HEXAGONbeta

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 2 / 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 ratio of the stable Faltings height to the conductor of its Jacobian, κn:=hF(Jac(Xn))/log⁡cond(Jac(Xn))\kappa_n := h_F(\mathrm{Jac}(X_n))/\log \mathrm{cond}(\mathrm{Jac}(X_n)). We prove four groups of results. (I) Closed forms for the Fermat tower: slope log⁡p4 n\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 \leq \frac{1}{3} for GL2\mathrm{GL}_2-type towers with σ→1\sigma \to 1, conditional on arithmetic BMY in general (Problem 10.1) and unconditional on modular towers, where the BMY input is a theorem (Dolce-Mercuri, Mayer) and where X0(N)X_0(N) attains equality asymptotically, so the constant cannot be improved. (III) The barrier is strictly stronger than BMY: κ≤13⇒R≤4σ−1≤3\kappa \leq \frac{1}{3} \Rightarrow R \leq \frac{4}{\sigma} - 1 \leq 3. (IV) The closure of the attainable set in the abelian-variety tower world is [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 (countably) filled by reindexed product convexity; the curve-tower interior filling remains open. 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

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References

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Version history

  1. v1Submitted by Zeng ChunpingInitial depositSupersededOct 25, 2026
  2. v2Submitted by Zeng ChunpingSubmitted as a new versionCurrentOct 29, 2026