Classical and Umbral Moonshine:Connections and P-Adic Properties

The classical theory of monstrous moonshine describes the unexpected connection between the representation theory of the monster group M, the largest of the sporadic simple groups, and certain modular functions, called Hauptmoduln. In particular, the n-th Fourier coefficient of Klein’s j-function is the dimension of the grade n part of a special infinite dimensional representation V♮ of the monster group. More generally the coefficients of Hauptmoduln are graded traces T_g of g∈M acting on V♮. Similar phenomena have been shown to hold for the Mathieu group M₂₄, but instead of modular functions, mock modular forms must be used. This has been conjecturally generalized even further, to umbral moonshine, which associates to each of the 23 Niemeier lattices a finite group, infinite dimensional representation, and mock modular form. We use generalized Borcherds products to relate monstrous moon-shine and umbral moonshine. Namely, we use mock modular forms from umbral moonshine to construct via generalized Borcherds products rational functions of the Hauptmoduln T_g from monstrous moonshine. This allows us to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group, and gives rise to identities relating dimensions of representations from umbral moonshine to values of T_g. We also show that the logarithmic derivatives of the Borcherds products are p-adic modular forms for certain primes p and describe some of the resulting properties of their coefficients modulo p.