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

Let $n \geq 3$. 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 class groups of not-necessarily-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 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. 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 $p \mid n > 2$.