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

Let j(z) = q−1 + 744 + 196884q + · · · denote the usual elliptic modular function on SL2(Z) (q := e throughout). We shall refer to a complex number τ of the form τ = −b+ √ b2−4ac 2a with a, b, c ∈ Z, gcd(a, b, c) = 1 and b −4ac < 0 as a Heegner point, and we denote its discriminant by the integer dτ := b − 4ac. The values of j at such points are known as singular moduli, and they play a substantial role in classical and modern number theory. For example, the theory of complex multiplication implies that if τ is a Heegner point with discriminant dτ , then j(τ) is an algebraic integer which generates a ring class field of Q( √ dτ ). Singular moduli also play an important role in Borcherds’ [B1, B2] recent work on the infinite product expansions of certain modular forms. A meromorphic modular form f on SL2(Z), by definition, has a Heegner divisor if its zeros and poles are supported at the cusp at infinity and Heegner points. In particular, Borcherds obtains an elegant description of the infinite product expansion of those meromorphic modular forms on SL2(Z) with a Heegner divisor. Here we consider the values of a specific sequence of elliptic modular functions jn, where j1 = j − 744. In an important recent paper [Z], Zagier expressed the traces of the values of jn at Heegner points in terms of Fourier coefficients of half integral weight modular forms. Here we consider the more general case of the sums of the values of jn over divisors of meromorphic modular forms. We show that the “traces” of these values (see Theorem 1) dictate the properties of modular forms on SL2(Z). This result