Showing papers on "Ring of integers published in 1979"

On BCH codes over arbitrary integer tings (Corresp.)

Abstract: Bose-Chadhuri-Hocquenghem (BCH) codes with symbols from an arbitrary finite integer ring are derived in terms of their generator polynomials. Tile derivation is based on the factorization of x^{n}-1 over the unit ring of an appropriate extension of the Finite integer ring. The construction is thus shown to be similar to that for BCH codes over finite fields.

Modular forms of degree $n$ and representation by quadratic forms. II

Abstract: Let S, T be positive definite integral matrices and suppose that T is represented by S over each p-adic integer ring Zp. We proved arithmetically in [3] that T is represented by S over Z provided that m > 2n + 3 and the minimum of T is sufficiently large. This guarantees the existence of at least one representation but does not give any asymptotic formula for the number of representations. To get an asymptotic formula we must employ analytic methods. As a generating function of the numbers of representations we consider the theta function

The ar property and chain conditions in group rings

Abstract: It is shown that ifJ is the ring of integers or a field andG a group such that the augmentation ideal of the group ringJG has the AR property then the ringJG and the groupG satisfy certain chain conditions.

ON WEIERSTRASS POINTS OF RIEMANN SURFACES ASSOCIATED WITH Γ(n, 2n)

Abstract: Introduction. Let f be a congruence subgroup of SL(2, R), and let R(F) be the Riemann surface associated with F, i.e., R(F) is the canonical compactification of F\H, where H is the upper half plane of C. If the genus of R(F) is not less than two, then we can ask the following problems: Problem 1. Is R(F) hyperelliptic ? Problem 2. Are the cusps of R(F) Weierstrass points? Historically, Problems 1 and 2 are completely solved for F = F(n) by H. Petersson [8] and by B. Schoeneberg [9] respectively. In the case of F0(n), partial solutions are given by J. Lehener and M. Newmann [6] and A.O.L. Atkin [2]. The purpose of this note Is to answer both problems in the case of F = F(n, 2n) (as for the definition of I , 2ri), see Definition 1). Our results are the following: 1. R(I , 2n)) is non-hyperelliptic for any positive integer ≪i^4 (see Theorem 4). 2. Every cusp of R(F(n, 2n)) is a Weierstrass point for any even integer n^4. But there is an example, where the opposite situation may occur if n is odd (see Theorem 6 and Remark 1). Notation. SL{2, R) (resp. SL(2, Z)) is the special linear group of degree two over the real number field R (resp. the rational integer ring Z), and PSL(2, R) is the projective special linear group of degree two over R. When J is a subgroup of SL(2, R), the image of J under the canonical homomorphism SL(2, R)―>PSL (2, R) is denoted by A. H* means the disjoint union of the upper half plane H, the rational numbers Q and {oo}.

Special Arithmetic Subgroups of Chevalley Groups

Abstract: Introduction. Let k be an algebraic number field, and G an almost simple, simply connected linear algebraic group defined over k. Assume G is embedded over k in GL(V) where V is a finite dimensional vector space defined over k. Let L C V(k) be a lattice, and for each prime p of the ring of integers let Lp be its closure in V(kp). We denote by r, r, the stabilizers in G(k), G(kp) of L, Lp. The group r is a special arithmetic group by definition if rp is a special maximal compact subgroup of G(kp) (in the sense of [5]) for

ON THE ORDER OF THE ERROI~ FUNCTION OF THE SQUAt~E-FULL INTEGERS

Abstract: Let L(x) denote the number of square full integers ~ x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that ~(3/2) ,, ~(2/3) 1,