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.

Local Zeta Functions of Certain Prehomogeneous Vector Spaces

Abstract: in which K is a p-adic field, |*|K is an absolute value on K, OK is the ring of integers of K, X is an affine space with an OK-structure, and f(x) is a polynomial on X. We take as f(x) the discriminant f(x1, x2) of the binary cubic form N(ux1 + vx2) in u, v, where x1, x2 are in a simple K-split Jordan algebra A of degree 3 with N as its generic norm, and we shall determine Z(s). In general if F(x) is a polynomial on an affine space V, equipped with a nondegenerate quadratic form Q(x), and if F(x, y) is the coefficient of a variable t in F(x + ty), then a morphism # : V -> V is defined as Q(x#, y) = F(x, y) for every x, y in V. In this notation the above N(x) can be characterized, up to a normalization, as a cubic form satisfying (x#)# = N(x)x; cf. Springer [18]. In general if Gi is a subgroup of GL(Vi) for i = 1, 2, . .. , then we shall denote by G1 0 G2 0.. the image of G1 x G2 x .. in GL(V1 0 V20 .. ). In this and other standard notations, the above f(x1, x2) becomes a generator of the ring of relative invariants of

An effective number geometric method of computing the fundamental units of an algebraic number field

Abstract: The Minkowski method of unit search is applied to particular types of parallelotopes permitting to discover algebraic integers of bounded norm in a given algebraic number field of degree n at will by solving successively 2n linear inequalities for one unknown each. Application is made to the unit search for all totally real number fields of minimal discriminant for n < 7. Introduction. Many methods have been devised for producing maximal sets of independent units of an integral domain 0 with a finite basis w,, . . ., In over the rational integer ring, Z. The number geometric methods devised so far always lead to a large number of linear inequalities for n unknowns simultaneously. It is the purpose of this note to reduce the application of number geometric methods to sequences of n pairs of linear inequalities, each only for one unknown. 1. Parallelotopes of Bounded Norm. Denoting by r, the number of distinct isomorphisms of 0 into the real number field, R, say d o R: Oi ( 1 < i

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Abstract: Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $O_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $O_K/2O_K$. The level $\ell$ theta function $\Th_{Ł_{\ell} (C)} $ of $C$ is defined on the lattice $Ł_{\ell} (C):= \set {x \in O_K^n : \rho_\ell (x) \in C}$, where $\rho_{\ell}:O_K \rightarrow O_K/2O_K$ is the natural projection. In this paper, we prove that: % i) for any $\ell, \ell^\prime$ such that $\ell \leq \ell^\prime$, $\Th_{\Lambda_\ell}(q)$ and $\Th_{\Lambda_{\ell^\prime}}(q)$ have the same coefficients up to $q^{\frac {\ell+1}{4}}$, % ii) for $\ell \geq \frac {2(n+1)(n+2)}{n} -1$, $\Th_{Ł_{\ell}} (C)$ determines the code $C$ uniquely, % iii) for $\ell < \frac {2(n+1)(n+2)}{n} -1$ there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to $\Th_{\La_\ell}(C)$.

The syntomic regulator for the K-theory of fields

Abstract: We define complexes analogous to Goncharov's complexes for the K-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K-theory, there is a map from the cohomology of those complexes to the K-theory of the ring under consideration. In case the ring is a localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K-theory with the syntomic regulator. The result can be described in terms of a p-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson's cyclotomic elements. The result is again given by the p-adic polylogarithm. This last result is related to one by Somekawa and generalizes work by Gros.