Real Quadratic Number Fields Research Articles

Proof of class number formulae by machine

As an attempt to follow the direction of machine proof, a personal computer LE0386/25 is used to prove some class number formulae for certain imaginary quadratic number fields \({\Bbb Q}\left( {\sqrt p } \right)\),q=3, 7, 11, 19, 23, 31, 43 and 47) if the real quadratic number field \({\Bbb Q}\left( {\sqrt { - p} } \right)\) has class number one for a primep = 4N 2 + 1 (N is a positive integer).

Unramified Quadratic Extensions of Real Quadratic Fields, Normal Integral Bases, and 2-AdicL-Functions

LetK=Q(m) be a real quadratic number field. In this article, we find a necessary and sufficient condition forKto admit an unramified quadratic extension with a normal integral basis distinct fromK(1), provided that the prime 2 splits neither inK/Q nor in Q(−m)/Q, in terms of a congruence satisfied by the value of a 2-adicL-function forKat 1.

Computation of Partial Zeta Values ats=0 over a Totally Real Cubic Field

In theory the problem of computing the special values of partial zeta functions of a totally real field was solved by Shintani. In practice it rarely works as the fundamental domain of the action of the relevant unit group is too cumbersome to apply in a computational setting. The object of this article is to give a computationally efficient version of Shintani's approach in the cubic case. The basic idea is to take the convex closure of the lattice points, in the first octant, and use the boundary points, of this convex set, to construct a more amenable fundamental domain. Thus the basic thrust is to arrive at an algorithm to find points on this boundary. For us the case of interest is whens=0. In this case we find a cubic analogue of the CF-formula for real quadratic number fields.

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

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.

Real Quadratic Number Fields with 2-Class Group of Type (2,2).

Real Quadratic Number Fields with 2-Class Group of Type (2,2).

A note on symmetry and ambiguity

We present the relationship between quadratic irrationals whose continued fraction expansion has symmetric period, and ambiguous ideal cycles in real quadratic number fields.

On the Values of Partial Zeta Functions of Real Quadratic Fields at Nonpositive Integers

The authors study the partial fraction decomposition of narrow class partial zeta functions of real quadratic number fields at nonpositive integer arguments and obtain an analogue of the classical v. Staudt/Clausen Theorem. The main results follow from the study of the residues of p-adic zeta functions associated with ray classes of modulus (p) for all rational primes p.

Capitulation in unramified quadratic extensions of real quadratic number fields

Letkbe an algebraic number field andCkits ideal class group (in the wider sense). SupposeKis a finite extension ofk. Then we say that an ideal class ofk capitulatesinKif this class is in the kernel of the homomorphisminduced by extension of ideals fromktoK(See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields,with distinct primesPi≡ 1 (mod 4), for which all the ideal classes of the 2-class group,Ck,2(the 2-Sylow subgroup ofCk), capitulate in an unramified quadratic extension ofk. In these examples,Ck,2is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

An integral transform and its applications

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

The Values of the Zeta-Function of a Class of Ideals and the Ankeny, Artin and Chowla Congruences for the Class Number of Real Quadratic Number-Fields

Let K = Q(√D) be a real quadratic number field with discriminant D > 0, χ = χD the Dirichlet character belonging to K (χ(n) = (D/n)) and p an arbitrary prime number ≥ 5. The natural numbers T and U are defined by the power ϵ = ϵ1 − χ(p)p0 = 12(T + U√D) of the fundamental unit ϵ0 > 1 of K. We obtain the congruence 1p(p − χ(p)) NK/Q(ϵ)TUhD ≡ 2Bp − 1,χ mod p between the class number hD of K and the (p − 1)st generalized Bernoulli number belonging to χ. In the case p | D we have ϵ = ϵ0 and the congruence (1) can be written in the form NK/Q(ϵ0) TUhD ≡ 2Bp − 1,χ mod p. Using the Kummer congruences for the generalized Bernoulli numbers it is easy to show that (2) is essentially the well known congruence of Ankeny, Artin, and Chowla for the class number of a real quadratic number field. Similar congruences are true for the prime number p = 3.

