The values and the coefficients of the modular function j(z) play a variety of important roles in number theory and representation theory. For example, its values generate class fields in algebraic number theory, and its coefficients are the degrees of the graded representation of the Monster group. In this lecture I will introduce a specific sequence of modular functions j_n whose arithmetic literally dictates the behavior of all modular forms on SL₂(Z). The corollaries include:

1) Borcherds type infinite products for generic forms;
2) Universal recursions for Fourier expansions of all forms;
3) p-adic class number formulas in number theory.