Infinite families of cyclotomic function fields with any prescribed class group rank

Abstract

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k = F-q(T) whose class groups can have arbitrarily large l(n)-rank, where F-q is the finite field of prime power order q. We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k(Lambda)(+) of cyclotomic function fields k(Lambda) whose ideal class groups have arbitrary l(n)-rank for n = 1, 2, and 3, where l is a prime divisor of q - 1. We also obtain a tower of cyclotomic function fields K-i whose maximal real subfields have ideal class groups of l(n)-ranks getting increased as the number of the finite places of k which are ramified in K-i get increased for i >= 1. Our main idea is to use the Kummer extensions over kwhich are subfields of k(Lambda)(+), where the infinite prime infinity of k splits completely. In fact, we construct the maximal real subfields k(Lambda)(+) of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k. We demonstrate our results by presenting some examples calculated by MAGMA at the end. (C) 2020 Elsevier B.V. All rights reserved.