p-group of type λ [23].…”