ON THE WITT GROUP OF A 2-ADIC GROUP RING

Abstract

Z2TT, where -n is a 2-group, endowed with the orientable involution f = g~1. In this paper we explicitly calculate the Witt group W(A) of Hermitian forms over A with respect to this involution. The difficult part of the calculation is the quotient W(A)/W™(A), where W°(A) denotes the subgroup of even Hermitian forms. To obtain W(A), one must observe that Wv(A)ssZ/2 and is a direct summand of W(A). This was pointed out by W. Pardon. The calculation is a generalization of the method of characteristi c elements, which one uses to compute W(Z)/W™(Z)ssZ/8. (See [4].) There one associates to every non-singular bilinear form over Z a characteristic element x in the underlying free Z-module and observes that {x, x) is an invariant of the isomorphism class of the form if reduced mod 8. Moreover, this invariant vanishes on even forms, hence the above isomorphism. In the case of A = Z2ir, we find that we must associate a collection of characteristi c elements Xu • • • > Xk> m tne underlying free A-module, one for each conjugacy class of involutions in n. Furthermore, the elements (xi, xj) are aU invariants in suitable quotient groups of A, and most of the work in this paper is the evaluation of the quotient groups. These invariants detect all but a small subgroup of W(A)/W°(A), i.e., the subgroup of Hermitian forms for which all these invariants vanish is isomorphic to TT^, ® ZJ2, where TI^, is the commutator quotient TT/[TT, IT].