Real Algebraic Number Field Research Articles

On arithmetic quotients of the group SL2 over a quaternion division k-algebra

Abstract Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces X G / Γ {X_{G}/\Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k-group G = SL M 2 ⁢ ( D ) {G=\mathrm{SL}_{M_{2}(D)}} , attached to a quaternion division k-algebra D. The group G is k-simple, of k-rank one, and non-split over k. Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ {Y_{\Gamma}\subset X_{G}/\Gamma} such that its closure Y ¯ Γ {\overline{Y}_{\Gamma}} is a compact manifold with boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} , and the inclusion Y ¯ Γ → X G / Γ {\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma} is a homotopy equivalence. The connected components Y [ P ] {Y^{[P]}} of the boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k-subgroups of G. We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k-algebra D is totally definite, that is, D ramifies at all archimedean places of k, we prove that the basis of this bundle is homeomorphic to the torus T s - 1 {T^{s-1}} of dimension s - 1 {s-1} , has the compact fibre T 4 ⁢ s {T^{4s}} , and its structure group is SL 4 ⁢ s ⁢ ( ℤ ) {\mathrm{SL}_{4s}(\mathbb{Z})} . We determine the cohomology of Y [ P ] {Y^{[P]}} . Secondly, if the quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ {v\in V_{k,\infty}} at which D v {D_{v}} splits over ℝ {\mathbb{R}} , that is, D v ≅ M 2 ⁢ ( ℝ ) {D_{v}\cong M_{2}(\mathbb{R})} , the fibre is homeomorphic to T 4 ⁢ s {T^{4s}} , but the base space of the bundle is more complicated.

Automorphism groups and normal forms in Normaliz

We discuss the computation of automorphism groups and normal forms of cones and polyhedra in Normaliz and indicate its implementation via nauty. The types of automorphisms include integral, rational, Euclidean and combinatorial, as well as algebraic for polytopes defined over real algebraic number fields. Examples treated in detail are the icosahedron and linear-ordering polytopes whose Euclidean automorphism groups are determined.

On the modular forms of weight 1/2 over algebraic number fields

Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of modular forms of weight 1/2 over imaginary quadratic fields.In this paper, we determine an explicit basis of the space of modular forms of weight 1/2, level c and character ψ over algebraic number fields. We prove our assertion using their arguments and Shimura's transformation formula of theta series over algebraic number fields.

Fields of Algebraic Numbers Computable in Polynomial Time. I

It is proved that the field of complex algebraic numbers has an isomorphic presentation computable in polynomial time. A similar fact is proved for the ordered field of real algebraic numbers. The constructed polynomially computable presentations are based on a natural presentation of algebraic numbers by rational polynomials. Also new algorithms for computing values of polynomials on algebraic numbers and for solving equations in one variable with algebraic coefficients are presented.

Zero-Sum Stochastic Games over the Field of Real Algebraic Numbers

We consider a finite state, finite action, zero-sum stochastic games with data defining the game lying in the ordered field of real algebraic numbers. In both the discounted and the limiting average versions of these games, we prove that the value vector also lies in the same field of real algebraic numbers. Our method supplies finite construction of univariate polynomials whose roots contain these value vectors. In the case where the data of the game are rational, the method also provides a way of checking whether the entries of the value vectors are also rational.

Polynomial Computability of Fields of Algebraic Numbers

We prove that the field of complex algebraic numbers and the ordered field of real algebraic numbers have isomorphic presentations computable in polynomial time. For these presentations, new algorithms are found for evaluation of polynomials and solving equations of one unknown. It is proved that all best known presentations for these fields produce polynomially computable structures or quotient-structures such that there exists an isomorphism between them polynomially computable in both directions.

The growth of class numbers of quadratic Diophantine equations

Let F be a totally real algebraic number field, with OF the ring of algebraic integers in F, let L be an OF-lattice on a d-dimensional (d≥2) positive definite quadratic space V over F and let u0 be a primitive vector in L. The main objective of this paper is to study when a is a fixed (non-unit) algebraic integer in OF and n is a positive rational integer, how the class numbers of lattices translation L+u0an grow as n tends to infinity.

The constructive membership problem for discrete two-generator subgroups of [formula omitted

We describe a practical algorithm to solve the constructive membership problem for discrete two-generator subgroups of SL2(R) or PSL2(R). This algorithm has been implemented in Magma for groups defined over real algebraic number fields.

On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

Abstract Let Γ ⊂ ℝ s be a lattice obtained from a module in a totally real algebraic number field. Let ℛ( θ , N) be the error term in the lattice point problem for the parallelepiped [−θ 1 N 1, θ 1 N 1] × . . . × [−θs Ns , θs Ns ]. In this paper, we prove that ℛ( θ , N)/σ(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ 1, . . . , θs ) is a uniformly distributed random variable in [0, 1] s , N = N 1 . . . . Ns and σ(ℛ, N) ≍ (log N)(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.

On the conjugates of certain algebraic integers

A well-known theorem, due to C. J. Smyth, asserts that two conjugates of a Pisot number, having the same modulus are necessary complex conjugates. We show that this result remains true for K-Pisot numbers, where K is a real algebraic number field. Also, we prove that a j-Pisot number, where j is a natural number, can not have more than 2j conjugates with the same modulus.

