Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Unramified class field theory of arithmetical surfaces

Abstract: Let k be an algebraic number field, (9k its ring of integers and V a non-empty open subscheme of Spec(Ck). Let X be a projective smooth geometrically irreducible scheme over k, and X a regular proper flat scheme over V such that X x Vk X. The purpose of this paper is to give an explicit description of the abelian fundamental groups ,ab (X) and 7T b(X), using some "idele class groups" attached to X and X by a K-theoretical method. Its outline is as follows: In general, for a noetherian scheme Z and i = 0 or 1, we define the group SKi(Z) to be the cokernel of

Discrete Logarithms and Local Units

Abstract: Let K be a number field and $\scr{O}\_{K}$ its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct l$^{e}$th powers in $\scr{O}\_{K}$ using smooth algebraic integers. This method makes use of approximations of the l-adic logarithm to identify l$^{e}$th powers. One version we give is successful if the class number of K is not divisible by l and if the units in $\scr{O}\_{K}$ which are congruent to 1 modulo l$^{e+1}$ are l$^{e}$th powers. A second version only depends on Leopoldt's conjecture. We use the technique of constructing l$^{e}$th powers to find discrete logarithms in a finite field of prime order. Our method for computing discrete logarithms is closely modelled after Gordon's adaptation of the number field sieve to this problem. We conjecture that the expected running time of our algorithm is L$\_{p}$[1/3; (64/9)$^{1/3}$ + o(1)] for p $\rightarrow \infty $, where L$_{p}$[s;c] = exp (c (log q)$^{s}$ (log log q)$^{1-s}$). This is the same running time as is conjectured for the number field sieve factoring algorithm.

Congruence Properties of Zariski‐Dense Subgroups I

Abstract: This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of K-rational points of G. Then what should it mean for F to be a 'large' subgroup? We might require F to be a lattice in G, to be arithmetic, to contain many elements of a specific kind, to have a large closure in some natural topology, et cetera. There are many theorems proving implications between conditions of'size' of this kind. We shall consider the case of G a K-simple group, usually Q-simple. The Zariski topology of G, for which the closed sets are those defined by the vanishing of polynomial functions, is very coarse. On the other hand, the various valuations v of K each give rise to a 'strong' topology on G(K); if v is non-archimedean, and Kv is the completion of K with respect to v, then G(KV) is a non-archimedean Lie group, and if £>„ is the ring of integers of Kv, the open subgroup G(£5 J of G(KV) of integral points is defined for all but finitely many v. The Zariski closure F of a subgroup F of G(K) is a /C-algebraic subgroup of G, while the strong closure Fy is a Lie subgroup of G(KV). For elementary reasons the dimension of Tv is at most the dimension of F. Our results are in the opposite direction: we show, under suitable conditions on G, that if F = G then for almost all v we have that Tv contains G(OV). To illustrate this with a specific example, if Ml 5. . . , Mk are matrices in SL2(Q) generating a group F which is Zariskidense in SL2, then for all sufficiently large prime numbers p we have F p = SL2(Zp). Further, this result is effective, in the sense that we could, in principle, check the hypothesis for given Mx,..., Mk, and derive a bound on p beyond which the conclusion holds. (It is perhaps worth remarking that this special case, which is the simplest application of our results, may be proved with much less effort than the general theorem.)

Integral p-adic Hodge theory

Abstract: We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of $\mathbf {C}_{p}$ . It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. Notably, this cohomology theory specializes to all other known $p$ -adic cohomology theories, such as crystalline, de Rham and etale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings’ almost purity theorem, along with a certain functor $L\eta $ on the derived category, defined previously by Berthelot–Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham–Witt complexes of Langer–Zink, and can be computed as a $q$ -deformation of de Rham cohomology.

Signal sets matched to groups

Abstract: Recently, linear codes over Z/sub M/ (the ring of integers mod M) have been presented that are matched to M-ary phase modulation. The general problem of matching signal sets to generalized linear algebraic codes is addressed based on these codes. A definition is given for the notion of matching. It is shown that any signal set in N-dimensional Euclidean space that is matched to an abstract group is essentially what D. Slepian (1968) called a group code for the Gaussian channel. If the group is commutative, this further implies that any such signal set is equivalent to coded phase modulation with linear codes over Z/sub M/. Some further results on such signal sets are presented, and the signal sets matched to noncommutative groups and the linear codes over such groups are discussed. >