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In this article, we combine Bhargava's geometry-of-numbers methods with thedynamical point-counting methods of Eskin--McMullen and Benoist--Oh to developa new technique for counting integral points on symmetric varieties lyingwithin fundamental domains for coregular representations. As applications, westudy the distribution of the $2$-torsion subgroup of the class group in thinfamilies of cubic number fields, as well as the distribution of the $2$-Selmergroups in thin families of elliptic curves over $\mathbb{Q}$. For example, ourresults suggest that the existence of a generator of the ring of integers withsmall norm has an increasing effect on the average size of the $2$-torsionsubgroup of the class group, relative to the Cohen--Lenstra predictions.

Counting integral points on symmetric varieties with applications to arithmetic statistics