The Leopoldt and Iwasawa Conjectures for CM fields (1403.7331v1)

Abstract: This is a two - part paper, in which we prove the following fact: let K be a CM field and L/K be a CM Z_p-extension. Then the Iwasawa mu-invariant of L vanishes. For the case when L is the cyclotomic Z_p extension, this is the Iwasawa conjecture stating that mu = 0 - a fact which had been proven for abelian fields by Ferrero and Washington. If L is not cyclotomic, than the Leopoldt conjecture fails for K. In this case we show that there is some auxiliar CM Z_p-extension for which mu does not vanish. This is a contradiction, showing that the Leopoldt conjecture must hold for CM fields. Concerning the ideas of the proof, they use class field theory and a concept of stability of Lambda modules under deformation by Thaine shifts, which is developed explicitly in the papers. This combined paper makes obsolete earlier versions and attempts to prove parts of this result, which are found on this arxiv.