论文标题
Weyl不变的代数构建$ E_8 $ JACOBI表格
Algebraic construction of Weyl invariant $E_8$ Jacobi forms
论文作者
论文摘要
我们研究Weyl不变的环$ e_8 $弱雅各比表格。王最近证明了该环不是多项式代数。我们将包含环的多项式代数视为子集，并阐明多项式代数的元素是Weyl不变的$ e_8 $弱的jacobi形式的必要条件。这是一种用于构建给定重量和索引的所有雅各比形式的新算法。该算法纯粹是代数，不需要傅立叶扩展。使用此算法，我们确定Weyl不变的免费模块的发电机$ e_8 $弱的jacobi形式的给定索引$ m $ for $ m \ le 20 $。我们还确定了$ m \ le 28 $的索引$ m $的自由模块的最低重量发生器。我们的结果支持太阳和王的下边界猜想，并明确证明存在Weyl不变的发电机$ e_8 $弱的jacobi形式的重量$ -4M $ $ -4M $和index $ m $，全部12美元\ le m \ le m \ le 28 $。
We study the ring of Weyl invariant $E_8$ weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient condition for an element of the polynomial algebra to be a Weyl invariant $E_8$ weak Jacobi form. This serves as a new algorithm for constructing all the Jacobi forms of given weight and index. The algorithm is purely algebraic and does not require Fourier expansion. Using this algorithm we determine the generators of the free module of Weyl invariant $E_8$ weak Jacobi forms of given index $m$ for $m\le 20$. We also determine the lowest weight generators of the free module of index $m$ for $m\le 28$. Our results support the lower bound conjecture of Sun and Wang and prove explicitly that there exist generators of the ring of Weyl invariant $E_8$ weak Jacobi forms of weight $-4m$ and index $m$ for all $12\le m \le 28$.
