The Height--Conductor Ratio on Curve Towers: Closed Forms for the Fermat Tower, the One-Third Barrier, and the Attainable Set
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…
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