Complex Pisot numbers in algebraic number fields

Let P(K) be the set of Pisot numbers generating a real algebraic number field K over the field of rationals Q. Then, a result of Meyer implies that P(K) is relatively dense in the interval [1,∞) and a theorem of Pisot gives that P(K) contains units, whenever K≠Q. In the present note, we prove analogous results for the set of complex Pisot numbers generating a non-real number field K′ over Q when K′ is neither a quadratic field nor a CM-field.

On the Lower Bound in the Lattice Point Remainder Problem for a Parallelepiped

Let $$ \Gamma \subset {\mathbb {R}}^s $$Γ?Rs be a lattice, obtained from a module in a totally real algebraic number field. Let G be an axis parallel parallelepiped, and let |G| be a volume of G. In this paper we prove that [Equation not available: see fulltext.]Thus the known estimate $$\det \Gamma \#(\Gamma \cap G)=|G| +O(\ln ^{s-1} |G|)$$detΓ#(Γ?G)=|G|+O(lns-1|G|) is exact. We obtain also a similar result for the low discrepancy sequence corresponding to $$\Gamma $$Γ.

Asymptotic behaviors of means of central values of automorphic L-functions for [formula omitted

Let A be the adele ring of a totally real algebraic number field F. We push forward an explicit computation of a relative trace formula for periods of automorphic forms along a split torus in GL(2) from a square-free level case done by Masao Tsuzuki [20], to an arbitrary level case. By using a relative trace formula, we study central values of automorphic L-functions for cuspidal automorphic representations of GL(2,A) corresponding to Maass forms with arbitrary level.

A generalization of Wantzel's Theorem, m-sectable angles, and the density of certain Chebyshev-polynomial images

The eponymous theorem of P.L. Wantzel [5] presents a necessary and sufficient criterion for angle trisectability in terms of the third Chebyshev polynomial T3, thus making it easy to prove that there exist non-trisectable angles. We generalize this theorem to the case of all Chebyshev polynomials Tm (Corollary 1.4.1). We also study the set m-Sect consisting of all cosines of m-sectable angles (see Section 1), showing that, when m is not a power of two, m-Sect contains only algebraic numbers (Theorem 1.1). We then introduce a notion of density based on the diophantine-geometric concept of height of an algebraic number and obtain a result on the density of certain polynomial images. Using this in conjunction with the Generalized Wantzel Theorem, we obtain our main result: for every real algebraic number field K, the set m-Sect ∩K has density zero in [−1,1]∩K when m is not a power of two (Corollary 1.5.1).

On the ideal class groups of the maximal cyclotomic extensions of algebraic number fields

We shall consider the maximal cyclotomic extension of a totally real finite algebraic number field and its ideal class group. We shall investigate the structure of the ideal class group with the action of the cyclotomic Galois group.

The constructive membership problem for discrete free subgroups of rank 2 of

AbstractWe exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank $2$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$. This algorithm, together with methods for checking whether a two-generator subgroup of $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$ is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.Supplementary materials are available with this article.

Operator algebras and conjugacy problem for the pseudo-Anosov automorphisms of a surface

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism f, we assign an AF-algebra A(f) (an operator algebra). It is proved that the assignment is functorial, i.e. every f', conjugate to f, maps to an AF-algebra A(f'), which is stably isomorphic to A(f). The new invariants of the conjugacy of the pseudo-Anosov automorphisms are obtained from the known invariants of the stable isomorphisms of the AF-algebras. Namely, the main invariant is a triple (L, [I], K), where L is an order in the ring of integers in a real algebraic number field K and [I] an equivalence class of the ideals in L. The numerical invariants include the determinant D and the signature S, which we compute for the case of the Anosov automorphisms. A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated.

Linear forms at a basis of an algebraic number field

It was proved by Cassels and Swinnerton-Dyer that the Littlewood conjecture in simultaneous Diophantine approximation holds for any pair of numbers in a cubic field. Later this result was generalized by Peck to a basis ( 1 , α 1 , … , α n ) of a real algebraic number field of degree at least 3. By transference, this result provides some solutions for the dual form of Littlewoodʼs conjecture. Here we find another solutions, and using Bakerʼs estimates for linear forms in logarithms of algebraic numbers, we discuss whether the result is best possible.

𝔬K0-quasi-abelian varieties with complex multiplication

We are interested in finding the relation between rings of endomorphisms of toroidal groups and algebraic number fields which are not totally real. For algebraic number fields K , we define real maximal algebraic number fields K 0 and 𝔬 K 0 -quasi-abelian varieties. We show that any simple 𝔬 K 0 -quasi-abelian variety is constructed by embeddings of a non-totally real algebraic number field K and it has finite dimensional cohomology groups.

L-Functions for GSp(2) × GL(2): Archimedean Theory and Applications

Abstract. Let Π be a generic cuspidal automorphic representation of GSp(2) defined over a totally real algebraic number field k whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product L-functions L(s,Π × σ) for an arbitrary cuspidal automorphic representation σ of GL(2). We also give an application to the spinor L-function of Π.