The distribution of $H_{8}$-extensions of quadratic fields

Abstract

We compute all the moments of a normalization of the function which counts unramified H 8 -extensions of quadratic fields, where H 8 is the quaternion group of order 8, and show that the values of this function determine a point mass distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for G in a large class of 2-groups, which we conjecture will give finite moments which determine a distribution. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified H 8 -extensions.

Nonabelian Cohen–Lenstra moments

“…). See also [AK16,Kly17] for further evidence that the answer to Question 6.1 is yes. The argument for Theorem 1.2 could be used to suggest a value for c ± G,G in the function field case, but it would not be clear how to make the analogous constant in the number field case.…”

4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields

“…, where the first contribution comes from the R ramified at l, and the second from the R unramified at l. 2 Meanwhile, the value of the average for p " c is p p `1 `p p `1, where the first contribution comes from R ramified at p and the second from R unramified at p. Lastly, we consider the case p 2 |c. Remarkably enough, one observes that the case p " 3 acquires a special status in the computation of this average: indeed 1 8 of the imaginary quadratics locally at 3 give the extension Q 3 pζ 3 q{Q 3 , and the result for them will be different than for the 1 8 totally ramified that locally at 3 become Q 3 p ? 3q.…”

“…If K is not strongly of type R, we set S 1 2 pKq to be the symbol ‚. We now proceed by offering a heuristic model for S 1 2 pKq as K varies among imaginary quadratic number fields of type R. Let R be an unramified ring at c and denote by G 2 a set of representatives of isomorphism classes of finite abelian 2-groups, viewed as C 2 -modules under the action of ´Id. Denote by S 2 pRq the union of the singleton t‚u and of the set of equivalence classes of pairs pG, θq, where G P G 2 , θ P Ą Ext Z 2 rC 2 s pG, W R r2 8 sq and the equivalence is defined as follows: two pairs pG 1 , θ 1 q, pG 2 , θ 2 q are identified if G 1 " G 2 and θ 1 , θ 2 are in the same Aut ring pRqˆAut ab.gr.…”

“…Let p be an odd prime. The average value of # ClpK, cqrps as K ranges over imaginary quadratic number fields with gcdpDpKq, cq " 1 and ordered by their discriminant is: (1) p #tl prime: l|c,l"1pmod pqu ´1 `´p `1 2 ¯#tl prime: l|c,l"1 or ´1pmod pqu ¯ if p 2 does not divide c,…”

“…The evidence provided in [21] is over function fields, by means of the approach of Ellenberg, Venkatesh and Westerland [6]. In a recent preprint, Alberts and Klys [1] offered evidence for the heuristics in Wood's work [21] over number fields using the approach of Fouvry and Klüners. It is interesting to note that in a previous work Klys [14] extended the work of Fouvry and Klüners to the p-torsion of cyclic degree p extensions.…”