HEXAGONbeta

Asymptotic closed forms for the Faltings height of Fermat Jacobians and the spectrum of logarithmic derivatives of L-functions

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

Contributed by Zeng Chunping ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

For the Fermat tower FN ⁣:xN+yN=1F_N\colon x^N+y^N=1, N=pnN=p^n, we assemble the complete asymptotic machine for the stable Faltings height hF(J(FN))h_F(J(F_N)): the main-chain closed form h^n=log⁡p4n−6log⁡A−plog⁡p4(p2−1)+Θ ⁣(nlog⁡ppn)\hat h_n=\frac{\log p}{4}n-6\log A-\frac{p\log p}{4(p^2-1)}+\Theta\!\bigl(\frac{n\log p}{p^n}\bigr); an exact spectral identity h(Φ)=μ(Φ)2−∑χm(Φ,χ)ρχh(\Phi)=\frac{\mu(\Phi)}{2}-\sum_\chi m(\Phi,\chi)\rho_\chi with ρχ=Rχ+log⁡fχ\rho_\chi=R_\chi+\log f_\chi; an arithmetic channel for the deviation functional through the Bernoulli-mass M1(Φ)=∑χm(Φ,χ)log⁡∣B1,χˉ∣M_1(\Phi)=\sum_\chi m(\Phi,\chi)\log|B_{1,\bar\chi}| (R2=0.78R^2=0.78, residual std 1.901.90); a profile law E[Mc′]=3c′−2Λ(n)\mathbb{E}[M_{c'}]=3^{c'-2}\Lambda(n) proved by a carry-counting identity; and the layerwise height closure mean hCol(n)=3n−1log⁡34(n−n∗)\mathrm{mean}\,h^{\mathrm{Col}}(n)=\frac{3^{n-1}\log 3}{4}(n-n^*) with n∗→5.69n^*\to 5.69, which explains the sign flip at n=5n=5 and the Δ9\Delta_9 explosion (h^9Fal=+7.24\hat h^{\mathrm{Fal}}_9=+7.24, twenty-five times outside the corridor 0.250.25--0.300.30). All identities are certified at the exact-arithmetic level on layers n≤12n\le 12 (21872187 orbits and 729729 odd characters at layer 77, statistical verdicts at layers 88--1212). The irreducible content of the unified wall is reduced to a bounded per-type functional and an explicit five-parameter residual kernel (R2=1.0000R^2=1.0000 on the orbit level).

Tools used

SageMath
SageMath

References

  1. P. Colmez, Périodes des variétés abéliennes à multiplication complexe , Ann.\ of Math.\ 138 (1993), 625--683. MR1239797.
  2. R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters , J. Reine Angew.\ Math.\ 385 (1988), 41--101. doi:10.1515/crll.1988.385.41.DOI
  3. B. H. Gross, On the periods of abelian varieties , arXiv:2005.04194; and Complex Multiplication lecture notes.arXiv
  4. C. Curilla and S. Wewers, Fermat curves of odd squarefree exponent , arXiv:1605.04548.arXiv
  5. A. Languasco and P. Moree, Euler's divergent series in arithmetic progressions , Ramanujan J., Kronecker limit formula in Eq. (10)--(12) (2020).
  6. L. C. Washington, Introduction to Cyclotomic Fields , 2nd ed., GTM 83, Springer, 1997.
  7. N. Aoki, Endomorphisms of jacobian varieties of Fermat curves , Compositio Math.\ 80 (1991), 85--102.
  8. T. Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces , Duke Math.\ J.\ 57 (1988), 151--173. MR0952229.

Version history

  1. v1Submitted by Zeng ChunpingInitial depositCurrentOct 31, 2026