A Generalized Problem Associated to the Kummer–Vandiver Conjecture

To discuss the validity of the Kummer–Vandiver conjecture, we consider a generalized problem associated to the conjecture. Let p be an odd prime number and \(\zeta _p\) a primitive p-th root of unity. Using new programs, we compute the Iwasawa invariants of \({\textbf{Q}}(\sqrt{d},\zeta _p)\) in the range \(|d|<200\) and \(200<p <1{,}000{,}000\). From our data, the actual numbers of exceptional cases seem to be near the expected numbers for \(p<1{,}000{,}000\). Moreover, we find a few rare exceptional cases for \(|d|<10\) and \(p>1{,}000{,}000\). We give two partial reasons why it is difficult to find exceptional cases for \(d=1\) including counter-examples to the Kummer–Vandiver conjecture.