Infinite families of cyclotomic function fields with any prescribed class group rank
Let \(k=\mathbb F_q(T)\) be the rational function field over the finite field \(\mathbb F_q\) of \(q\)-elements. For a polynomial \(N(t)\in \mathbb F_q[T]\), let \(\Lambda_N\) be the set of roots of \(N(\phi+\mu_T)(u)=0\) in the algebraic closure of \(k\), where \(\phi(u)=u^q,\mu_T(u)=Tu\). The field \(k(\Lambda_N)\) is called the \(N\)-th cyclotomic function field. Its maximal real subfield is denoted by \(k(\Lambda_N)^+\). In this article, for a prime divisor \(\ell\) of \(q-1\) and \(n=1,2,3\), the authors explicitly construct infinite families of the maximal real subfields of cyclotomic function fields whose ideal class groups have arbitrary \(\ell^n\)-rank, and further give a tower of cyclotomic function fields whose maximal real subfields have ideal class groups of \(\ell^n\)-ranks getting increased. The main idea is as follows. Let \(K\) be a Kummer extension over \(k\) of degree \(\ell\), \(Cl_K(\ell)\) the \(\ell\)-Sylow subgroup of the ideal class group of \(K\) and \(\sigma\) a generator of the Galois group of \(K\) over \(k\). Then \(\lambda_n=\dim_{\mathbb F_q}Cl_K(\ell)^{(\sigma-1)^{n-1}}/Cl_K(\ell)^{(\sigma-1)^n}\) is a lower bound for \(\ell^n\)-rank of \(Cl_K\). For a monic polynomial \(N\) of degree divisible by \(\ell\), the ideal class group of the Kummer extension \(k(\sqrt[\ell]N)\) is canonically isomorphic to a subgroup of the ideal class group of \(k(\Lambda_N)^+\). Therefore, the problem is reduced to construction of Kummer extensions with the desired \(\lambda_n (n=1,2,3)\). They offer algorithms to compute \(\lambda_3\) for given Kummer extensions and to give cyclotomic function fields whose \(\ell^n\)-rank of the ideal class group is greater than or equal to the given integer \(\lambda_3\). The algorithm for \(\lambda_1\) and \(\lambda_2\) has already appeared in [\textit {C.Wittmann}, Finite Fields Appl. 13, No. 2, 327--347 (2007; Zbl 1176.11061)]. They obtain the tower with the described property by using their result that for polynomials \(P_1,P_2\in\mathbb F_q\), the ideal class group of \(k(\Lambda_{P_1})^+\) is a subgroup of that of \(k(\Lambda_{P_1P_2})^+\).
scientific article; zbMATH DE number 7357198
scientific article