{"id":"hexagon/2611.00085","pdf":"https://hexagonmath.org/pdf/2611.00085v1","url":"https://hexagonmath.org/abs/2611.00085v1","tags":["math.AG","msc:11G05","msc:11D75","msc:11G50","msc:11F80","msc:14H52","Diophantine equations","Faltings height","Frey curve","Szpiro conjecture"],"score":0,"title":"The Discriminant Structure of Frey Curve Isogeny Classes: Crater–Peak Duality, the Volcano Law, and a Mass Criterion","domain":"math.NT","source":"hexagon","authors":["Zeng Chunping"],"reviews":0,"subject":"math.NT","summary":"Let $a,b,c$ be coprime positive integers with $a+b=c$, and let $E: y^2=x(x-a)(x+b)$ be the associated Frey curve. We determine the structure of minimal discriminants across the $\\mathbb{Q}$-isogeny class $\\mathcal{C}(E)$ by closed formulas in $(a,b,c)$. The three $2$-torsion points of $E$ are rational, so $E$ is always the centre of its class; the first-level quotients via Velu's formula have monomial model discriminants $\\{16(abc)^2,\\ 256a^4bc,\\ 256ab^4c,\\ 256abc^4\\}$, an exact collapse forced by the identity $u=4f(r)$ in Velu's algorithm. We prove the Crater--Peak Law: among the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["SageMath"],"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}