Generalized Kac--Moody algebras, automorphic forms and Conway's group. I (Q1826709)

The fake monster algebra carries a natural action of an extension of the automorphism group of the Leech lattice. The author shows that the corresponding twisted denominator identities are infinite products with multiplicities given by coefficients of modular forms. This suggests that the identities are images of vector-valued modular forms under Borcherds singular theta correspondence and therefore are automorphic forms. The author also conjectures that the identities have singular weights. The conjecture is proved for elements with square-free level and non-trivial fixpoint lattice, and it is promised to treat more complicated cases in forthcoming papers. The construction presented in the paper gives, as a bonus, a family of generalized Kac-Moody algebras, associated to the elements of square-free order in Mathieu's group \(M_{23}\). Finally, it is shown that twisted denominator identities of the fake monster algebra and the fake monster superalgebra often come from the same automorphic form.