Formules de genres et conjecture de Greenberg

Greenberg's well knownconjecture, ( GC) for short, asserts that the Iwasawa invariants. and mu associated to the cyclotomic Zp- extension of any totally real number field F should vanish. In his foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for ( GC) to hold, in two seemingly opposite cases, when p is undecomposed, resp. totally decomposed in F. In this article we present an encompassing approach covering both cases and resting only on `` genus formulas ``, that is ( roughly speaking) on formulas which express the order of the Galois ( co-) invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg.