The Additive Characters of the Witt Ring of an Algebraic Number Field

Abstract

For an algebraic number field K there is a similarity between the additive characters defined on the Witt ring W(K), [20], [11], [17], [14, p. 131], and the local root numbers associated to a real orthogonal representation of the absolute Galois group of K, [18], [5]. Using results of Deligne and of Serre, [16], we shall derive in (5.3) a formula expressing the value, at a prime in K, of the additive character on a Witt class in terms of the rank modulo 2, the stable Hasse-Witt invariant and the local root number associated to the real quadratic character defined by the square class of the discriminant. Thus we are able to separate out the contributions made to the value of the additive character by each of the standard Witt class invariants.