Gauss Sums and the Local Classification of Hermitian Forms

The trigonometric studied by [2], in connection with cyclotomy, were applied to quadratic forms by Dirichlet [1] in his determination of the class number of some binary forms, and these so-called Gauss sums have since become fundamental in studying certain questions in the arithmetic theory of quadratic forms. In [10], O'Meara used to classify local integral quadratic forms, in particular showing, as Theorem 4 of [10], that the conditions in Theorem 2 of [9], involving Hasse invariants, could be replaced by equality of sums. In the present paper, we shall develop a theory of for hermitian forms, and use this to classify hermitian forms over the integers of a local field, thus providing an alternate classification to that given in [4]. Our main result is the following. Let F be a local field of characteristic not 2, with an involution a-* a*; then equality of sums, together with the typeand u -invariants of [4], gives a classification for integral hermitian forms (with respect to the given involution) over F, under integral equivalence. Further, if F is non-dyadic, i. e., 2 is a unit, then the u -invariants are unnecessary; and if F is an unramified extension of the fixed subfield under the involution, then equality of alone suffices for the classification. The need to know when the of a particular lattice vanish will become evident as we proceed. This question was first considered by Minkowski ([8], pp. 50-58) for rational p-adic forms and Hiecke ([3], pp. 218-249) for unary forms over p-adic completions of algebraic number fields, later generalized by O'Meara ([10], pp. 695-698) to quadratic forms in any number of variables over any local field (of characteristic not 2). We shall study the same question for hermitian forms, obtaining in ? 7 results that are more precise than can be given in the quadratic case; as we observed in [4], the apparent complication of working with a pair of fields often yields simpler results in the hermitian case than the quadratic.