Abstract: The intent of this paper is to determine the first flat cohomology groups of certain finite fiat group schemes which are defined over the spectrum of the ring of integers in a local number field. We discover that the first cohomology groups are isomorphic to certain subgroups of the group of units in the ring modulo p-th powers. Our main result, Theorem 1, was announced in [M-R, Prop. 9.3]. I would like to express my thanks to Professor Barry Mazur for his generous interest and encouragement in this work. Throughout we will consistently use the following notation: K is a local number field with ring of integers R; U is the group of units in R, ord is the additive valuation which takes R surjectively to Z; U(M { u C U: ord (1 - u) ? i}, the residue field k of R is assumed to have characteristic p, and we shall regard P. = Z/pZ as being a subfield of k; the number of elements in kc is q =- pf; e = e(K/Qp) will denote the absolute ramification index of K over Q,. We will always assume that K contains the p-th roots of unity; among other things this implies that -p is a p - 1-st power in R and that m = e/ (p -11 is an integer. Ks will denote a fixed separable closure of K. All our group schemes will be flat over Spec (R) and will be considered as inducing sheaves for the (fppf)- or (fpqf)-site over Spec(R) [SGA 3, IV 6.3].

Abstract: Let t (E, p, B, F) be a fiber bundle F. If B and F are dominated by finite complexes, it is not difficult to show that E is also dominated by a finite complex. Hence the (unreduced) Wall invariants (cf. [12] or [6]) w (F) E KoZ7ri (F). w (E) EK0Zw1 (E), and w (B) E KoZr1 (B) are all defined and it is natural to ask what relationships there are among these invariants. It is the object of this note to study the relationship between w (B), p*qw(E) E KoZ71 (B), and orientation phenomena of the bundle. The point of departure for this study is the following theorem due to R. G. Swan [11] (cf. also [3; pp. 563-565]):

Abstract: where g is an analytic function referred to as the associate of *. The domain of g, W, is an axiconvex subset of the complex plane, C, meaning that whenever g is contained in w then the entire line segment joining g and C is in o. The domain of j is the axisymmetric region obtained by rotating o abolut the real axis. Marden (4, p. 142) employs methods drawn from the analytic theory of polynomials in one complex variable to study A. S. P. and G. A. S. P. whose associates are rational functions. In particular. the set of points in C where the rational function assumes an assigned value is used to determine a pair of cones in E3 and more generally El, n > 4, where the corresponding A. S. P. or G. A. S. P. omits this value.

Abstract: where Z is a multiplier for G. This series, which will converge in this case if q > 1, represent an automorphic form of dimension q. (Cf. Lehner [4], Chapter III) }. The purpose of the present paper is to extend Knopp's Lemma to the case q > 1. Knopp's proof involves substantial use of the Ber's spaces of automorphic forms. Since the character of these spaces in the case q ? 2 and 1 1 at every stage of Knopp's proof, the proof turns out to be valid. However, verfication of two of the resulting statements requires techniques far outside Knopp's paper. We begin by reproducing IKnopp's proof of the Main Lemma after which