On λ-invariants of number fields and étale cohomology

For an odd prime p we prove a Riemann-Hurwitz type formula for odd eigenspaces of the standard Iwasawa modules over F ( μ p ∞), the field obtained from a totally real number field F by adjoining all p -power roots of unity. We use a new approach based on the relationship between eigenspaces and etale cohomology groups over the cyclotomic ℤ p -extension F ∞ of F . The systematic use of etale cohomology greatly simplifies the proof and allows to generalize the classical result about the minus-eigenspace to all odd eigenspaces.