100个秩三根基的普查中的Petersson刚性格
Petersson-Rigid Lattices in a Census of 100 Rank-Three Root Bases
- Department of Mathematics, Chung-Ang University（中央大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文枚举100个秩三根基，计算障碍空间，筛选出3个Petersson刚性格，推广了前人的影子范数和Petersson标量不变量并验证数值精度。
AI中文摘要：
第一篇论文完整计算了一族四个格，其行列式绝对值|det|分别为12、24、36和72。本文将其推广，对44个可对称化的秩三双曲Cartan矩阵及其56个深度一编辑进行枚举，共得到100个根基，并在权重5/2预算内计算了其中98个的障碍空间S_{5/2}(ρ_L)：10个为空，34个无阻碍，54个有阻碍。这10个空障碍的格是Bruinier、Ehlen和Freitag所称的简单格，对应符号(2,3)。在他们的15个此类格中，本文的10个实现了其中5个；其余5个需要超过3个生成元，根本无法成为秩三格的判别式形式，3个不满足|det L|=2k²，还有2个并非根基。在秩三双曲领域内，简单格恰好是指数k≤4的Feingold-Frenkel邻域。判别式群L'/L带有有限二次型，其等距变换的有限群作用于ρ_L的权重3/2尖点形式，即最低反对称 rung。本文对该作用绝对不可约且维度至少为2的格进行了调查，因此Petersson配对可由单个标量确定。仅该条件就将普查范围缩小至3个：第一篇论文的L₄，以及|det|为40和88的两个新格。出现的四元数判别式为6、10和22，对应Shimura曲线X^D的亏格为0。刚性未使用任何四元数输入，因此本文将其记录为观察结果而非特征描述。第一篇论文的两个不变量——影子范数||Ξ||²和Petersson标量t——在那里均为单点。与第一篇论文不同的是，三个刚性格的特殊面显现：||Ξ||²对应L(f,1)的推广，而第一篇论文中是L(f,2)；t对应椭圆曲线的虚周期，而第一篇论文中是Γ(1/3)，两者均精确到26至58位数字。
英文摘要：
The first paper worked out one family of four lattices in full -- $|\det| = 12, 24, 36$ and $72$. This paper generalizes it. We enumerate the 44 symmetrizable rank-three hyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, and compute the obstruction space $S_{5/2}(ρ_L)$ of the 98 within our weight-$5/2$ budget: 10 vacuous, 34 unobstructed, 54 obstructed. The vacuous ten are the lattices Bruinier, Ehlen and Freitag call simple, read in signature $(2,3)$. Of their fifteen, our ten realize five; of the rest, five need more than three generators and cannot be the discriminant form of a rank-three lattice at all, three fail $|\det L| = 2k^2$, and two are simply not root bases. Within the rank-three hyperbolic world the simple lattices are exactly the Feingold-Frenkel neighbours of index $k \le 4$. The discriminant group $L'/L$ carries a finite quadratic form, and the finite group of its isometries acts on the weight-$3/2$ cusp forms for $ρ_L$, the bottom antisymmetric rung. We survey the lattices on which that action is absolutely irreducible of dimension at least two, so that the Petersson pairing there is pinned down up to a single scalar. The condition alone cuts the census to three: $L_4$ of the first paper, and two new ones at $|\det| = 40$ and $88$. The quaternion discriminants that 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 it as an observation, not a characterisation. Both invariants of the first paper -- the shadow norm $\|Ξ\|^2$ and the Petersson scalar $t$ -- were single points there. Three rigid lattices instead of one are where their special faces come off: $\|Ξ\|^2$ generalizes against $L(f,1)$ where the first paper read $L(f,2)$, and $t$ against an elliptic curve's imaginary period where it read $Γ(1/3)$. Both numerically, to 26-58 digits.
