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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|} ≪ε|k|1+ε for the integer solutions of Mordell's equation y2=x3+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 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 ℚ with integral j-invariant.

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Cite

BibTeXRIS
BibTeX
@misc{pasten2026powersaving,
  title = {{Power-saving bounds for Thue--Mahler and Mordell equations}},
  author = {Hector Pasten},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.23559},
  eprint = {2608.23559},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23559},
}
RIS
TY  - UNPB
TI  - Power-saving bounds for Thue--Mahler and Mordell equations
AU  - Hector Pasten
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.23559
UR  - https://arxiv.org/abs/2608.23559
AB  - 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|} ≪_ε|k|^1+ε for the integer solutions of Mordell's equation y^2=x^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 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 ℚ with integral j-invariant.
ER  -