Rational Quaternion Algebra Research Articles

Generating hyperbolic isometry groups by elementary matrices

We consider three families of groups: the Bianchi groups where is the ring of integers of an imaginary, quadratic field; the groups where is a -order of a definite, rational quaternion algebra with an orthogonal involution; and the groups where is an order of a definite, rational quaternion algebra. We show that such groups are generated by elementary matrices if and only if is semi-Euclidean (or -semi-Euclidean), which is a generalization of the usual notion of a Euclidean ring. The proofs are surprisingly simple and proceed by considering fundamental domains of Kleinian groups.

Hida theory for special orders

This note is devoted to the study of families of quaternionic modular forms arising from orders defined by Pizer. In this situation, the Hecke eigenspaces are 2-dimensional contrary to the classical case of Eichler orders. The main result is a Control Theorem in the spirit of Hida, interpolating these 2-dimensional Hecke eigenspaces. We restrict our attention to a definite rational quaternion algebra ramified at a single odd prime [Formula: see text].

Some properties of the norm in a division quaternion algebra

In this paper we provide some applications of the norm form in some quaternion division algebras over rational field, and we give some properties of Fibonacci sequence and Fibonacci sequence in connection to quaternion elements. We provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four‐square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quaternions defined on some division quaternion algebras, and we define and give properties of some special quaternions by using Fibonacci sequences.

The twisted Euclidean algorithm: Applications to number theory and geometry

We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution that admit such an algorithm. We give two applications: first, any order that admits such an algorithm has class number 1; second, we show how the existence of such an algorithm relates to the problem of constructing explicit Dirichlet domains for Kleinian subgroups of the isometry group of hyperbolic 4-space.

On units in orders in 2-by-2 matrices over quaternion algebras with rational center

We generalize an algorithm established in earlier work [21] to compute finitely many generators for a subgroup of finite index of an arithmetic group acting properly discontinuously on hyperbolic space of dimension 2 and 3, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators of finite index subgroups of a discrete subgroup of Vahlen’s group, i.e. a group of 2-by-2 matrices with entries in the Clifford algebra satisfying certain conditions. The motivation comes from units in integral group rings and this new algorithm allows to handle unit groups of orders in 2-by-2 matrices over rational quaternion algebras. The rings investigated are part of the so-called exceptional components of a rational group algebra.

Quaternion orders and sphere packings

We introduce an analog of Bianchi groups for rational quaternion algebras and use it to construct sphere packings that are analogs of the Apollonian circle packing known as integral crystallographic packings.

Hasse principle violations for Atkin-Lehner twists of Shimura curves

Let D > 546 D > 546 be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields l / Q l/\mathbb {Q} such that the twist of the Shimura curve X D X^D by the main Atkin-Lehner involution w D w_D and l / Q l/\mathbb {Q} violates the Hasse Principle over Q \mathbb {Q} . More precisely, the number of squarefree d d with | d | ≤ X |d| \leq X such that the quadratic twist of ( X D , w D ) (X^D,w_D) by Q ( d ) \mathbb {Q}(\sqrt {d}) violates the Hasse Principle is ≫ \gg X / log α D ⁡ X X/\log ^{\alpha _D} X and ≪ X / log β D ⁡ X \ll X/\log ^{\beta _D} X for explicitly given 0 > β D > α D > 1 0 > \beta _D > \alpha _D > 1 such that α D − β D → 0 \alpha _D - \beta _D \rightarrow 0 as D → ∞ D \rightarrow \infty .

Some special number sequences obtained from a difference equation of degree three

In this paper we present applications of special numbers obtained from a difference equation of degree three. As a particular case of this difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these quaternion elements, we can define a set with an interesting algebraic structure, namely, an order on a generalized rational quaternion algebra. Another presented application is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes).

Systoles of arithmetic hyperbolic surfaces and $3$-manifolds

Our main result is that for any positive real number x0, the set of commensurability classes of arithmetic hyperbolic 2– or 3–manifolds with fixed invariant trace field k and systole bounded below by x0 has density one within the set of all commensurability classes of arithmetic hyperbolic 2– or 3–manifolds with invariant trace field k. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is Q, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound to instead grow with the area/volume. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser–Sarnak and Katz–Schaps–Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3–manifolds with a small area totally geodesic surface.

