On the ideal class group problem for global fields
The ideal class group problem is the classical question: given a finite abelian group \(G\), does there exist a number field \(K\) having \(G\) as the ideal class group of \(K\)? This is an open problem. \textit{G. Cornell} [Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math. 751, 82-88 (1979; Zbl 0423.12009)] proved that given \(G\) a finite abelian group, there exists a cyclotomic field \(K_1\) such that \(G\) is a quotient of the ideal class group of \(K_1\). Furthermore, \textit{O. Yahagi} [Tokyo J. Math. 1, 275-283 (1978; Zbl 0403.12011)] proved that if \(G\) is an \(\ell\)-group, there exists a field \(K_2\) such that \(G\) is the \(\ell\)-Sylow subgroup of the ideal group of \(K_2\). For function fields, \textit{H. Stichtenoth} [Arch. Math. 32, 336-340 (1979; Zbl 0404.12011)] proved that if \(G\) is an abelian group of exponent \(n\) and cardinality \(\geq n ^{2 (48n/e) ^4}\), then there does not exist any smooth projective irreducible algebraic curve \(X\), defined over any finite field having \(G\) as its divisor class group of degree zero. In this paper the author considers the analogous problem for \(S\)-class groups of number fields and also for geometric ideal class groups of congruence function fields. He proves a reduction lemma that states that if \(K\) is a global field, \(S\) is a finite set of places (containing the archimedian places) and \(L\) is an intermediate field \(K \subset L \subset K _S ^{\text{Hilb}}\), then there exists a finite set \(S'\) of places of \(K\) such that \(L = K _{S\cup S'}^{\text{Hilb}}\). Using this lemma and Cornell's theorem in the number field case and results of \textit{B. Anglès} [J. Number Theory 70, 146-159 (1998; Zbl 0911.11052)] in the function field case, the author proves that the problem has an affirmative answer in both cases.
scientific article; zbMATH DE number 1309264
scientific article