Real Algebraic Number Field Research Articles

Weakly o-minimal expansions of ordered fields of finite transcendence degree

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K ⊆ ℝ is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond-Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded.

Lattices of some solvable Lie groups and actions of products of affine groups

We consider solvable Lie groups which are isomorphic to unimodularizations of products of affine groups. It is shown that a lattice of such a Lie group is determined, up to commensurability, by a totally real algebraic number field. We also show that the outer automorphism group of the lattice is represented faithfully in the automorphism group of the number field. As an application, we obtain a classification of codimension one, volume preserving, locally free actions of products of affine groups.

A generalization of Voronoi's reduction theory and its application

We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.

Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers

Let {K m } m ≥ 4 be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial f m (x) = x 3 − mx 2 − (m + 1)x − 1, where m is an integer with m ≥ 4. In this paper, we will apply Siegel’s formula for the values of the zeta function of a totally real algebraic number field at negative odd integers to K m , and compute the values of the Dedekind zeta function of K m .

Coincidence Site Modules in 3-Space

The coincidence site lattice (CSL) problem and its generalization to ℤ-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.

ε-Pisot numbers in any real algebraic number field are relatively dense

An algebraic integer is called an ε-Pisot number ( ε>0) if its Galois conjugates have absolute value less then ε. Let K be any real algebraic number field. We prove that the subset of K consisting of ε-Pisot numbers which have the same degree as that of the field is relatively dense in the real line R . This has some applications to non-stationary products of random matrices involving Salem numbers.

Towards faster real algebraic numbers

This paper presents a new encoding scheme for real algebraic number manipulations which enhances current Axiom’s real closure. Algebraic manipulations are performed using different instantiations of sub-resultant-like algorithms instead of Euclidean-like algorithms. We use these algorithms to compute polynomial gcds and Bezout relations, to compute the roots and the signs of algebraic numbers. This allows us to work in the ring of real algebraic integers instead of the field of real algebraic numbers avoiding many denominators.

An application of algebraic sieve theory

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieri’s theorem on primes in arithmetic progressions.

A note on a lemma of Shimura

The subject of this paper is a quaternion algebra over a quadratic extension of a totally real algebraic number field. We generalize an important technical lemma of Shimura, on which a certain period of automorphic forms depends.

Some Applications of a Bombieri-Type Theoremin Totally Real Number Fields

The primary concern of this paper is to present three further applications of a multi-dimensional version of Bombieri’s theorem on primes in arithmetic progressions in the setting of a totally real algebraic number field K. First, we deal with the order of magnitude of a greatest (relative to its norm) prime ideal factor of \(\), where the product runs over prime arguments ω of a given irreducible polynomial F which lie in a certain lattice point region. Then, we turn our attention to the problem about the occurrence of algebraic primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. Finally, we give further contributions to the binary Goldbach problem in K.

On Arithmetic Properties of the Solutions of a Universal Differential Equation at Algebraic Points

In 1981, L. A. Rubel found an explicit algebraic differential equation (ADE) of order four such that every real continuous function on the real line can be uniformly approximated by the C∞-solutions of this ADE. It is shown that an ADE of order five exists, where the C∞-solutions additionally satisfy some algebraic properties in the sense of C. L. Siegel's results from the analytical theory of numbers. For instance, all the solutions and their derivatives are transcendental at algebraic points, and large sets of these numbers are linearly independent over the field of real algebraic numbers.

Jacobi forms over totally real number fields

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.

Uniform bounds on complexity and transfer of global properties of Nash functions

We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian.

A note on class number 1 criteria for totally real algebraic number fields

A note on class number 1 criteria for totally real algebraic number fields

On Local Objects Attached to and Functions

Weil indices are attached individually to local Well representations of local symplectic groups and, collectively (and in a product formula), to the adélic Weil Θ funtional. Therefore Weil indices can be attached to classical ζ functions. Root numbers, or local constants, are attached to the decomposition of L and functions into local factors. Since there is a classical integral-transform connection between ζ and ζ there must be an according conection between Weil indices and local constants. We prove that this is so for the prototypical case of the Dedekind ζ funtion, and, indeed, that this relationship is given in terms of an equation involving an integral of dimension nplus1 where n is the degree of the underlying (totally real) algebraic number field

On adelic automorphic forms with respect to a quadratic extension

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

Theta correspondence and Hecke operators relative to a quadratic extension

Introduction. Let E/F be a totally real quadratic extension of a totally real algebraic number field. To a suitably defined automorphic form h defined with respect to a quaternion algebra BE over E, we can associate a Hilbert modular form I(z, h) defined with respect to the field F . Such a lifting has been explicitly considered in the author’s recent paper [D99], via a convolution with a theta function, and the Fourier coefficients of the theta lift have been computed in terms of certain periods of the original form. The purpose of the current paper is to establish an explicit formula relating the actions of the Hecke operators on the original automorphic form and its theta lift. This result is analogous to theorems of Shimura in, for example, [Sh82] and [Sh88], where he has shown commutativity properties with respect to these Hecke operators. In addition to their inherent interest, these commutativity results are necessary in obtaining algebraicity results concerning Shimura’s period invariants as well as certain special L-values which naturally arise. The reader can consult Shimura’s [Sh88] for further motivation in this regard. See also [Y]. It is the author’s hope to discuss applications in this direction in the near future. Since it is essential for our purposes to consider everything explicitly, the quantity of technical details is considerable. In the interest of space, and also in order not to obscure essential ideas, a large amount of computation has been suppressed. Instead we have endeavored to point out key ingredients in the computations and to give precise references. It is then hoped that

On lattice bases with special properties

In this paper we introduce multiplicative lattices in (ℝ >0 ) r and determine finite unions of suitable simplices as fundamental domains for sublattices of finite index. For this we define cyclic non-negative bases in arbitrary lattices. These bases are then used to calculate Shintani cones in totally real algebraic number fields. We mainly concentrate our considerations to lattices in two and three dimensions corresponding to cubic and quartic fields.

On a Theta Correspondence with Respect to a Quadratic Extension

Let F be a totally real algebraic number field, and let E be a totally real quadratic extension of F. In this article we establish a theta correspondence between certain automorphic forms defined with respect to a quaternion algebra over E and Hilbert modular forms defined with respect to F. Given such a quaternionic form, say h, the main theorem expresses the Fourier coefficients of its theta lift in terms of periods of h. The results in this paper generalize some theorems of Shimura.

On fundamental domains of arithmetic Fuchsian groups

Let K be a totally real algebraic number field and O an order in a quaternion algebra A over K. Assume that the group O^1 of units in O with reduced norm equal to 1 is embedded into PSL_{2}(R) as an arithmetic Fuchsian group. It is shown how Ford's algorithm can be effectively applied in order to determine a fundamental domain of O^1 as well as a complete system of generators of O^1.