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

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

Contributed by Zeng Chunping ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

For a tower of curves XnX_n over a number field with fixed bad-reduction set and gn→∞g_n \to \infty, we study the stable Faltings height to conductor ratio of its Jacobian, κn:=hF(Jac⁡(Xn))/log⁡cond⁡(Jac⁡(Xn))\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 log⁡p4n\frac{\log p}{4} n and second-order intercept −6log⁡A-6 \log A (AA the Glaisher--Kinkelin constant), yielding κFermat=18\kappa_{\mathrm{Fermat}} = \frac{1}{8}; the orbit stratification is strictly increasing and converges to 18\frac{1}{8} from below. (II) A one-third barrier κ≤13\kappa \le \frac{1}{3} for GL2\mathrm{GL}_2-type towers, with three independent proofs (systole growth on X1X_1, short geodesic counting on X0X_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 X0(N)X_0(N), hence cannot be improved. (III) The barrier is strictly stronger than BMY: κ≤13⇒R≤4σ−1≤3\kappa \le \frac{1}{3} \Rightarrow R \le \frac{4}{\sigma} - 1 \le 3. (IV) The attainable set in the tower world is exactly [112,13][\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 SS-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)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.

Provenance statement

The mathematical content originates from the author's working manuscripts. An AI assistant (large language model) was used for: English translation and typesetting; reconstructing formulas that had been garbled during PDF text extraction (each reconstruction verified by the author against the original pages); independent numerical cross-checks (exact counts and residue computations); and editorial restructuring, including the revision-history appendix. All AI-assisted modifications were reviewed by the author before inclusion. No theorem in the paper is due to the AI.

Tools used

SageMath
SageMath

References

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Version history

  1. v1Submitted by Zeng ChunpingInitial depositCurrentOct 25, 2026