Anderson's module for cyclotomic fields of prime conductor
In 1996, \textit{G. Anderson} [J. Number Theory 60, 142-164 (1996; Zbl 0872.11047)] introduced a module which combines in a nice way two modules which are important in the theory of abelian fields: the so-called Stickelberger quotient \(S\), and the quotient \(E/C\) of units by cyclotomic units. The former quotient is intimately related to the minus class group of the corresponding abelian field, and the latter to the plus class group. Away from the two-primary part, Anderson's module simply decomposes as a direct sum of the two parts just described, but in the two-primary part, the extension which arises is in general nontrivial, and new questions arise. The author of the paper under review considers the case of a cyclotomic field \(L\) of prime conductor \(l\) and studies its Anderson module \(\mathcal A\). From the above general discussion one is led to conclude that \(\mathcal A\) should be closely related to the whole class group Cl\((L)\). Anderson proved that these two abelian groups have the same order. The author shows a nice result pointing in this precise direction: the modules \(\mathcal A\) and Cl\((L)\) have the same Tate cohomology over \(G\) (the Galois group of \(L\)). Then it is shown that the Pontryagin dual \({\mathcal A}^*\) of the finite module \(\mathcal A\) is cyclic over \(Z[G]\), so \({\mathcal A}^*\) is isomorphic to \(Z[G]/J\) for a unique ideal \(J\). More intrinsically, \(J\) can be gotten as the (first) Fitting ideal of \(\mathcal A\). (\(\mathcal A\) itself need not be cyclic. Similar statements concerning \(E/C\) can be made.) He puts forward a conjecture that \(J\) annihilates Cl\((L)\), and the even stronger conjecture that \(J\) is the Fitting ideal of Cl\((L)\). The latter seems to be a difficult problem, although there is a lot of support for it, in the reviewer's opinion. The author proves that 2\(J\) annihilates Cl\((L)\). An interesting report on explicit calculations concludes the paper.
scientific article; zbMATH DE number 1109651