Computation of the Iwasawa invariants of certain real abelian fields. (Q1429808)

Let \(k\) be a finite extension of \(\mathbb{Q}\) and let \(k_\infty\) be the cyclotomic \(\mathbb{Z}\)-extension of \(k\). Further, let \(k_n\) be the subfield of \(k_\infty\) with degree \(p^n\) over \(k\). Write the order of the \(p\)-Sylow subgroup of the ideal class group of \(k_n\) as \(p^{e_n(k)}\). The well-known theorem of Iwasawa is that \(e_n(k)=\lambda_p (k)n+\mu_p (k) p^n+\nu_p(k)\) for all sufficiently large \(n\), where \(\lambda_p(k)\), \(\mu_p(k)\) and \(\nu_p(k)\) are invariants. In this note, estimates of \(e_n(k)\) are obtained by consideration of Gauss sums; under some conditions on auxiliary prime numbers, these estimates become exact. In particular, let \(k=\mathbb{Q}(\sqrt f,\zeta_p+ \zeta_p^{-1})\) where \(f\) is the discriminant of a totally real quadratic field and \(\zeta_p\) is a \(p\)th root of unity. Then the author gives computations of the Iwasawa invariants in the range \(1<f<2000\) and \(5\leq p<10000\).