The Discriminant Structure of Frey Curve Isogeny Classes: Crater–Peak Duality, the Volcano Law, and a Mass Criterion
Source: hexagonmath.org/abs/2611.00085v1
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…
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