Note on a theorem of Professor X

TL;DR

This paper refines Siegel's 1926 technique to establish tighter bounds on the number of integral solutions to hyperelliptic equations over number fields, improving previous bounds by executing descents more efficiently.

Contribution

It optimizes Siegel's method to derive sharper bounds on integral points, advancing the understanding of hyperelliptic curve solutions over number fields.

Findings

Improved bounds on the number of integral solutions to hyperelliptic equations.

Enhanced descent technique that delays root reduction for better bounds.

Comparison showing bounds are tighter than previous results by Evertse-Silverman and Bombieri-Gubler.

Abstract

Between his arrival in Frankfurt in $1922$ and and his proof of his famous finiteness theorem for integral points in $1929$, Siegel had no publications. He did, however, write a letter to Mordell in $1926$ in which he explained a proof of the finiteness of integral points on hyperelliptic curves. Recognizing the importance of this argument (and Siegel's views on publication), Mordell sent the relevant extract to be published under the pseudonym "X". The purpose of this note is to explain how to optimize Siegel's $1926$ technique to obtain the following bound. Let $K$ be a number field, $S$ a finite set of places of $K$, and $f\in \mathfrak{o}_{K,S}[t]$ monic of degree $d\geq 5$ with discriminant $\Delta_f\in \mathfrak{o}_{K,S}^\times$. Then: $$\#|\{(x,y) : x,y\in \mathfrak{o}_{K,S}, y^2 = f(x)\}|\leq 2^{\mathrm{rank}\,\mathrm{Jac}(C_f)(K)}\cdot O(1)^{d^3\cdot ([K:\mathbb{Q}] + \#|S|)}.$$ This improves bounds of Evertse-Silverman and Bombieri-Gubler from $1986$ and $2006$, respectively. The main point underlying our improvement is that, informally speaking, we insist on "executing the descents in the presence of only one root (and not three) until the last possible moment".