math.NT — Number Theory

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

Contributed by Zeng Chunping

For the Fermat tower $F_N\colon x^N+y^N=1$, $N=p^n$, we assemble the complete asymptotic machine for the stable Faltings height $h_F(J(F_N))$: the main-chain closed form $\hat h_n=\frac{\log p}{4}n-6\log A-\frac{p\log p}{4(p^2-1)}+\Theta\!\bigl(\frac{n\log p}{p^n}\bigr)$; an exact spectral identity $h(\Phi)=\frac{\mu(\Phi)}{2}-\sum_\chi m(\Phi,\chi)\rho_\chi$ with $\rho_\chi=R_\chi+\log f_\chi$; an arithmetic channel for the deviation functional through the Bernoulli-mass $M_1(\Phi)=\sum_\chi m(\Phi,\chi)\log|B_{1,\bar\chi}|$ ($R^2=0.78$, residual std $1.90$); a profile law $\mathbb{E}[M_{c'}]=3^{c'-2}\Lambda(n)$ proved by a carry-counting identity; and the layerwise height closure $\mathrm{mean}\,h^{\mathrm{Col}}(n)=\frac{3^{n-1}\log 3}{4}(n-n^*)$ with $n^*\to 5.69$, which explains the sign flip at $n=5$ and the $\Delta_9$ explosion ($\hat h^{\mathrm{Fal}}_9=+7.24$, twenty-five times outside the corridor $0.25$--$0.30$). All identities are certified at the exact-arithmetic level on layers $n\le 12$ ($2187$ orbits and $729$ odd characters at layer $7$, statistical verdicts at layers $8$--$12$). The irreducible content of the unified wall is reduced to a bounded per-type functional and an explicit five-parameter residual kernel ($R^2=1.0000$ on the orbit level).