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 ≫ X17 18. We also obtain a better bound on the number of imaginary quadratic fields with 3-rank at least two, which is ≫ X2 3; the best-known lower bound so far is X1 3. For finding these effective lower bounds, we use the Scholz criteria and the parametric families of quadratic fields K1 and K2 (defined as follows) with escalatory case. We find new infinite families of quadratic fields K1 = (a12 - a1 b13) and K2 = (a22 - b23), where ai and bi are integers subject to certain conditions for i = 1, 2. More specifically, we find a complete criterion for the 3-rank difference between K1 and its associated quadratic field K1 to be one; this is the escalatory case. We also obtain a sufficient condition for the family K2 and its associated family K2 to have escalatory case. We illustrate some selective implementation results on the 3-class group ranks of Ki and Kĩ for i = 1, 2. © 2023 World Scientific Publishing Company.