---
title: "The average number of integral points on elliptic curves is bounded"
authors:
  - "Levent Alpoge"
year: 2014
venue: "arXiv preprint"
arxiv: "1412.1047"
doi: null
source: "https://arxiv.org/abs/1412.1047"
topics:
  - "arithmetic-geometry"
  - "number-theory"
  - "elliptic-curves"
seed_rank: 1211
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 10
lineage: arithmetic-geometry
cites:
  - title: "The second moment of the number of integral points on elliptic curves is bounded"
    url: "https://arxiv.org/abs/1807.03761"
    year: 2018
    arxiv: "1807.03761"
    doi: null
see:
  - "1209-the-second-moment-of-the-number-of-integral-points-on-elliptic"
---

# The average number of integral points on elliptic curves is bounded

## One-sentence takeaway

When Weierstrass models E_{A,B}/Q are ordered by naive height, the average of #E(Z) is < 66 (at most 8/9 on the minimalist conjecture) — the Harvard senior thesis / Fay Prize paper, and the last solo journal-track result he delayed on purpose.

## Why it's lovely

Silverman / Helfgott / Helfgott-Venkatesh sphere-packing plus Bhargava-Shankar Selmer averages plus local heights for "most" curves. Same method on y^2=x^3+B, y^2=x^3+Ax, y^2=x^3-D^2 x.

## Problem

How many integral points does a random elliptic curve have?

## Design

Small / medium (spherical codes) / large (height gap) split; average 2^{rk}.

## Evidence

arXiv:1412.1047, v3 Feb 2015. Senior thesis "The average elliptic curve has few integral points" is this paper — do not remint the thesis.

## Limitations

Quasiminimal Weierstrass models, not everywhere-integral Neron models. First moment only (see 1209).

## Implications for Broadside

The origin of the Alpoge integral-points spine.

## Bottom line

Mint 1412.1047; skip the thesis PDF.

## Links

- arXiv: [1412.1047](https://arxiv.org/abs/1412.1047)