On the distribution of ${Cl}(K)[l^\infty]$ for degree $l$ cyclic fields

Abstract

Using a recent breakthrough of Smith [22], we prove that l 1 -class groups of cyclic degree l fields have the distribution conjectured by Gerth under GRH.

The p-rank $ε$-conjecture on class groups is true for towers of p-extensions

“…This restriction seems essential because of the existence of very rare fields giving exceptional large invariants M as shown in [7,8,30] for class groups (or [19] for torsion groups T ). This is also justified, in the framework of p-class groups, by the Koymans-Pagano density results [28] as analyzed in [14] for F p Q ; indeed, in any relative degree p cyclic extension, the algorithm defining the filtration (M h ) h≥0 is a priori unbounded, giving possibly large # M contrary to the p-ranks (or the # M[p r ] as seen in Corollary 3.7 which allows to take r ≫ 0, but constant regarding the familly F p e κ ). All the previous results on p-rank ε-inequalities fall within the framework of "genus theory" at the prime p for p-extensions; the case of degree d number fields, when p ∤ d, is highly non-trivial.…”

“…However, this suggests well, considering also the density results of [28], that the strong ε-conjecture is true "for almost all" elements of F p e κ .…”

“…. , then the related density results of Koymans-Pagano [28], all these questions being in relation with the classical heuristics/conjectures on class groups [1,4,5,9,13,31].…”

Successive maxima of the non-genus part of class numbers

“…The concept of ε-conjecture comes from the work of Ellenberg-Venkatesh [10], in close relation with the heuristics and conjectures of Cohen-Lenstra-Martinet. Many developments have followed as [1,7,9,11,14,16,21,22,26,28], to give an order of magnitude of various invariants attached to the class group of a number field K, according to the function ( |D K | ) ε of its discriminant, for any ε > 0. We shall now emphasize some of these ε-conjectures and precise our purpose, although this paper is not concerned by the q-class groups Cℓ K ⊗ Z q , for each prime q, except that genus theory involves the q-parts of the class group when q | d, as for the degree p cyclic extensions for which the p-Sylow subgroup Cℓ K ⊗ Z p may be very large: (i) The p-rank ε-conjecture for number fields claims that for all ε > 0:…”

“…, at least for sparse families of fields when considering the Koymans-Pagano density results [16] (see Section 2).…”