On the class number one problem for nonnormal quartic {CM}-fields

We give explicit upper bounds for the discriminants of the non-normal quartic CM-fields with class number one, and for the discriminants of the dihedral octic CM-fields with class number one. These upper bounds are too large to enable us to achieve the determination of these number fields. Nevertheless, whenever a real quadratic number field k is fixed, we can explain how to determine the non-normal quartic CM- fields or the dihedral octic CM-fields with class number one and with real quadratic subfield k.

On a class of insoluble binary quadratic diophantine equations

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

Non-Isomorphic Equivalent Azumaya Algebras

AbstractWe explicitly describe an infinite collection of pairs of Azumaya algebras over the ring of integers of real quadratic number fields K which are maximal orders in the usual quaternion algebra over K, hence Brauer equivalent, but are not isomorphic. The result follows from an identification of the groups of norm one units, using the classification of Coxeter.

On the period length of the generalized Lagrange algorithm

The generalized Lagrange algorithm is a number geometric generalization of Lagrange's continued fraction method for computing fundamental unit and class number of real quadratic number fields. This algorithm yields a system of fundamental units and the class number of an arbitrary algebraic number field by means of computing cycles of reduced ideals. In this paper we prove that the cardinality of a cycle of reduced ideals in an ideal class of an order of an algebraic number field is O( R), where R is the regulator of this order, and where the O-constant only depends on the degree of the field. We also give a lower bound on this cardinality.

The conjugacy problem of the modular group and the class number of real quadratic number fields

We shall discuss the conjugacy problem of the modular group, and show how its solution, in conjunction with a theorem of Olga Taussky can be used to compute the class number of certain real quadratic number fields.

On unramified Am-extensions of quadratic number fields

Number fields such as described in the title play a rôle in the study of Artin L-functions and automorphic forms for the groups SL2 over rings of integers in quadratic extensions of ℚ. They are also of some interest on their own. We have not found many examples in the literature. Lang [4] mentions an unramified A5-extension of a real quadratic number field which is due to E. Artin.

On the theory of Hilbert modular functions I. Arithmetic groups and Eisenstein series

One of the objectives of this paper is to prepare for a relatively elementary treatment of the reciprocity laws for special values of Hilbert modular functions. Our attention was drawn to this subject by a paper of Eichler [ 151, in which a significant portion of the theory of complex multiplication and reciprocity laws for special values of the j-function was developed by elementary methods, and by a modified version of Eichler’s treatment due to Chevalley [lo]. Eichler’s paper caused us to look more carefully at Hecke’s thesis and Habilitationsschrift (Nos. 1 and 4 of his Mathematische Werke), where it was shown that the special values of arithmetic modular functions for the Hilbert modular group of a real quadratic number field generate abelian extensions of certain CM-fields, and a reciprocity law for some of these special values was established. Subsequently Sugawara [39] studied the fields generated by division values of suitably normalized abelian functions with complex multiplication. However, both of Hecke’s papers, though basically correct in their thrust, contained mistakes in function theory, and Sugawara’s work, though useful in some ways, was marred by deficiencies. Therefore, in the case of modular functions of several variables it remained for Shimura and Taniyama [37,40] to make pioneering breakthroughs in developing a general theory of complex multiplication based on the theory of abelian varieties and their fields of moduli. This was developed much further by Shimura [36] in providing a general theory of canonical models and deriving from it a very general reciprocity law for special values of modular functions of several variables on classical domains of symplectic type. After

Selberg trace formula for the Hilbert modular group of a real quadratic algebraic number field

In the paper the Selberg trace formula is derived for the Hilbert modular group of a real quadratic algebraic number field with one class of ideals, acting on the product of two upper semiplanes.

Congruences between cusp forms and the fundamental units of real quadratic number fields

Congruences between cusp forms and the fundamental units of real quadratic number fields

Fundamental units and cycles in the period of real quadratic number fields. I

Fundamental units and cycles in the period of real quadratic number fields. I