The mean number of 3-torsion elements in ray class groups of quadratic fields

Abstract

We determine the average number of 3-torsion elements in the ray class groups of fixed (integral) conductor c of quadratic fields ordered by absolute discriminant, generalizing Davenport and Heilbronn's theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor c is taken to be a squarefree integer having very few prime factors none of which are congruent to 1 mod 3. Additionally, we compute the second main term for the number of 3-torsion elements in ray class groups with fixed conductor of quadratic fields with bounded discriminant.

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

“…In this section we make a certain choice for f in Heuristic assumption 2.4 with the aim of stating conjectures for the average of p-torsion of ray class groups. These statements were previously proved for p " 3 by Varma [18]. In fact, the present paper partly began as an effort to fit her results into a general heuristic framework.…”

“…For p " 3 this conjecture was recently proved by Varma [18] using geometry of numbers. In [18, §1] she asked whether one can formulate an extension of the Cohen-Lenstra heuristic that explains her result.…”

“…Organization of the material. The remainder of this paper is organized as follows: In §2 we explain our heuristic model for the distribution of the p-part of ray class sequences of imaginary quadratic number fields, for odd primes p. We draw several conjectures from this heuristic principle and verify its consistency with the theorems of Varma [18] in the imaginary quadratic case.…”

“…Building on the recent improvement of the work of Bhargava, Shankar and Tsimerman [2], they established their conjecture for the average size of the 3-torsion. Incidentally, the work [2] is also employed by Varma [18] on the average 3-torsion of ray class groups, which is placed in a general conjectural framework by the present paper.…”