The 2026 landscape of weak Szpiro: an audit of Soulé's theorems, the constant $144$, and the two barriers
Year: 2026
Type: preprint
Abstract: We record two points concerning the preprint arXiv:2611.05217v1, which proves an effective Mordell theorem for genus-2 curves (Theorem A), a weak uniform abc theorem $c \le A(\varepsilon) N^{24+\varepsilon}$ (Theorem B), and a weak Szpiro inequality $\log|\Delta(E)| \le (144(1+\log 24)+\varepsilon)\log N(E)$ for semistable $E/\mathbb{Q}$ (Theorem 3). First, two applications of Corollary 1 in §6.3 use the corollary outside its stated hypothesis $\deg = 0$; we give the three-line repair and confirm that all estimates and constants are unaffected. Second, the factor $(1+\log 24)$ in Theorem 3 comes from a single crude bound $\omega(m) \le 1+\log m$ in Moret-Bailly's Theorem 6.4; replacing it by the classical sharp bound $\omega(m) \le (1+o(1))\log m / \log\log m$ improves the constant to $144+\varepsilon$ with the same proof. As a complement we report direct recomputation of the intersection numbers of Proposition 8 ($\omega^2 = 12k$, $B^2 = -16k$, $L^2 = -20k$) and the torsion count of Proposition 9(vi) ($\deg R^1\phi_*(B)_{\mathrm{tor}} \ge 8k$); they check out as stated.
Author: Zeng, Chunping
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225777781
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