Computation of the Iwasawa invariants of certain real abelian fields

Abstract

Let p be a prime number and k a finite extension of Q . It is conjectured that the Iwasawa invariants λ p ( k) and μ p ( k) vanish for all p and totally real number fields k. Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant ν p ( k) by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to k= Q ( f ,ζ p+ζ p −1) in the range 1< f<200 and 5⩽p<10 000 .