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].

NOTE ON THE p-DIVISIBILITY OF CLASS NUMBERS OF AN INFINITE FAMILY OF IMAGINARY QUADRATIC FIELDS

“…Yu also demonstrated that for sufficiently large X, there are >> X 1 2 −ǫ such fields with discriminant −D, where D ≤ X. In another work [50], Yu established that the number of real quadratic fields with discriminant ≤ X and class number divisible by n is >> X 1 n −ǫ for any ǫ > 0 and any odd n. Siyun Lee, Yoonjin Lee, and Jinjoo Yoo [35] have made advancements in providing effective lower bounds on the number of imaginary quadratic fields with absolute discriminants less than or equal to X and ideal class groups having a 3-rank of at least one, which is >> X 17 18 . Additionally, they determined that the number of imaginary quadratic fields with a 3-rank of at least two is >> X…”