Average $2$-Torsion in Class Groups of Rings Associated to Binary $n$-ic Forms

Abstract

Let n ≥ 3 be an integer. In this paper, we prove several theorems concerning the average behavior of the 2-torsion in class groups of rings defined by integral binary n-ic forms having any fixed odd leading coefficient and ordered by height. Specifically, we compute an upper bound on the average size of the 2-torsion in the class groups of maximal orders arising from such binary forms; as a consequence, we deduce that most such orders have odd class number. When n is even, we compute corresponding upper bounds on the average size of the 2-torsion in the oriented and narrow class groups of maximal orders; moreover, we obtain an upper bound on the average number of non-trivial 2-torsion elements in the ordinary, oriented, and narrow class groups of notnecessarily-maximal orders, where we declare a 2-torsion class to be trivial if it is represented by a 2-torsion ideal. We further prove that each of these upper bounds is in fact an equality, conditional on a conjectural uniformity estimate that is known to hold when n = 3.To prove these theorems, we first answer a question of Ellenberg by parametrizing square roots of the class of the different of a ring arising from a binary form in terms of the integral orbits of a certain representation. Since the set of square roots of the class of the different is a torsor for the 2-torsion in the class group, this new parametrization allows us to count 2-torsion classes.Our theorems extend recent work of Bhargava-Hanke-Shankar in the cubic case and of Siad in the monic n-ic case to binary forms of any degree having any fixed odd leading coefficient. When n is odd, our result demonstrates that fixing the leading coefficient has the surprising effect of augmenting the average 2-torsion in the class group, relative to the prediction given by the heuristics of Cohen-Lenstra-Martinet-Malle. When n is even, analogous heuristics are yet to be formulated; together with Siad's results in the monic case, our theorems are the first of their kind to describe the average behavior of the p-torsion in class groups of n-ic rings where p | n > 2.

The second moment of the size of the $2$-Selmer group of elliptic curves

“…, and so we may forget about I altogether. Moreover, condition (b) in Theorem 2.3 holds trivially because f 0 is a unit in K. We thus obtain the following immediate consequence of Theorem 2.3: 38,Prop. 26]).…”

“…The parametrization in Theorem 2.7 takes on a similarly simple form when R = Z p for a prime p and R f is the maximal order in its algebra of fractions K f . Indeed, by imitating the proof of [38,Theorem 21], we deduce the following consequence of Theorem 2.7: Corollary 2.9. Let R = Z p for a prime p, and suppose that R f is the maximal order in K f .…”

“…, n} and f 2 has unit leading coefficient. It then follows from [38,Lemma 19] that R f ≃ R f1 ×R f2 . Thus, any fractional ideal I of R f may be expressed as a product I 1 × I 2 of fractional ideals I i of R fi .…”

“…To prove the theorem, we first show that the elements of Sel 2 (J f ) correspond to square roots of the ideal class of the inverse different of R f . We then apply a parametrization first introduced by the third-named author in [38], which relates square roots of the class of the inverse different of R f to pairs (A, B) of n-ary quadratic forms over Z such that det(xA + yB) = f mon (x, y). This parametrization is inspired by a related parametrization due to Wood (see [43]).…”

“…Then, in §2.2, we study orbits of SL n /µ 2 on pairs of n-ary quadratic forms (A, B) for fixed A. In §2.3, we define the ring R f associated to a binary n-ic form f , and in §2.4, we recall a parametrization from [38] of square roots of the ideal class of the inverse different of R f in terms of orbits of the aforementioned representations. We conclude in §2.5 by describing the image of the parametrization discussed in §2.4.…”