Quadrics in arithmetic statistics

TL;DR

This paper introduces a novel combination of the circle method and Bhargava's counting technique to analyze arithmetic statistics problems involving quadratic invariants, demonstrated through specific Selmer group averages.

Contribution

It develops a new method integrating the circle method with Bhargava's approach for counting orbits on quadratic subvarieties, expanding tools in arithmetic statistics.

Findings

Computed average sizes of 2-Selmer groups for specific elliptic curve families.

Introduced a smoothed version of Bhargava's counting method.

Derived averages over constrained families from unconstrained cases.

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.