指标为4的奇异重量康威不变雅可比形式
Singular-weight Conway-invariant Jacobi forms of index four
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中文总结 AI 辅助
本文解决了指标4的奇异重量Co₀不变雅可比形式空间的精确维数问题，证明其维数为6，并构造了该空间的一组自然基。
中文摘要 AI 辅助
设Λ为李格（Leech lattice），Co₀为Λ的自同构群。孙与王证明，奇异重量12、指标4的Co₀不变全纯雅可比形式空间J^Co₀_{12,Λ,4}满足4≤dim J^Co₀_{12,Λ,4}≤9，并留下其精确维数未决。本文证明dim J^Co₀_{12,Λ,4}=6。在奇异重量下，θ分解将该空间与ℂ[Λ/4Λ]中同时为Co₀和Weil不变的子空间等同。康威对称性与T不变性将问题简化为12维的迷向轨道和空间。该空间上的投影Weil S算子满足由4级Hecke代数得到的通用关系S(S+(1/2)I)(S−I)=0，等价于其关联的整特征标矩阵K满足K(K+2²³I)(K−2²⁴I)=0。结合该关系、已知的指标4形式、模2约化及来自A₃⁸深孔的特征标数据，将剩余可能简化为有限精确计算。对已知指标3形式Φ₁₂,₃的最终挠率求值确定了最后所需的特征标值，精确排除后得到唯一可允许分支，维数为6。本文还从具有根系D₆⁴和D₄⁶的尼迈耶格（Niemeier lattices）的显式标记中构造了两个康威平均θ形式，与孙和王之前展示的四个形式一起，构成J^Co₀_{12,Λ,4}的自然基。
英文摘要
Let $Λ$ be the Leech lattice and let $\mathrm{Co}_0=\operatorname{Aut}(Λ)$. Sun and Wang proved that the space of $\mathrm{Co}_0$-invariant holomorphic Jacobi forms of singular weight $12$ and index $4$ satisfies \[ 4\leq \dim J^{\mathrm{Co}_0}_{12,Λ,4}\leq 9, \] and left its exact dimension open. We prove \[ \dim J^{\mathrm{Co}_0}_{12,Λ,4}=6. \] At singular weight, theta decomposition identifies this space with the simultaneous $\mathrm{Co}_0$- and Weil-invariant subspace of $\mathbb{C}[Λ/4Λ]$. Conway symmetry and $T$-invariance reduce the problem to a twelve-dimensional space of isotropic orbit sums. On this space the projected Weil $S$-operator satisfies the universal relation \[ S\left(S+\frac{1}{2}I\right)(S-I)=0, \] obtained from the level-$4$ Hecke algebra. Equivalently, the associated integral character matrix $K$ satisfies \[ K(K+2^{23}I)(K-2^{24}I)=0. \] Combining this relation with known index-$4$ forms, reduction modulo $2$, and character data obtained from the $A_3^8$ deep hole reduces the remaining possibilities to a finite exact calculation. A final torsion evaluation of the known index-$3$ form $Φ_{12,3}$ determines the last required character value, and exact elimination leaves a unique admissible branch, of dimension $6$. We also construct two Conway-averaged theta forms from explicit markings of the Niemeier lattices with root systems $D_6^4$ and $D_4^6$. Together with the four forms previously exhibited by Sun and Wang, they give a natural basis of $J^{\mathrm{Co}_0}_{12,Λ,4}$.
