The heuristics of Cohen-Lenstra-Martinet-Malle give us precise predictions for the average sizes of the torsion subgroups of class groups of number fields enumerated by discriminant. Although very few cases of these heuristics have been proven, significant evidence has been gathered to support them. For example, for the family of number rings defined by binary n-ic forms ordered by height, Ho-Shankar-Varma proved that the average 2-torsion in the class group is less than or equal to the prediction of Cohen-Lenstra-Martinet-Malle, with equality conditional on certain plausible tail estimates. Moreover, their results are robust in the sense that the averages remain unchanged upon passing to any subfamily of rings defined by a finite set of congruence conditions.
It is thus natural to wonder whether the averages remain robust upon passing to families of number rings defined by global conditions. In this direction, Bhargava-Hanke-Shankar surprisingly discovered that the average 2-torsion in the class groups of cubic rings defined by binary cubic forms with fixed leading coefficient is larger than the prediction of Cohen-Lenstra-Martinet-Malle. Recently, Siad generalized the methods of Bhargava-Hanke-Shankar to prove that the average 2-torsion in the class groups of rings defined by monic binary forms of any fixed odd degree is larger than expected. In this talk, we calculate the average 2-torsion in class groups of rings defined by binary forms of any fixed odd degree having any fixed leading coefficient. To do this, we first answer a question of Ellenberg by parametrizing square roots of the ideal class of the different of ring defined by a binary form in terms of the integral orbits of a representation. Our new parametrization reduces the problem of studying non-monic forms to one of studying monic forms, where Siad's counting results apply. We conclude the talk by explaining how our methods can be used to prove that most Diophantine equations of the form $y^2 = F(x,z)$, where F is an integral binary form of odd degree, have no primitive integer solutions.