Orders at infinity of modular forms with Heegner divisors

Abstract

Abstract. Borcherds described the exponents a(n) in product expansionsof meromorphic modular forms with a Heegner divisor. His description does not directly give any information about h, the order of vanishing at infinity of f . We give p-adic formulas for h in terms of generalized traces given by sums over the zeroes and poles of f . Specializing to the case of the Hilbert class polynomial f = H d (j(z)) yields p-adic formulas for class numbers that generalize past results of Bruinier, Kohnen and Ono. We also give new proofs of known results about the irreducible decomposition of the supersingular polynomial S p (X).