Conway invariant Jacobi forms on the Leech lattice

TL;DR

This paper investigates Conway-invariant Jacobi forms related to the Leech lattice, constructing generators, analyzing their properties, and applying these to modular differential equations, orbit decompositions, and lattice relations.

Contribution

It explicitly constructs generators of Conway-invariant Jacobi forms for the Leech lattice and explores their algebraic and modular properties.

Findings

Derived modular linear differential equations for generators

Decomposed products of Leech lattice orbits

Calculated intersections between orbits and vectors

Abstract

In this paper we study Jacobi forms associated with the Leech lattice $\Lambda$ which are invariant under the Conway group $\mathrm{Co}_0$. We determine and construct generators of modules of both weak and holomorphic Jacobi forms of integral weight and fixed index $t\leq 3$. As applications, (1) we find the modular linear differential equations satisfied by the holomorphic generators; (2) we determine the decomposition of many products of orbits of Leech vectors; (3) we calculate the intersection between orbits and Leech vectors; (4) we derive some conjugate relations among orbits modulo $t\Lambda$.