{"id":"hexagon/2610.00544","pdf":"https://hexagonmath.org/pdf/2610.00544v1","url":"https://hexagonmath.org/abs/2610.00544v1","tags":["math.AG","msc:14G40","msc:11G50","Brumer--Kramer conductor bound","Faltings height","Fermat tower","Heisenberg tower","Szpiro conjecture","arithmetic Bogomolov--Miyaoka--Yau","conductor"],"score":0,"title":"The Height--Conductor Ratio on Curve Towers: Closed Forms for the Fermat Tower, the One-Third Barrier, and the Attainable Set","domain":"math.NT","source":"hexagon","authors":["Zeng Chunping"],"reviews":0,"subject":"math.NT","summary":"For a tower of curves $X_n$ over a number field with fixed bad-reduction set and $g_n \\to \\infty$, we study the stable Faltings height to conductor ratio of its Jacobian, $\\kappa_n := h_F(\\operatorname{Jac}(X_n))/\\log \\operatorname{cond}(\\operatorname{Jac}(X_n))$. We prove four groups of results. (I) Closed forms for the Fermat tower: slope $\\frac{\\log p}{4} n$ and second-order intercept $-6 \\log A$ ($A$ the Glaisher--Kinkelin constant), yielding $\\kappa_{\\mathrm{Fermat}} = \\frac{1}{8}$; the orbit stratification is strictly increasing and converges to $\\frac{1}{8}$ from below. (II) A…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["SageMath"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}