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

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 $\log \max{|x|,|y|} \ll_ε|k|<sup>{1+ε}$ for the integer solutions of Mordell's equation $y<sup>2=x<sup>3+k$ ($k$ a non-zero integer) by reducing the exponent $1+ε$ to $1/2+ε$; this is the first power-saving improvement without restrictions on $k$ in more than 50 years. Secondly, for integer squares and cubes of size $O(T)$ we improve the known unconditional separation lower bound $(\log T)<sup>{1-o(1)}$ obtained by Stark in 1973 to $(\log T)<sup>{2-o(1)}$. 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.