p-adic properties of values of the modular j-function

The values of j(z) and its coefficients play many important roles in mathematics. For example, its values generate class fields and its coefficients appear as dimensions of a graded representation of the Monster via the Moonshine phenomenon. In a recent paper, Kaneko [K] produced an interesting connection between the values of j(z) at Heegner points, the so-called singular moduli, and its coefficients. Using recent formulas of Zagier, he systematically expresses the coefficients c(n) in terms of singular moduli. In this paper we want to illustrate some peculiar p-adic properties of certain values of the j-function in connection with class numbers and the integer 720. To motivate our first result, we begin by recalling some classical formulas for values of the Riemann zeta-function at negative odd integers. To state these formulas, we begin by recalling some standard facts and notation. If k ≥ 4 is even, then let Ek(z) denote the usual weight k Eisenstein series for the full modular group SL2(Z)