LAUSR

For a totally real number field which is abelian over Q, in which an odd prime p is totally split, and which verifies certain rather mild conditions on the cohomology of the circular units, we show that a weak form of Greenberg’s conjecture for p holds true. This fills a gap — pointed out by Rene Schoof — in a previous proof (see Sur la conjecture faible de Greenberg dans le cas abelien p-decompose, Int. J. Number Theory 2(1) (2006) 49–64), and also extends the original result (semi-simplicity is no longer required).