Faltings heights of Shimura curve Jacobians at levels divisible by 2 and 3
Year: 2026
Type: preprint
Abstract: Let $J_0^D(M)$ be the Jacobian of the Shimura curve $X_0^D(M)$, $D>1$, of genus $g$, and $N=DM$. The companion paper (rStation math_4) proved $h_F(J_0^D(M))\ge(\frac13-o(1))\,g\log N$ for squarefree $N$ prime to $6$. This excludes the levels used in Pasten's Shimura-curve approach to abc, where $2^r$ ($r\le8$) and $3$ divide $N$. We remove the restriction for bounded $2$-part: if $M=M_0 2^r3^s$ with $M_0$ squarefree and prime to $6D$, $r\le r_0$, $s\le1$, and $2,3$ allowed to divide $D$ (including $D=6$), then $12\,h_F(J/\mathbf Q)\ge4(g-1)[\sum_{q\mid D,q\ge5}\frac q{q-1}\log q+\sum_{q\mid D,q\le3}\frac{3q-1}{4(q-1)}\log q+\sum_{p\mid M_0}\frac p{p+1}\log p]-C_{r_0}(g+2^{\omega(N)}\log N)$, hence $h_F\ge(\frac13-o_{r_0}(1))\,g\log N$ uniformly. At $2$ and $3$ the models are neither semistable nor normal after base change, and elliptic sections can meet singular points. Instead of computing these fibres we prove that the completed local rings, with group action, forgetful map and elliptic germs, have finitely many isomorphism types, and we use three sign facts: the Artin conductor is non-negative, $\pi^*L\cdot V\ge0$, and $V^2$ is bounded below. As a consequence, the lattice ceiling and the model-relative obstruction of the companion paper extend to all factorizations in Pasten's Theorem 16.7, so to all abc triples with many prime factors: inside the stated relaxation model the Petersson-ratio channel has exponent at least $2/3$. We claim nothing about the stable height, which is left open, and nothing about the abc conjecture. The manuscript was produced with AI research agents and has not been reviewed by a human expert. Archival copy of version 1 of this paper, published on rStation on 2 November 2026 (rStation math paper 5, v1). paper.md is the manuscript exactly as published there; the PDF is typeset from it with the published title and abstract. Files: paper.md — SHA-256 0566e63b752cd05142ed33713a4e1f40ad25b180ab9a404f0c663ec0c3cd2ec6 faltings-heights-of-shimura-curve-jacobians-at-levels-divisible-by-2-and-3.pdf — SHA-256 bf8aa34577e34744a4205304881a7922362984dcfd8538f85210707a71f31226
Author: Li, Hongmin
Institutions: Institute of Science Tokyo, The University of Tokyo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225580239