Showing papers on "Ring of integers published in 1973"
TL;DR: In this article, it was shown that the first cohomology groups are isomorphic to certain subgroups of the group of units in the ring modulo p-th powers.
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].
16 citations
TL;DR: In this article, necessary and sufficient conditions for representing certain classes of primes by given quadratic forms are found by generalizing techniques of rational number theory, and the main result is that if m = 5 or 13, and if p is a rational prime such that ( − 1 p ) = 1 = ( m p ), then a necessary and necessary condition that x 2 + 4 my 2 = p for some rational integers x and y is that [ ϵ m p ] = 1, where ϵm denotes the fundamental unit of the field Q(m 1
16 citations
TL;DR: In this article, it was shown that the genus of a quadratic form of a number field over a rational field determines the ramification of primes in the field, up to integral equivalence.
14 citations
TL;DR: In this article, the authors generalize Pontryagin's theory to a theory of locally compact modules over appropriate rings, and characterize in general those modules which contain a lattice and characterize the adele rings.
12 citations
TL;DR: In this paper, it was shown that the group of universal norms of a formal group corresponding to an elliptic curve of one of the three main types defined over a quasilocal field is trivial.
Abstract: In this paper we prove that the group of universal norms of a formal group corresponding to an elliptic curve of one of the three main types defined over a quasilocal field [11] is trivial. Applications are also indicated.Bibliography: 12 items.
5 citations
TL;DR: In this article, the product of n copies of the maximal ideal of the ring of integers 0 of a local field of characteristic 0 with an algebraically closed residue field k of characteristic p>0 is defined as an n-parametric commutative formal group over 0.
Abstract: Let H be the group obtained by taking the product of n copies of the maximal ideal of the ring of integers 0 of a local field of characteristic 0 with an algebraically closed residue field k of characteristic p>0, and let the composition law be defined as for an n-parametric commutative formal group over 0. Let the kernel of multiplication by p in H be finite. A filtration pmH (m≥0 is an integer) in H is introduced whose properties allow us to obtain an exact sequence of proalgebraic groups 0→Z p r →Ws→H→ 0, where Zp and W are the additive groups of p-adic integers and Witt vectors of infinite length over k, respectively; r≥0 and s>0 are integers.
1 citations
TL;DR: In this article, an algorithm for computing units in Z[α] where α is a real root of a monic irreducible polynomial with integer coefficients is presented.
1 citations