Nonabelian Cohen–Lenstra moments

Abstract

In this paper we give a conjecture for the average number of unramified Gextensions of a quadratic field for any finite group G. The Cohen-Lenstra heuristics are the specialization of our conjecture to the case that G is abelian of odd order. We prove a theorem towards the function field analog of our conjecture, and give additional motivations for the conjecture including the construction of a lifting invariant for the unramified Gextensions that takes the same number of values as the predicted average and an argument using the Malle-Bhargava principle. We note that for even |G|, corrections for the roots of unity in Q are required, which can not be seen when G is abelian.

The 2‐Selmer group of a number field and heuristics for narrow class groups and signature ranks of units

“…Remark 7.7. There are other predictions for Prob(rk 2 C K = ρ): one due to Venkatesh-Ellenberg involving the Schur multiplier [47, § 2.4], work of Garton [24] accounting for roots of unity, and theorems of Wood in the function field case [50] that suggest predictions in the number field case. It is possible that these different perspectives all agree with Conjecture 7.3, but this has not yet been established.…”

Cohen-Lenstra heuristics and local conditions

“…We then will use the existence of a Hurwitz scheme parametrizing such extensions (as they are equivalently curves with a map to the line), which comes from work of Ellenberg, Venkatesh, and Westerland [EVW12], building on work of Romagny and Wewers [RW06]. Unlike in [BW17] and [Woo17b], in this paper we also use the homological stability results of Ellenberg, Venkatesh, and Westerland [EVW16] to have a bound on the ith cohomology groups of the Hurwitz schemes that is exponential in i.…”

Unramified extensions over low degree number fields

“…For ample k, we give an affirmative answer to both questions for arbitrary finite groups G. Namely, e(k(x), G) = ge(G) and G is the Galois group of an unramified k-regular extension L/M over a quadratic extension M/k(x), see Section 3. We note that further support for an affirmative answer follows from [20], which gives the desired extensions of k(x) when k is a sufficiently large finite field and ge(G) = 2. Moreover, such results are expected in cases where ge(G) > 2 as well, in view of [8], cf.…”

“…The distribution of (p-parts of) class groups over imaginary quadratic fields has been extensively studied, and is expected to be uniform by the Cohen-Lenstra heuristics. These heuristics were recently generalized to pro-p groups by Boston-Bush-Hajir [5], to pro-odd groups by Boston-Wood [6] and to finite groups by Wood [20]. Very little is known in the nonabelian case concerning the mere existence of such extensions, that is concerning: Question 1.…”