Orders at infinity of modular forms with Heegner divisors

Borcherds described the exponents $a(n)$ in product expansions $f = q^h \prod_{n = 1}^{\infty} (1-q^n)^{a(n)}$ of 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 $f = \mathcal H_d(j(z))$ 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 $S_p(X)$.