On θ-congruent numbers on real quadratic number fields

Let k=Q(d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d→Q(d) is a bijection from the set off all square-free integers d ≠ 0, 1 to the set of all quadratic fields Q(d)={x+yd|x,y∈Q}. Furthermore, integral basis element of algebraic integer’s ring in real quadratic fields is determined by either wd=d=[a0;a1,a2,⋯,al(d)−1,2a0¯] in the case of d ≡ 2,3(mod 4) or wd=1+d2=[a0;a1,a2,⋯,al(d)−1,2a0−1¯] in the case of d ≡ 1(mod 4) where l(d) is the period length of continued fraction expansion.The purpose of this paper is to obtain classification of some types of real quadratic fields Q(d), which include the specific form of continued fraction expansion of integral basis element wd, for which has all partial quotient elements are equal to each other and written as ξs (except the last digit of the period) for ξ positive even integer where period length is l=l(d) and d ≡ 2,3(mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain...

We discuss the computational application of Heegner points to the study of elliptic curves over Q, concentrating on the curves E d : Dy2 = x3 − x arising in the “congruent number” problem. We begin by briefly reviewing the cyclotomic construction of units in real quadratic number fields, which is analogous in many ways to the Heegner-point approach to the arithmetic of elliptic curves, and allows us to introduce some of the key ideas in a more familiar setting. We then quote the key results and conjectures that we shall need concerning elliptic curves and Heegner points, and show how they yield practical algorithms for finding rational points on E d and other properties of such curves. We conclude with a report on more recent work along similar lines on the elliptic curves x3 + y3 = A.

To date, the only non-group structure that has been suitably employed as the key space for Diffie-Hellman-type cryptographic key exchange is the infrastructure of a real quadratic (number or function) field. We present an implementation of a Diffie-Hellman-type protocol based on real quadratic number field arithmetic that provides a significant improvement in performance over previous versions of this scheme. This dramatic speed-up is achieved by replacing the ordinary multiplication and reduction procedures for reduced ideals by a new version of the NUCOMP algorithm due to Shanks.

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

We construct rational Poncelet configurations, which means finite sets of pairwise distinct [Formula: see text]-rational points [Formula: see text] in the plane such that all [Formula: see text] are on a fixed conic section defined over [Formula: see text], and moreover the lines [Formula: see text] are all tangent to some other fixed conic section defined over [Formula: see text]. This is done for [Formula: see text] in which case only [Formula: see text] and [Formula: see text] are possible, and for certain real quadratic number fields [Formula: see text]; here moreover [Formula: see text] and [Formula: see text] and [Formula: see text] occur, but no further new values of [Formula: see text]. In fact, for every pair [Formula: see text] presented here, we show that infinitely many such tuples [Formula: see text] exist. The construction uses elliptic curves [Formula: see text] over [Formula: see text] such that the group [Formula: see text] is infinite and moreover contains a point of exact order [Formula: see text]. As an aside, a formulation of Mazur’s theorem/Ogg’s conjecture in terms of arbitrary genus one curves over the rational numbers (so not necessarily containing any rational point) is presented, since this occurs naturally in the context of Poncelet configurations.

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

We develop methods for the rapid computation of the regulator of a real quadratic congruence function field K = k(x)(√D). By extending Shanks' infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step–giant step method for evaluating the regulator of K in O(|D|¼) polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in O(|D|⅕) polynomial operations. By implementing both methods on a computer, we found that the O(|D|⅕) algorithm tends to be far better than the baby step–giant step algorithm in those cases where the regulator exceeds 108.

Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y2 = x3 − N2x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more …

A positive integer is called a congruent number if it is the area of a rightangled triangle, all of whose sides have rational length. The problem of determining which positive integers are congruent is buried in antiquity (see Chapter 9 of Dickson [6]), with it long being known that the numbers 5, 6, and 7 are congruent. Fermat proved that 1 is not a congruent number, and similar arguments show that also 2 and 3 are not congruent numbers. No algorithm has ever been proven for infallibly deciding whether a given integer n ≥ 1 is congruent. The reason for this is that it can easily be seen that an integer n ≥ 1 is congruent if and only if there exists a point (x, y), with x and y rational numbers and y = 0, on the elliptic curve ny2 = x3− x. Moreover, assuming n to be square free, a classical calculation of root numbers shows that the complex L-function of this curve has zero of odd order at the center of its critical strip precisely when n lies in one of the residue classes of 5, 6, or 7 modulo 8. Thus, in particular, the unproven conjecture of Birch and Swinnerton-Dyer predicts that every positive integer lying in the residue classes of 5, 6, and 7 modulo 8 should be a congruent number. The aim of this paper is to prove the following partial results in this direction.

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields Q ( t 4 − n 2 ) \mathbb {Q}(\sqrt {t^4-n^2}) as t t varies over positive integers for a congruent number n n . Furthermore, we provide lower bounds on class numbers of Richaud-Degert type real quadratic fields of the form Q ( n 2 k 4 − 1 ) \mathbb {Q}(\sqrt {n^2k^4-1}) for positive integers k k and congruent numbers n n whose elliptic curves have algebraic rank greater than 2 2 .

Sums of nonvanishing integral squares in real quadratic fields

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

Explicit computation of Gross–Stark units over real quadratic fields

This was later extended by Deshouillers and Iwaniec [9] and their work was then used by Bombieri et al. [2, 3] to prove some spectacular results in number theory. In an unpublished note Zagier gave a new proof of the Kuznietsov trace formula (cf. Iwaniec [13]). His method can be generalized to the GL(n) case as pointed out by Friedberg [10] and was later carried out by Joyner [18] in order to extend the Kuznietsov trace formula to real quadratic number fields. The generalization to the group GL(n) was also done by Goldfeld [11] from which he expressed the principal cuspidal contribution to the Selberg trace formula in terms of special values of quadratic L-functions. An adelic version of the Kuznietsov trace formula was formulated by the author in [24] and by Jacquet and Rallis [14]. Comparing the Kuznietsov

We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such that there exists an element with exactly 1–5, 7, and 11 partitions.