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 [Formula: see text] and whose ideal class groups have 3-rank at least one, which is [Formula: see text]. We also obtain a better bound on the number of imaginary quadratic fields with 3-rank at least two, which is [Formula: see text]; the best-known lower bound so far is [Formula: see text]. For finding these effective lower bounds, we use the Scholz criteria and the parametric families of quadratic fields [Formula: see text] and [Formula: see text] (defined as follows) with escalatory case. We find new infinite families of quadratic fields [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are integers subject to certain conditions for [Formula: see text]. More specifically, we find a complete criterion for the 3-rank difference between [Formula: see text] and its associated quadratic field [Formula: see text] to be one; this is the escalatory case. We also obtain a sufficient condition for the family [Formula: see text] and its associated family [Formula: see text] to have escalatory case. We illustrate some selective implementation results on the 3-class group ranks of [Formula: see text] and [Formula: see text] for [Formula: see text].