Local and global universal norms in the cyclotomic -extension of an algebraic number field

Abstract

For an algebraic number field and a prime we study the subgroups of global universal norms and of everywhere locally universal norms in the cyclotomic -extension of in the pro--completion of the group of -units , where is the set of all places over . Assuming that the -adic Schanuel conjecture holds, we prove the finiteness of the index , whence we obtain a conditional proof of a conjecture in [1] on the Iwasawa module. We also obtain an unconditional proof of all these results in the particular case when is a Galois extension of with symmetric Galois group , contains an imaginary quadratic field, and is a prime such that the decomposition subgroup of its prime divisor coincides with the Sylow -subgroup of .