Petersson-Rigid Lattices in a Census of 100 Rank-Three Root Bases

The first paper worked out one family of four lattices in full — |det| = 12, 24, 36and 72. This paper generalizes it. We enumerate the 44 symmetrizable rank-threehyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, andcompute the obstruction space S5/2(rho_L) of the 98 that lie within our weight-5/2budget: 10 vacuous, 34 unobstructed, 54 obstructed. The vacuous ten are the lattices Bruinier, Ehlen and Freitag call simple, read insignature (2,3). They classified them — there are 15 — and our ten realize five of thefifteen; of the other ten, five need more than three generators and cannot be thediscriminant form of a rank-three lattice at all, three fail |det L| = 2k^2, and twoare simply not root bases. Within the rank-three hyperbolic world the simple latticesare exactly the Feingold-Frenkel neighbours of index k <= 4. The discriminant group L'/L carries a finite quadratic form, and the finite group ofits isometries acts on the weight-3/2 cusp forms for rho_L, the bottom antisymmetricrung of the weight tower. We survey the lattices on which that action is absolutelyirreducible of dimension at least two, so that the Petersson pairing on that space ispinned down up to a single scalar. The condition alone cuts the census to three: L_4 ofthe first paper, and two new ones at |det| 40 and 88. The three quaternion discriminantsthat occur are 6, 10 and 22, the three for which the Shimura curve X^D has genus zero.Rigidity uses no quaternion input, so we record this as an observation about the censusand not as a characterisation. Both invariants of the first paper — the shadow norm ||Xi||^2 and the Petersson scalart — were single points there. Three rigid lattices instead of one are where theirspecial faces come off: the irrational part of each generalizes, ||Xi||^2 againstL(f,1) where the first paper read L(f,2), and t against an elliptic curve's imaginaryperiod where it read Gamma(1/3). Both numerically, to between 26 and 58 digits. --- Files. V2.pdf is the paper. 100.zip is the reproduction archive, version 3.5: thecensus in machine-readable form, every script, and a single-command checker (verify.py,29 assertions) with its reference output. Its entry point is CENSUS.tsv, one table withone row per root basis; START_HERE.md indexes the files. dossier_1213_v3_5.md is thecompanion falsification dossier, which states each claim about the chain (1,2,1,3) witha grade, a reproduction recipe and the known weaknesses, and includes a tamper batteryshowing the checks are not vacuous. All vector-valued modular form computation is performed by weilrep (Brandon Williams,https://github.com/btw-47/weilrep) at commit e3a7784. Nothing here reimplements it. Licensing. The paper and the dossier are CC-BY-4.0. The scripts in 100.zip are derivedworks of weilrep and carry GPL-2.0-or-later; see LICENSE inside the archive.