Conway invariant Jacobi forms on the Leech lattice

Abstract

Abstract In this paper, we study Jacobi forms associated with the Leech lattice Λ which are invariant under the Conway group Co 0 {\operatorname{Co}_{0}} . We determine and construct generators of modules of both weak and holomorphic Jacobi forms of integral weight and fixed index t ≤ 3 {t\leq 3} . As applications, (i) we find the modular linear differential equations satisfied by the holomorphic generators; (ii) we determine the decompositions of many products of orbits of Leech vectors; (iii) we calculate the intersections between orbits and Leech vectors; (iv) we derive some conjugate relations among orbits modulo t ⁢ Λ {t\Lambda} .