The $T$ and $T^*$ components of $\Lambda$ - modules and Leopoldt's conjecture

Abstract

The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. A conjecture, due to Gross and Kuz'min will be shown here to be in a deeper sense a dual of Leopoldt's conjecture with respect to the Iwasawa involution. We prove both conjectures for arbitrary number fields $\K$. The main ingredients of the proof are the Leopoldt reflection, the structure of quasi - cyclic $\Z_p[ \Gal(\K/\Q) ]$ - modules of some of the most important $\Lambda[ \Gal(\K/\Q) ]$ - modules occurring ($T$ acts on them like a constant in $\Z_p$), and the Iwasawa skew symmetric pairing. There a simplified presentation of the Iwasawa linear space and the proofs of the Conjectures of Leopoldt and Gross-Kuz'min can be found, together with a proof of $lambda^+ = 0$ for CM fields. The present paper is at present the only one which presents the approach for non CM extensions. This will be in time incorporated in the exposition of Snoqit, allowing the proofs of all mentioned conjectures for general number fields. Only then will the present paper become obsolete.