Ideal class groups of imaginary quadratic fields

Abstract

© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.Let d be a square-free positive integer and CL (- d) the ideal class group of the imaginary quadratic field Q(-d). In this paper, we show that given any odd integer g≥ 3 and any integers r≥ 1 , s with 0 ≤ s≤ r, there are infinitely many d such that CL (- d) contains an element of order g and CL (- d) / CL (- d) 4≅ (Z/ 2 Z) r-s× (Z/ 4 Z) s.