---
title: "The second moment of the number of integral points on elliptic curves is bounded"
authors:
  - "Levent Alpoge"
  - "Wei Ho"
year: 2018
venue: "arXiv preprint"
arxiv: "1807.03761"
doi: null
source: "https://arxiv.org/abs/1807.03761"
topics:
  - "arithmetic-geometry"
  - "number-theory"
  - "elliptic-curves"
seed_rank: 1209
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 9
lineage: arithmetic-geometry
cites:
  - title: "The average number of integral points on elliptic curves is bounded"
    url: "https://arxiv.org/abs/1412.1047"
    year: 2014
    arxiv: "1412.1047"
    doi: null
see:
  []
---

# The second moment of the number of integral points on elliptic curves is bounded

## One-sentence takeaway

Over any number field, the second moment of the number of S-integral points on elliptic curves is bounded; the pointwise bound is 2^{rk} O(1)^s, averaged via Selmer averages, including a new orbit-count with unbounded weights.

## Why it's lovely

Upgrade of the senior-thesis / 1412.1047 first-moment theorem to a second moment, with Wei Ho, uniformly in K and S.

## Problem

How many S-integral points does a typical elliptic curve have? First moment bounded (1211); second?

## Design

New pointwise bound in terms of rank, class group, [K:Q], and primes whose square divides Delta. Average 2^{rk} from Selmer averages. Equality-case moments use coregular representations with unbounded weights.

## Evidence

arXiv:1807.03761, v3 2 Nov 2022.

## Limitations

Still a preprint. Moments up to log2 5 on the full Weierstrass family.

## Implications for Broadside

See 1211 for the first-moment ancestor.

## Bottom line

Mint 1807.03761.

## Links

- arXiv: [1807.03761](https://arxiv.org/abs/1807.03761)