---
title: "Note on a theorem of Professor X"
authors:
  - "Levent Alpoge"
year: 2021
venue: "arXiv preprint"
arxiv: "2109.14328"
doi: null
source: "https://arxiv.org/abs/2109.14328"
topics:
  - "arithmetic-geometry"
  - "number-theory"
seed_rank: 1205
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 8
lineage: arithmetic-geometry
cites:
  []
see:
  - "1200-conditional-algorithmic-mordell"
---

# Note on a theorem of Professor X

## One-sentence takeaway

Optimise Siegel's 1926 pseudonymous letter (published as "X") to bound S-integral points on y^2=f(x), deg f>=5, by 2^{rk Jac} * O(1)^{d^3([K:Q]+#S)} — better than Evertse-Silverman and Bombieri-Gubler — by delaying the three-root descent.

## Why it's lovely

Siegel wrote Mordell a letter, asked not to publish, Mordell published it as X. Alpoge executes the descents "in the presence of only one root until the last possible moment."

## Problem

Uniform bounds for hyperelliptic integral points in terms of rank and (d, K, S), not height.

## Design

Optimised Siegel 1926 descent.

## Evidence

arXiv:2109.14328, 29 Sep 2021.

## Limitations

Degree >=5, monic, S-unit discriminant. Not a full effective Faltings.

## Implications for Broadside

The historical-note card in the effectivity spine.

## Bottom line

Mint 2109.14328.

## Links

- arXiv: [2109.14328](https://arxiv.org/abs/2109.14328)