Infinite families of class groups of quadratic fields with 3-rank at least one: quantitative bounds

Abstract

We improve an effective lower bound on the number of imaginary quadratic fields whose absolute discriminants are less than or equal to X and whose ideal class groups have 3-rank at least one, which is >> X 17/18. We also obtain a better bound on the number of imaginary quadratic fields with 3-rank at least two, which is >> X 2/3; the best-known lower bound so far is X 1/3. For finding these effective lower bounds, we use the Scholz criteria and the parametric families of quadratic fields K-1 and K-2 (defined as follows) with escalatory case. We find new infinite families of quadratic fields K-1 = Q(root a(1)(2) - a(1)b(1)(3)) and K-2 = Q(root a(2)(2) - b(2)(3)), where a(i) and b(i) are integers subject to certain conditions for i = 1, 2. More specifically, we find a complete criterion for the 3-rank difference between K-1 and its associated quadratic field <(K)over tilde(1) to be one; this is the escalatory case. We also obtain a sufficient condition for the family K-2 and its associated family <(K)over tilde(2) to have escalatory case. We illustrate some selective implementation results on the 3-class group ranks of K-i and (K) over tilde (i) for i = 1, 2.