On a new type of -adic regulator for algebraic number fields (the -adic regulator without logarithms)

Abstract

For an algebraic number field such that a prime splits completely in , we define a regulator that characterizes the subgroup of universal norms from the cyclotomic -extension of in the completed group of -units of , where consists of all prime divisors of . We prove that the inequality follows from the -adic Schanuel conjecture and holds for some Abelian extensions of imaginary quadratic fields. We study the connection between the regulator and the feeble conjecture on the -adic regulator, and define analogues of the Gross regulator.