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arXiv · 2608.23559

Power-saving bounds for Thue--Mahler and Mordell equations

Hector Pasten
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Abstract

We prove a new effective bound for cubic Thue--Mahler equations with power-saving dependence on the regulator. This has some applications. First, we improve Stark's bound $\log \max\{|x|,|y|\} \ll_\epsilon |k|^{1+\epsilon}$ for the integer solutions of Mordell's equation $y^2=x^3+k$ ($k$ a non-zero integer) by reducing the exponent $1+\epsilon$ to $1/2+\epsilon$; this is the first power-saving improvement without restrictions on $k$ in more than 50 years. Secondly, for integer squares and cubes of size $\asymp T$ we improve the known unconditional separation lower bound $(\log T)^{1-o(1)}$ obtained by Stark in 1973 to $(\log T)^{2-o(1)}$. Finally, we obtain a power-saving improvement in the conductor aspect of the strongest currently available bounds for Frey's height conjecture (a strengthening of Szpiro's conjecture) in the case of elliptic curves over $\mathbb{Q}$ with integral $j$-invariant.

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Hector Pasten. 2026-08-24. Power-saving bounds for Thue--Mahler and Mordell equations. https://arxiv.org/abs/2608.23559

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