Mathieu moonshine and Borcherds products

TL;DR

This paper proves that certain infinite products derived from Mathieu moonshine are Borcherds products on specific lattices, clarifying their modular properties and divisors, and identifying when they are lifts of Siegel modular forms.

Contribution

It establishes that these products are meromorphic Borcherds products on particular lattices, confirming Cheng's conjecture and analyzing their divisors and lift properties.

Findings

Products are meromorphic Borcherds products on $U(N_g)  U  A_1$

Divisors of the products are explicitly computed

Identification of conjugacy classes where products are additive lifts

Abstract

The twisted elliptic genera of a $K3$ surface associated with the conjugacy classes of the Mathieu group $M_{24}$ are known to be weak Jacobi forms of weight $0$. In 2010, Cheng constructed formal infinite products from the twisted 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 $N_g$ the associated product is a meromorphic Borcherds product on the lattice $U(N_g)\oplus U \oplus A_1$ 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.