Exponents of class groups and elliptic curves

We show that the number of elliptic curves over Q with conductor. N is much less than (epsilon) N1/4 + epsilon, and for almost all positive integers N , this can be improved to much less than (epsilon) N-epsilon. The second es timate follows from a theorem of Davenpart and Heilbronn on the average siz e of the 3-class groups of quadratic fields. The fil st estimate follows fr om the fact that the 3-class group of a quadratic field Q(rootD) has size m uch less than (epsilon) \D\(1/4 + epsilon), a non-trivial improvement over the Brauer-Siegel estimate. (C) 2001 Academic Press.