On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Abstract

It is conjectured that within the class group of any number field, for every integer ℓ ≥ 1, the ℓ-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the ℓ-torsion conjecture has relied crucially on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the ℓ-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the ℓ-torsion conjecture to be true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments." Pierce, Turnage-Butterbaugh, and Wood 7 Known results toward Conjecture 1.1, and relation to GRH 606 8 Appendices 611 References 614 1/2+ε K , 1 We will use Vinogradov's notation: A ≪ B denotes that there exists a constant C such that |A| ≤ CB, and A ≪ κ B denotes that C may depend on κ.

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:…”

A zero density estimate for Dedekind zeta functions

“…Restricting k ∈ F to squarefree discriminants D k allows Pierce, Turnage-Butterbaugh, and Wood to show that the second maximum is [20,19] for recent comprehensive accounts.…”

Genus theory and ε-conjectures on p-class groups

“…After the Cohen-Lenstra-Martinet, Adam-Malle, Delaunay-Jouhet, Gerth, Koymans-Pagano,... heuristics, conjectures, or density statements, on the order and structure of Cℓ K ⊗ Z p [4,5,1,30,8,15,28], many authors study and prove inequalities of the form # (Cℓ K ⊗ F p ) ≤ C d,p,ε • ( √ D K ) c+ε , with positive constant c < 1 as small as possible (e.g., under GRH, the inequality Proposition 1]; see also [14, § 1.1] for more examples and comments). The various links between this ε-conjecture and the above classical heuristics (or results) are described in [33,§ 1.1,Theorem 1.2,Remark 3.3].…”

“…For a general history upon today about such inequalities, we refer to some recent papers of the bibliography (e.g., [9,10,14,33,43]) in which the reader can have a more complete list of recent contributions. For short, we shall call "p-rank ε-conjecture" the case:…”