Singular-weight Conway-invariant Jacobi forms of index four

Abstract

Let $\Lambda$ be the Leech lattice and let $\mathrm{Co}_0=\operatorname{Aut}(\Lambda)$. Sun and Wang proved that the space of $\mathrm{Co}_0$-invariant holomorphic Jacobi forms of singular weight $12$ and index $4$ satisfies \[ 4\leq \dim J^{\mathrm{Co}_0}_{12,\Lambda,4}\leq 9, \] and left its exact dimension open. We prove \[ \dim J^{\mathrm{Co}_0}_{12,\Lambda,4}=6. \] At singular weight, theta decomposition identifies this space with the simultaneous $\mathrm{Co}_0$- and Weil-invariant subspace of $\mathbb{C}[\Lambda/4\Lambda]$. 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 $\Phi_{12,3}$ 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 $J^{\mathrm{Co}_0}_{12,\Lambda,4}$.