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Quadrics in arithmetic statistics

Levent Alpöge

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.

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 -Selmer groups in the families and . 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 from knowledge of the averages over "unconstrained" families.

Paper Structure

This paper contains 16 sections, 14 theorems, 95 equations.

Table of Contents

  1. Introduction.
  2. Main theorem.
  3. Main technique.
  4. Acknowledgments.
  5. Smoothing in Bhargava's counting method.
  6. Proof of Theorem \ref{['all six families']} for $k\equiv 1\mkern4mu({\operator@font mod}\mkern6mu6)$.
  7. Reduction to point counting.
  8. Avoiding a calculation of local densities.
  9. The uniformity estimate.
  10. Point counting.
  11. Proof of Theorem \ref{['all six families']} for $k\equiv 2\mkern4mu({\operator@font mod}\mkern6mu6)$.
  12. Reduction to point counting.
  13. The uniformity estimate.
  14. Point counting.
  15. Proof of Theorem \ref{['all six families']} for $k\equiv 5\mkern4mu({\operator@font mod}\mkern6mu6)$.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k\in \mathbb{Z}$. Let $\mathcal{B}\subseteq \mathbb{Z} - \{0\}$ be a set of positive density defined by congruence conditions. Then: with equality if e.g.We write "with equality if e.g." to emphasize that this is not an if and only if statement.$\mathcal{B}\subseteq \mathbb{Z}-\{0\}$ is defined by finitely many congruence conditions.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 4.1: The "tail" estimate.
  • Lemma 4.2: The "bulk" estimate.
  • Lemma 4.3
  • proof : Proof of Theorem \ref{['all six families']} for $k\equiv 1\mkern4mu({\operator@font mod}\mkern6mu6)$ assuming Lemmas \ref{['the tail estimate']} and \ref{['the bulk estimate']}.
  • proof : Proof of Lemma \ref{['the tail estimate']}.
  • proof : Proof of Lemma \ref{['the bulk estimate']}.
  • Lemma 5.1
  • Lemma 5.2
  • proof : Proof of Lemma \ref{['the tail estimate for squares']}.
  • ...and 19 more