The arithmetic of the values of modular functions and the divisors of modular forms

Abstract

We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(x) defined by the q-expansion ∼n=0∞jn(x)qn:=E 4(z)2E6(z)/Δ(z)·1/j(z)-x. They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas. © Foundation Compositio Mathematica 2004.