HEXAGONbeta

The Discriminant Structure of Frey Curve Isogeny Classes: Crater–Peak Duality, the Volcano Law, and a Mass Criterion

Primarily human-written textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Zeng Chunping ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

Let a,b,ca,b,c be coprime positive integers with a+b=ca+b=c, and let E:y2=x(x−a)(x+b)E: y^2=x(x-a)(x+b) be the associated Frey curve. We determine the structure of minimal discriminants across the Q\mathbb{Q}-isogeny class C(E)\mathcal{C}(E) by closed formulas in (a,b,c)(a,b,c). The three 22-torsion points of EE are rational, so EE is always the centre of its class; the first-level quotients via Velu's formula have monomial model discriminants {16(abc)2, 256a4bc, 256ab4c, 256abc4}\{16(abc)^2,\ 256a^4bc,\ 256ab^4c,\ 256abc^4\}, an exact collapse forced by the identity u=4f(r)u=4f(r) in Velu's algorithm. We prove the Crater--Peak Law: among the first-level vertices the minimal and maximal discriminants are min⁡{16(abc)2, 256min⁡(a,b)4max⁡(a,b)c}\min\{16(abc)^2,\,256\min(a,b)^4\max(a,b)c\} and 256abc4256abc^4, the exponent 44 landing on the smallest entry (crater) and the largest (peak); for star classes these are the class extrema, while on square or quartic triggers the extrema are the minimum and maximum over ten explicit candidates. The class size ∣V∣∈{4,6,8}|V|\in\{4,6,8\} is decided by the square and quartic character of acac and bcbc (107 samples, zero failures), matching the isogeny-torsion graph classification via the lcm of the isogeny matrix. When acac or bcbc is a square the class extends to depth 22 with model discriminants 210ACu8v22^{10}ACu^8v^2 and −210ACu2v8-2^{10}ACu^2v^8, and to depth 33 on quartic triggers with discriminants 211pq u16v4e2^{11}pq\,u^{16}v^4e and 211pq u4v16e2^{11}pq\,u^4v^{16}e. On the multiplicative bad-prime set π\pi we prove the Volcano Law: each vertex carries an integer height vector k(V)∈Zπk(V)\in\mathbb{Z}^{\pi} with vp(ΔVmin⁡)=2kp(V)vp(Δcratermin⁡)v_p(\Delta^{\min}_V)=2^{k_p(V)}v_p(\Delta^{\min}_{\mathrm{crater}}), and along every 22-edge k↦k+σk\mapsto k+\sigma with σ∈{±1}π\sigma\in\{\pm1\}^{\pi} independently per prime (578 edge-prime samples, zero exceptions), via the Tate parameterization. For odd primes p∣min⁡(a,b)p\mid\min(a,b) we have vp(j)=−2vp(min⁡)v_p(j)=-2v_p(\min), so the entire law is read off the jj-invariant. The peak curve is explicit (y2=x3−2(b−a)x2+c2xy^2=x^3-2(b-a)x^2+c^2x), and the jj-denominator exponent is a theorem: log⁡ den(j)/log⁡ num(j)=(α+2)/3+O(1/log⁡c)\log\,\mathrm{den}(j)/\log\,\mathrm{num}(j)=(\alpha+2)/3+O(1/\log c), with the exact discrepancy identity verified to machine precision on 21852185 samples --- the Pasten meeting point in its precise form; we also show the Pasten mechanism is information-free on the Frey family. As an application, for Ax+By=CzA^x+B^y=C^z we give an explicit two-branch formula for the crater ratio ρc\rho_c; and with α=log⁡min⁡(a,b)/log⁡c\alpha=\log\min(a,b)/\log c we prove σmax⁡→(5+α)q\sigma_{\max}\to(5+\alpha)q, so Szpiro is equivalent on the Frey family to the mass bound q≤6/(5+α)q\le 6/(5+\alpha), sharp at α=1\alpha=1 where Szpiro and abc coincide, while the residual gap lives at α=0\alpha=0, the Pillai/SS-unit corner. A Sage-verified table of 1717 classes (Kraus minimization agreeing with Sage minimal_model() throughout) gives σmax⁡>6\sigma_{\max}>6 in 14 of 17 rows (non-refutations: Szpiro allows O(1)/log⁡NO(1)/\log N slack). All absolute σ\sigma values are certified against Sage; structural statements are proved or machine-verified symbolically. We prove no conjecture: abc, Szpiro, Beal and the height conjecture remain open.

Tools used

SageMath
SageMath

References

  1. A. Barrios, E. González-Jiménez, M. Novak, Isogeny graphs of elliptic curves in characteristic zero , arXiv:2608.02494.arXiv
  2. A. Barrios, Explicit classification of isogeny graphs of rational elliptic curves , Int.\ J.\ Number Theory 19 (4), 913--936, 2023.
  3. H. Chiloyan, Á. Lozano-Robledo, A classification of isogeny-torsion graphs of $\Q$-isogeny classes of elliptic curves , Trans.\ London Math.\ Soc.\ 8 (1), 1--34, 2021.
  4. H. Darmon, A. Granville, On the equations zm=F(x,y)z^m=F(x,y) and Axp+Byq=CzrAx^p+By^q=Cz^r , Bull.\ London Math.\ Soc.\ 27 , 513--543, 1995.
  5. M. A. Kenku, On the number of $\Q$-isomorphism classes of elliptic curves in each $\Q$-isogeny class , J.\ Number Theory 15 (2), 199--202, 1982.
  6. B. Mazur, Rational isogenies of prime degree , Invent.\ Math.\ 44 , 129--162, 1978.
  7. H. Pasten, The largest prime factor of n2+1n^2+1 and improvements on subexponential ABC , Invent.\ Math.\ 236 , 373--385, 2024.
  8. C. L. Stewart, K. Yu, On the abc conjecture, II , Duke Math.\ J.\ 108 (1), 169--181, 2001.
  9. J. Vélu, Isogénies entre courbes elliptiques , C.\ R.\ Acad.\ Sci.\ Paris Sér.\ A-B 273 , A238--A241, 1971.

Version history

  1. v1 · 24 pagesSubmitted by Zeng ChunpingInitial depositCurrentNov 04, 2026