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.

The big de Rham–Witt complex

Abstract: This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a λ-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a λ-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the λ-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kahler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to etale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

Formal groups and zeta-functions

Abstract: Cl be an elliptic curve over the rational number field Q, uniformized by automorphic functions with respect to some congruence modular group T0(N). In the language of formal groups results of Eichler [3] and Shimura [14] imply that a formal completion C1 of Cl (as an abelian variety) is isomorphic over Z' to a formal group whose invariant differential has essentially the same coefficients as the zeta-f unction of Q. In this paper we prove that the same holds for any elliptic curve C over Q (th. 5). This follows from general theorems which allow us explicit construction and characterization of certain important (one-parameter) formal groups over finite fields, p-adic integer rings, and the rational integer ring (th. 2 and th. 3). The proof of th. 5 depends only on the fact that the Frobenius endomorphism of an elliptic curve over a finite field is the inverse of a zero of the numerator of the zeta-function, and implies a general relation between the group law and the zetafunction of a commutative group variety. In fact it is remarkable that the p-f actor of the zeta-function of C for bad p also can be given a clear interpretation from our point of view (cf. th. 5). Moreover, we prove that the Dirichlet L-function with conductor D has the same coefficients as the canonical invariant differential on a formal group isomorphic, over the ring of integers in Q(\/ D), to the algebroid group x+y+\/T) xy (th. 4). In this way the zeta-function of a commutative group variety may be characterized as the L-series whose coefficients give a normal form of its group law.

Approximating rings of integers in number fields

Abstract: In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic mimber field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for facto- ring integers. Applying a variar.t of a Standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the "best guess" one has for the ring of integers. This best guess is probably often correct. Our main cor.cern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are "tarne" in our more general sense. As a byproduct, we prove complexity results that elaborate upon a result of Chistov. The paper also includes a section that discusses polynomial time algorithms related to finitely generated abelian groups.

New classes of algebraic interleavers for turbo-codes

Abstract: In this paper we present classes of algebraic interleavers that permute a sequence of bits with nearly the same statistical distribution as a randomly chosen interleaver. When these interleavers are used in turbo-coding, they perform equal to or better than the average of a set of randomly chosen interleavers. They are based on a property of quadratic congruences over the ring of integers modulo powers of 2.