On a certain invariant for real quadratic 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...

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

The aim of this paper is to obtain the real quadratic fields $\mathbb{Q}\left(\sqrt{d}\right)$ including $\omega_d = \left[a_0;;\overline{\underbrace{\gamma,\gamma,\dots,\gamma}_{l-1},a_l}\right] $ where $l = l\left(d\right)$ is the period length and $\gamma$ is a positive odd integer. Moreover, we have considered a new perspective to determine the fundamental units $\epsilon_d$ and got important results on Yokoi's invariants $n_d$ and $m_d$ [since they satisfy necessary and sufficient conditions related to Ankeny-Artin-Chowla conjecture (A.A.C.C), give bounds for fundamental units and so on...] for such types of fields.

Explicit computation of Gross–Stark units over real quadratic fields

Well-rounded twists of ideal lattices from real quadratic fields

Gauss Problem for Function Fields

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

Uniform distribution and real fields

Sums of nonvanishing integral squares in real quadratic fields

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.

Fundamental units of real quadratic fields of odd class number

Congruence function fields of genus g and class number g + 1 g + 1 are fully classified. As an application we determine explicitly the real quadratic function fields with this property and of odd characteristic for which the ring of integers is a unique factorization domain.

Article The Gauss-Hasse hypothesis on real quadratic fields with class number one was published on January 1, 1982 in the journal Journal für die reine und angewandte Mathematik (volume 1982, issue 335).

The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k 0-regular primes and the index of k 0-irregularity, we discuss how to compute these indices when k 0 is a real quadratic field. Finally, we present the results of some preliminary computations, and show that the frequency of various indices seems to agree with those predicted by a heuristic argument.

We study the density of the setof real quadratic fields for which the norm of the fundamental unit equals −1 inside the set of real quadratic fields containing elements of norm −1. A conjectural density is derived from a single heuristic assumption, and experimental data supporting this assumption are given. We finally discuss how close one can get to proving such conjectural densities.