On the class groups of imaginary abelian fields (Q908960)

Let p be an odd prime, \(\chi\) an odd, p-adic Dirichlet character and K the cyclic imaginary extension of \({\mathbb{Q}}\) associated to \(\chi\). We define a ``\(\chi\)-part'' of the Sylow p-subgroup of the class group of K and prove a result relating its p-divisibility to that of the generalized Bernoulli number \(B_{1,\chi^{-1}}\). This uses the results of Mazur and Wiles in Iwasawa theory over \({\mathbb{Q}}\). The more difficult case, in which p divides the order of \(\chi\) is our chief concern. In this case the result is new and confirms an earlier conjecture of G. Gras.