Quadrics in arithmetic statistics

Abstract

We (re)introduce the circle method into arithmetic statistics. More specifically, we combine the circle method with Bhargava's counting technique in order to give a general method that allows one to treat arithmetic statistical problems in which one is trying to count orbits on a subvariety of affine space defined by the vanishing of a quadratic invariant. We explain this method by way of example by computing the average size of $2$-Selmer groups in the families $y^2=x^3+B$ and $y^2=x^3+B^2$. In the course of the argument we introduce a smoothed form of Bhargava's aforementioned method, as well as a trick with which we formally deduce that the above averages are $3$ from knowledge of the averages over "unconstrained" families.

25 pages, comments welcome as usual! v2: added dedication to Chrysi Notskas and a requested clarification that, as usual, no journal will receive this paper (due to my ideology and their rot and obsolescence). Enjoy