Showing papers on "Ring of integers published in 1985"

Relative Lubin-Tate groups

Abstract: We construct a class of formal groups that generalizes LubinTate groups. We formulate the major properties of these groups and indicate their relation to local class field theory. The aim of this note is to introduce a certain family of formal groups generalizing Lubin-Tate groups. Although the construction, basic properties and relation with local class field theory are all similar to Lubin-Tate theory, the author is unaware of previous references to these groups. We remark, however, that they are complementary in some sense to the formal groups studied by Honda in [2]. Since we want to keep this note short, all the proofs are omitted. The reader who is acquainted with Lubin-Tate theory as in [4 or 5] will be able to supply them without any difficulties. I would like to acknowledge my debt to K. Iwasawa. His beautiful exposition of local class field theory [3] motivated this note. 1. Let k be a finite extension of Qp, v: kx -Z the normalized valuation (normalized in the sense that v(kX) = Z), 0 and p its ring of integers and maximal ideal, and k = 0/ the residue field, a finite field of characteristics p and q elements. kalg denotes an algebraic closure of k and kUr the maximal unramified extension of k in it. We also fix a completion of kalg, Q, and let K be the closure of k"r in it. We write (p for the Frobenius automorphism of kur/k, characterized by p(x) = mod pur, for all x E Our. It extends by continuity to an automorphism of K/k, still denoted by p. If k' is another finite extension of Qp, the corresponding objects will be denoted by ', e.g. (p', q', etc. If A is any ring, A[[X,... ., Xn]] will denote the power series ring in Xi. If f and g are elements of it, f -g mod deg m means that the power series f g involves only monomials of degree at least m. 2. Fix the field k. For each integer d let Ed be the set of all ( E k, v(s) = d. Fix also d > 0 and let k' be the unique unramified extension of k of degree d. Let E Ed and consider = {f E 0'[[X]]If _ 7r'Xmoddeg2, Nk,/k(1r') = ( and f-Xq mod O'}. THEOREM 1. For each f E 7e there is a unique one-dimensional commutative formal group law Ff E 0'[[X, Y]] satisfying FJ' o f = f o Ff. In others words, f is a homomorphism of Ff to Fr. Received by the editors March 2, 1984. 1980 Mathematics Subject Claification. Primary 12B25.

The VLSI design of a single chip for the multiplication of integers modulo a Fermat number

Abstract: Multiplication is central in the implementation of Fermat number transforms (FNT) and other residue number algorithms. There is need for a good multiplication algorithm which can be realized easily on a VLSI chip. In this paper, the Leibowitz multiplier [1] is modified to realize multiplication in the ring of integers modulo a Fermat number. The advantage of this new algorithm over Leibowitz's algorithm is that Leibowitz's algorithm takes modulo after the product of multiplication is obtained. Hence time is wasted. In this new algorithm, modulo is taken in every bit operation when performing multiplication. Therefore no time is wasted in this respect. Furthermore, this algorithm requires only a sequence of cyclic shifts and additions. The design for this new multiplier is regular, simple, expandable, and therefore, suitable for VLSI implementation.

Rational algebraic $K$-theory of certain truncated polynomial rings

Abstract: In this paper we derive a formula for rationalized algebraic K-theory of certain overrings of rings of integers in number fields. Truncated polynomial algebras are examples. Our method is homological calculation which is facilitated by some basic rational homotopy theory and interpreted in terms of the cyclic homology theory of algebras invented by Alain Connes. The object of this paper is to compute, in terms of A. Connes' cyclic homology and the rational algebraic K-theory of a ring of integers (9 in a number field k = (9 ? z Q, the rationalized K-theory of an augmented (9-algebra A (9 which is finitely generated projective as an (9-module and whose augmentation ideal A is nilpotent. The standard example to have in mind is A = d [T]/(T + 1), a truncated polynomial algebra. Our main result may be stated as follows: THEOREM 1. For q > 1 and A as above, Kq(A) ? Q-Kq((9) C Q D Vq where Vq is a rational vector space of dimension d dimk HCq-i(k S9 A) and d is the degree of k over Q. This formula for Kq(A) ? Q follows that obtained by C. Soule in [9] for the case A = (9[T]/(T2), the dual numbers over (9. Using a classical Lie algebra homology calculation based on invariant theory, but no cyclic homology, he found that for the dual numbers dimQ V = d, or = 0, if q is odd, or even. In ?2 of this paper we sketch a computation in cyclic homology relevant to the case A = C9[T]/(T + 1) following the steps of a computation of the rational algebraic K-theory of the space CPn shown to us by T. Goodwillie. It turns out that dim Vq = n d, or = 0, depending on whether or not q is odd, or even. The proof of this theorem, together with the proof of the theorem of Loday and Quillen relating Lie algebra homology and cyclic homology, actually gives when Z = (9a chain of natural isomorphisms linking the rationalized relative K-group Kq + 1 (A Z ) = 7Tq (fibre(BGL + (A) -BGL(Z)+)) ? Q Received by the editors May 18, 1984 and, in revised form, December 28, 1984. 1980 Mathematics Subject Classification. Primary 18F25, 55P62. 1 Partially supported by NSF grant MSC-80-002 396. ?01985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

On computing the discriminant of an algebraic number field

Abstract: Letf(x) be a monic irreducible polynomial in Z[x], and r a root of f(x) in C. Let K be the field Q(r) and X the ring of integers in K. Then for some k E Z, disc r = k2 disc M. In this paper we give constructive methods for (a) deciding if a prime p divides k, and (b) if p I k, finding a polynomial g(x) E Z[x] so that g(x) i 0 (mod p) but g(r)/p E M.

Dirichlet series and automorphic forms on unitary groups

Abstract: In a special case our unitary group takes the form C = { g E GL( p + 2, C) |t gRg = R } Here S O O R= 0 0 1 O -1 0 is a skew-Hermitian matrix with entries in an imaginary quadratic number field K We suppose that -iR has signature (p + 1,1) This group acts naturally on the symmetric domain D = { u E CP, s E C:|Im(7) > -2twSw} If r = G n SLf p + 2, dK) with dK the ring of integers in K, then an automorphic formf(up, ) with respect to r has an expansionErg,(w) e2X'r- The functions g, ( e) are theta functions Given another automorphic form g( w, ) with an expansion , h, ( w ) e 2X" we define a Dirichlet series Er is a certain positive definite inner product on the space of theta functions The series is obtained as an integral of Rankin type: A fg (Im(-) + 21tWSW) awdwdvd Pr\D with Pr C r a subgroup of 4'translations" The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over r \ D In the case p = 1 our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type

On a generalization of Hamburger’s theorem

Abstract: The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grossencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grossencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grossencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.