The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders

Abstract

We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a quadratic order O we distinguish between the two groups: Cl 3 (O), the group of ideal classes of order 3; and I 3 (O), the group of ideals of order 3. We determine the mean values of both |Cl 3 (O)| and |I 3 (O)|, as O ranges over any family of orders defined by finitely many (or in suitable cases, even infinitely many) local conditions.As a consequence, we prove the surprising fact that the mean value of the difference |Cl 3 (O)|− |I 3 (O)| is equal to 1, regardless of whether one averages over the maximal orders in complex quadratic fields or over all orders in such fields or, indeed, over any family of complex quadratic orders defined by local conditions. For any family of real quadratic orders defined by local conditions, we prove similarly that the mean value of the difference |Cl 3 (O)| − 1 3 |I 3 (O)| is equal to 1, independent of the family.

The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

“…Let N (T ; X) denote the number of irreducible binary cubic forms contained in T , up to SL 2 (Z)-equivalence, having absolute discriminant at most X. Then we have the following theorem counting SL 2 (Z)-classes of integer-matrix binary cubic forms of bounded reduced discriminant, which easily follows from the work of Davenport [18] and Davenport-Heilbronn [19] (see [13,Theorem 19] for this deduction):…”

“…Recently, in a more direct proof of Davenport and Heilbronn's theorem on 3‐torsion elements in class groups of quadratic fields was given. This proof used a count of integer‐matrix binary cubic forms (), together with a direct correspondence between 3‐torsion ideal classes in quadratic fields and integer‐matrix binary cubic forms as studied in (see Section for a description of this correspondence over any Dedekind domain).…”

Nonabelian Cohen–Lenstra moments

“…This is a local condition at 2 on the quadratic fields. It is predicted that such local conditions do not affect the Cohen-Lenstra heuristics and in particular the conjectured averages in the case that G is abelian (see [BV15,BV16,Woo17b]).…”

Cohen-Lenstra heuristics and local conditions

“…We reduce the problem to counting cubic and quadratic extensions with certain local conditions, and then use the work of Davenport and Heilbronn [DH71] to count cubic extensions. This strategy and the computation of local masses follows along similar lines to the proof of [BV16,Corollary 4]. For a number field K, let O K denote its ring of integers.…”