Mathieu moonshine and Borcherds products - Wang, Haowu; Williams, Brandon (Communications in number theory and physics, 2025)
The twined elliptic genera of K3 a surface associated with the conjugacy classes of the Mathieu group M24 are known to be weak Jacobi forms of weight 0 . In 2010, Cheng constructed formal infinite products from the twined elliptic genera and conjectured that they define Siegel modular forms of degree two. In this paper we prove that for each conjugacy class of level N9 the associated product is a meromorphic Borcherds product on the lattice U(Ng) ⊗ U ⊗ A1 in a strict sense. We also compute the divisors of these products and determine for which conjugacy classes the product can be realized as an additive (generalized Saito-Kurokawa) lift.