Normal forms for real quadratic forms

We investigate the non-diagonal normal forms of a quadratic form on , in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of . Received: 27 June 2008     

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem

Using the Markoff theorem on indefinite binary quadratic forms it is shown that every element of the class group of any real quadratic order of discriminant d has an ideal of norm less than or equal to . This bound is best possible as there are quadratic orders of discriminant d which have ideal classes where the least norm is . These discriminants are explicitly given and their connection to the Markoff conjecture is highlighted.     

二次函数域理想类数问题的一个注记

应用F_q[t]上的Pell方程这一初等方法重新证明一个已知的结果:实二次函数域F_q(t)(D~(1/2))理想类数为1时,D只能为P或QR,其中P,Q,R是F_q[t]中的首一不可约多项式且Q,R次数为奇数.     

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property, e.g. fields withm=(z ⁿ +t−1)²+4t(witht|z ⁿ −1), which contains the well-known fields withm=4z ⁿ +1 andm=4z ²ⁿ +4 as special cases. Project supported by the National Natural Science Foundation of China.     

Pell identities via a quadratic field

In this article, we derive a number of identities involving both Pell numbers and binomial coefficients. We also consider briefly the potential for obtaining Pell identities involving trinomial coefficients and beyond. A key point is the simplicity of the derivations, and indeed this work can lead on to a number of interesting explorations for first-year undergraduates.     

Heuristics and related results on class groups of real quadratic fields

The fact is studied that the ideal class numbersh of types of real quadratic fields usually contain a fixed prime numberp as a factor, and the reason is found to be existing there a kind of prime ideals whosepth powers are principal. A modification of the Cohen-Lenstra Heuristics for the probability that in this situation the class numberh is actually a multiple ofp then is presented: Prob (p|h)=1-(1-p ^-1)(1-P ^-2)⋯. This idea is also extended to predict the probability that the classP represented by the above prime ideal is actually of orderp: Prob (_o(P)=p) =1/p. Both of these predictions agree fairly well with the numerical data. Project supported by the National Natural Science Foundation of China.     

On the exceptional fields for a class of real quadratic fields

In this paper, we give a lower bound exp(2.2 × 10~8 ) for those discriminants of real quadratic fields Q(√ d) with d= N~2-4 and h(d)=1.     

Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes

Let X(t), , be a centered real-valued stationary Gaussian process with spectral density f(λ). The paper considers a question concerning asymptotic distribution of Toeplitz type quadratic functional Q _T of the process X(t), generated by an integrable even function g(λ). Sufficient conditions in terms of f(λ) and g(λ) ensuring central limit theorems for standard normalized quadratic functionals Q _T are obtained, extending the results of Fox and Taqqu (Prob. Theory Relat. Fields 74: 213–240, 1987), Avram (Prob. Theory Relat. Fields 79:37–45, 1988), Giraitis and Surgailis (Prob. Theory Relat. Fields 86: 87–104, 1990), Ginovian and Sahakian (Theory Prob. Appl. 49:612–628, 2004) for discrete time processes.      

Computation of -invariants of real quadratic fields

Let be a real quadratic field and an odd prime number which splits in . In a previous work, the author gave a sufficient condition for the Iwasawa invariant of the cyclotomic -extension of to be zero. The purpose of this paper is to study the case of this result and give new examples of with , by using information on the initial layer of the cyclotomic -extension of .

     

Real and imaginary quadratic representations of hyperelliptic function fields

A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.

     

On the 2-primary part of tame kernels of real quadratic fields

Let \(F = Q\left( {\sqrt p } \right)\), where p = 8t+1 is a prime. In this paper, we prove that a special case of Qin’s conjecture on the possible structure of the 2-primary part of K ₂ O _F up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K ₂ O _F , which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.     

实二次域上理想类的Zeta-函数在-3处的值

用简单连分数给出了实二次域理想类的Zeta-函数在-3处值的一个具体的计算公式.     

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for their ideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic function fields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal class numbers of these function fields are divisible by n.     

Computing discrete logarithms in real quadratic congruence function fields of large genus

The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields.

     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of is examined for degree-four monic polynomials when is a finite field of characteristic greater than 3 with or and is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the -Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.