---
title: "Quadrics in arithmetic statistics"
authors:
  - "Levent Alpoge"
year: 2021
venue: "arXiv preprint"
arxiv: "2110.03947"
doi: null
source: "https://arxiv.org/abs/2110.03947"
topics:
  - "arithmetic-geometry"
  - "number-theory"
  - "elliptic-curves"
seed_rank: 1204
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 9
lineage: arithmetic-geometry
cites:
  - title: "Integers expressible as the sum of two rational cubes"
    url: "https://arxiv.org/abs/2210.10730"
    year: 2022
    arxiv: "2210.10730"
    doi: null
see:
  - "1201-integers-expressible-as-the-sum-of-two-rational-cubes"
---

# Quadrics in arithmetic statistics

## One-sentence takeaway

Reintroduce the circle method into Bhargava-style orbit counting so you can count on a quadric cut out by a quadratic invariant; worked example: average 2-Selmer is 3 on y^2=x^3+B and y^2=x^3+B^2.

## Why it's lovely

The method paper behind the two-cubes work. Smoothed Bhargava counting plus a deduction that the constrained average is 3 from the unconstrained one.

## Problem

Arithmetic statistics on invariant quadrics — the locus the usual geometry-of-numbers wants to avoid.

## Design

Circle method times Bhargava. Formal deduction of the average 3 from unconstrained families.

## Evidence

arXiv:2110.03947, v2 14 Oct 2024.

## Limitations

Still a preprint. The spectacular application is 1201.

## Implications for Broadside

See 1201 for the Davenport application.

## Bottom line

Mint 2110.03947 as the method; do not remint as two-cubes.

## Links

- arXiv: [2110.03947](https://arxiv.org/abs/2110.03947)