1. Introduction. It is well known that there are infinitely many quadratic number fields with class number divisible by a given integer n (see Nagell [8] (1922) for imaginary fields and Yamamoto [11] (1970) and Weinberger [10] (1973) for real fields). A related question concerns the n-rank of the field, that is, the greatest integer r for which the class group contains a subgroup isomorphic to (Z/nZ) r . In [11], Yamamoto showed that infinitely many imaginary quadratic number fields have n-rank ≥ 2 for any positive integer n ≥ 2. In 1978, Diaz y Diaz [2] developed an algorithm for generating imaginary quadratic fields with 3-rank at least 2, and Craig [1] showed in 1973 that there are infinitely many real quadratic number fields with 3-rank at least 2 and infinitely many imaginary quadratic number fields with 3-rank at least 3. A few examples of higher 3-rank have also been found (see for instance Llorente and Quer [6,9] who found in 1987/1988 three imaginary quadratic number fields with 3-rank 6). In this paper, we give infinite, simply parameterized families of real and imaginary quadratic fields with 3-rank 2. Although the existence of such fields has been known, the fields here are much easier to describe, and the parameterization yields a new lower bound on the number of fields with discriminant < x and 3-rank ≥ 2 (see [7]).The main result is as follows:

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).