---
title: "The average number of rational points on genus two curves is bounded"
authors:
  - "Levent Alpoge"
year: 2018
venue: "arXiv preprint"
arxiv: "1804.05859"
doi: null
source: "https://arxiv.org/abs/1804.05859"
topics:
  - "arithmetic-geometry"
  - "number-theory"
seed_rank: 1210
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 average number of rational points on genus two curves is bounded

## One-sentence takeaway

Genus-2 curves over Q with a marked Weierstrass point, ordered by height, have bounded average #C(Q), by the same small / Mumford-gap plus sphere-packing / Bombieri-Vojta trichotomy as the elliptic integral-point paper, plus Bhargava-Gross 2-Selmer.

## Why it's lovely

Faltings on average, with an explicit 1.872^{rk} for non-small points.

## Problem

Average #C(Q) on a universal family of genus-2 curves.

## Design

Small points by hand; medium by Mumford gap plus Kabatiansky-Levenshtein; large by Bombieri-Vojta. Average the 2^{rk} via Bhargava-Gross.

## Evidence

arXiv:1804.05859, 16 Apr 2018. Title on arXiv omits "odd"; homepage says "odd genus two."

## Limitations

Marked Weierstrass point. Medium-height step uses Stoll's genus-2 local heights.

## Implications for Broadside

The g=2 sibling of 1211.

## Bottom line

Mint 1804.05859.

## Links

- arXiv: [1804.05859](https://arxiv.org/abs/1804.05859)