Local Units Modulo Gauss Sums Original Research Article

For a prime numberpand a number fieldk, letk∞/kbe the cyclotomic imagep-extension. LetA∞be the projective limit of thep-part of the ideal class group of each intermediate field ofk∞/k. Whenkis totally real, it is conjectured thatA∞is finite, namely that the characteristic polynomial char(A∞) ofA∞as aΛ-module is 1. We give an interpretation of char(A∞) (and hence, of the conjecture) in terms ofp-adic behaviour of certain Gauss sums whenkis a real abelian field (satisfying some conditions). Whenk=image(cos(2π/p)), similar results are already obtained by Coleman [3], Kaneko and the author [9].