Local and global universal norms in the cyclotomic Z(l)-extension of an algebraic number field

摘要

For an algebraic number field K and a prime l we study the subgroups of global universal norms U-S,U-1(K) and of everywhere locally universal norms U-S,U-2(K) in the cyclotomic Z(l)-extension K-infinity of K in the pro-l-completion of the group of S-units U-S(K)[l], where S is the set of all places over l. Assuming that the l-adic Schanuel conjecture holds, we prove the finiteness of the index (U-S,U-2(K) : U-S,U-1(K)), whence we obtain a conditional proof of a conjecture in [1] on the Iwasawa module.