RESEARCH ANNOUNCEMENTS Continued Fractions of Functions Connec …

The theory of continued fractions is very useful in studying real quadratic number fields (see ［2-5］).E. Artin in ［1］ introduced continued fractions of functions to study quadratic function fields, using formal Laurent expansions, which isessentially the theory of completion of the function fields at the infinite valuation. Here we first re-developthe theory of continued fractions of functions in a more elementary and manipulable manner mainly using long division of polynomials; and then study properties of the continued fractions, which will have important applications in studying quadratic function fields obtaining remarkable results on unit groups, class groups, and class numbers.     

The Fundamental Units ofAlgebraic Real Quadratic Function Fields

A general theorem on the fundamental unit E of any algebraic quadratic function field K isgiven here; and then the fundamental unit E is exhibited explicitly for sixteen types of four seriesof quadratic function fields K.Suppose that k = Fq.(T) is the rational function field in the indeterminate (variable) T overF,, the finite field with q elements (q is a power of odd prime number p). Let R = Fq[T] bethe polynomial ring of T over F., which is also said to be the ring (domain) of integers…     

The Fundamental Units of Algebraic Real Quadratic Function Fields

A general theorem on the fundamental unit of any algebraic quadraticfunction field K is given here; and then the fundamental unit isexhibited explicitly for sixteen types of four series of quadratic function fields K.     

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.     

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields K=k(D)over k=Fq(T)．For five series of real quadratic function fields K,the bounds of h(D)are given more explicitly,e．g.,if D=F^2 C．then h(D)≥degF／degP;if D=(SG)^2 cS．then h(D)≥degS／degP;if D=(A^m a)^2 A,then h(D)≥degA／degP,where P is an irreducible polynomial splitting in K,c∈Fq．In addition,three types of quadratic function fields K are found to have ideal class numbers bigger than one．     

The Continued Fraction Algorithm and Regulator for Quadratic Function Fields of Characteristic 2

The continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied. Recently, these ideas have even been used to produce cryptosystems. Much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic. We will explore these ideas, and show that the situation is very similar to the odd characteristic case. This exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2.     

实二次域的一个同余式

本文证明了关于实二次域的类数和某类特征和的同余式,同时给出某类实二次域的类数可除性的一个判别法则．     

Cryptography in Quadratic Function Fields

We describe severalcryptographic schemes in quadratic function fields of odd characteristic.In both the real and the imaginary representation of such a field,we present a Diffie-Hellman-like key exchange protocol as wellas a public-key cryptosystem and a signature scheme of ElGamaltype. Several of these schemes are improvements of systems previouslyfound in the literature, while others are new. All systems arebased on an appropriate discrete logarithm problem. In the imaginarysetting, this is the discrete logarithm problem in the idealclass group of the field, or equivalently, in the Jacobian ofthe curve defining the function field. In the real case, theproblem in question is the task of computing distances in theset of reduced principal ideals, which is a monoid under a suitableoperation. Currently, the best general algorithms for solvingboth discrete logarithm problems are exponential (subexponentialonly in fields of high genus), resulting in a possibly higherlevel of security than that of conventional discrete logarithmbased schemes.     

Dihedral Congruence Primes and Class Fields of Real Quadratic Fields

We show that for a real quadratic field F the dihedral congruence primes with respect to F for cusp forms of weight k and quadratic nebentypus are essentially the primes dividing expressions of the form ε^k−1₊±1 where ε₊ is a totally positive fundamental unit of F. This extends work of Hida. Our results allows us to identify a family of (ray) class fields of F which are generated by torsion points on modular abelian varieties.     

Indivisibility of Class Numbers of Real Quadratic Fields

Let D denote the fundamental discriminant of a real quadratic field, and let h(D) denote its associated class number. If p is prime, then the Cohen and Lenstra Heuristics give a probability that ph(D). If pgt;3 is prime, then subject to a mild condition, we show that

On Singular Points of Meromorphic Functions Determined by Continued Fractions

It is shown that Leighton’s conjecture about singular points of meromorphic functions represented by C-fractions K^∞_n=1(a _n z ^αn /1) with exponents α₁, α₂,... tending to infinity, which was proved by Gonchar for a nondecreasing sequence of exponents, holds also for meromorphic functions represented by continued fractions K^∞_n=1(a _n A _n (z)/1), where A₁,A₂,... is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.     

Periodicity Criterion for Continued Fractions of Key Elements in Hyperelliptic Fields

Doklady Mathematics - The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field $$L = mathbb{Q}(x)(sqrt f )$$ has a more complex nature than the...     

On Real Quadratic Number Fields and Simultaneous Diophantine Approximation

 Here we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over ℚ. Such a condition has implications to simultaneous diophantine approximation. In particular, we deduce an improvement to Dirichlet’s Theorem in this context which, as an immediate consequence, shows the Littlewood Conjecture to hold for all numbers α and β both from . Specifically, for all such pairs we have . (Received 10 August 1998; in revised form 23 November 1998)     

Cyclotomic Units and Greenberg's Conjecture for Real Quadratic Fields

We give new examples of real quadratic fields for which the Iwasawa invariant and are both zero by calculating cyclotomic units of real cyclic number fields of degree 18.

     

Fractals and Number Systems in Real Quadratic Number Fields

In this paper we compute the box counting dimension of sets, that are related to number systems in real quadratic number fields. The sets under discussion are so-called graph-directed self affine sets. Contrary to the case of self similar sets, for self affine sets there does not exist a general theory for the determination of the box counting dimension. Thus we are forced to construct the covers, needed for its calculation, directly. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Primes of Simple Imaginary Quadratic Fields and Real Euclidean Quadtic Fields

As is well known,an Euclidean fiels is a simple field.As regards the quadratic field Q(√D),where Q is the ring of rational integers and D is a square-free rational integer,we have know that an imaginary quadratic field Q(√D) is simple iff D∈｛-1,-2,-3,-7,-11,-19,-43,-67,-163｝,and a real quadratic field is Euclidean iff D∈｛2,3,5,6,,7,11,13,17,19,21,29,33,37,41,57,73｝.This paper will discuss the primes of Q(√D) when D belongs to the set QD=｛-1,-2,-3,-7,-11,-19,-43,-67,-163,2,3,5,6,,7,11,13,17,19,21,29,33,37,41,57,73｝.     

A Capitulation Problem and Greenberg's Conjecture on Real Quadratic Fields

We give a sufficient condition in order that an ideal of a real quadratic field capitulates in the cyclotomic -extension of by using a unit of an intermediate field. Moreover, we give new examples of 's for which Greenberg's conjecture holds by calculating units of fields of degree 6, 18, 54 and 162.

     

SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, which holds true for both real and imaginary fields K. Then several series of function fields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, are given, for which the class groups H(K) are proved to contain cyclic subgroups of order n.