math.NT — Number Theory

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

Contributed by Zeng Chunping

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 one-third barrier $\kappa \le \frac{1}{3}$ for $\mathrm{GL}_2$-type towers, with three independent proofs (systole growth on $X_1$, short geodesic counting on $X_0$, and a function-field Mason comparison); the barrier is identified with Parshin's arithmetic Bogomolov--Miyaoka--Yau constant and is asymptotically attained by the modular tower $X_0(N)$, hence cannot be improved. (III) The barrier is strictly stronger than BMY: $\kappa \le \frac{1}{3} \Rightarrow R \le \frac{4}{\sigma} - 1 \le 3$. (IV) The attainable set in the tower world is exactly $[\frac{1}{12}, \frac{1}{3}]$: the lower end is unconditional (Zhang, Wilms) and attained by Heisenberg towers, the upper end is the barrier and is attained by modular towers, and the interior is densely filled by reindexed product convexity. We further record three structural negative results: the archimedean channel is absorbed into the discriminant channel by the product formula; totally rational branch-point hyperelliptic families are closed off by Evertse's theorem; and the Belyi/PU route is likewise closed by $S$-unit equations. The discriminant channel is recast in the thickness language with an explicit upper bound (unconditional for automorphic towers, conditional on the analyticity of $L(A,s)$ in general), and a fully unconditional alternative based on the Brumer--Kramer conductor bound moves---but does not close---the remaining gap. All conditional dependencies are explicitly flagged in the body; all corrections are recorded in the revision history, Appendix B.