Local Points on Shimura Coverings of Shimura Curves at Bad Reduction Primes

Let XD be the Shimura curve associated with an indefinite rational quaternion algebra of reduced discriminant D>1. For each prime l|D, there is a natural cyclic Galois covering of Shimura curves XD,l → XD constructed by adding certain level structure at l. The main goal of this note is to study the existence of local points at primes p≠l of bad reduction on the intermediate curves of these coverings and their Atkin–Lehner quotients.

Presentations for Quaternionic S-Unit Groups

We give an algorithm for presenting S-unit groups of an order in a definite rational quaternion algebra B such that for every p ∈ S at which B splits, the localization of at p is maximal, and all left ideals of of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamilton’s quaternions over the rational field . To our knowledge, this provides the first explicit presentations of an S-arithmetic lattice in a semisimple Lie group with S large. We also include some discussion and experimentation related to the congruence subgroup problem, which is open for S-units of the Hurwitz order when S contains at least two odd primes.

Nonuniform Fuchsian codes for noisy channels

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.

The Shimura covering of a Shimura curve: automorphisms and étale subcoverings

Let X D be the Shimura curve associated with an indefinite rational quaternion algebra of discriminant D , and let p be a prime dividing D . In their investigations on the arithmetic of X D , Jordan and Skorobogatov introduced a covering X D , p of X D whose maximal étale quotient is referred to as the Shimura covering of X D at p . The goal of this note is to describe the group of modular automorphisms of the curve X D , p and its quotients. As an application, we construct cyclic étale Galois coverings of Atkin–Lehner quotients of X D .

A NOTE ON CONTROL THEOREMS FOR QUATERNIONIC HIDA FAMILIES OF MODULAR FORMS

We extend a result of Greenberg and Stevens on the interpolation of modular symbols in Hida families to the context of non-split rational quaternion algebras. Both the definite case and the indefinite case are considered.

Intersection theory on Shimura surfaces II

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over ℚ associated to a rational quaternion algebra into the Shimura surface associated to the base change of the quaternion algebra to a real quadratic field. After extending the associated moduli problems over ℤ we obtain an arithmetic threefold with a embedded arithmetic surface, which we view as a cycle of codimension one. We then construct a family, indexed by totally positive algebraic integers in the real quadratic field, of codimension two cycles (complex multiplication points) on the arithmetic threefold. The intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the Fourier coefficients of a (very particular) Hilbert modular form of weight 3/2. The results are higher dimensional variants of results of Kudla-Rapoport-Yang, which relate intersection multiplicities of special cycles on the integral model of a Shimura curve to Fourier coefficients of a modular form in two variables.

A Mordell inequality for lattices over maximal orders

In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that the 16-dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structures.

Parametrization of Abelian K-surfaces with quaternionic multiplication

We prove that the Abelian K-surfaces whose endomorphism algebra is a rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin–Lehner involutions. To cite this article: X. Guitart, S. Molina, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves X0(D, N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut (X0(D, N)) is a 2-elementary abelian group which contains the group of Atkin–Lehner involutions W0(D, N) as a subgroup of index 1 or 2. It is conjectured that Aut (X0(D, N))=W0(D, N) except for finitely many values of (D, N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik–Drinfeld theory on their rigid analytic uniformization at primes p| D.

Heights and Determinants over Quaternion Algebras#

In Liebendörfer (2004) we defined a height function for matrices over a positive definite rational quaternion algebra. In this article, we prove that this height can be expressed, like the well-known height over number fields, as a product of local factors involving the maximal minors of the matrix. Part of our proof uses a quaternionic determinant invented by Moore (1922) and yields a non-commutative analogue of the Cauchy–Binet formula. The new formula of the height enables us to improve a result of Liebendörfer (2004).

Failure of the Hasse Principle for Atkin–Lehner Quotients of Shimura Curves over Q

We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin–Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin–Lehner involutions. A particular example is the quotient of the Shimura curve X23∙107 attached to the indefinite rational quaternion algebra of discriminant 23∙107 by the Atkin–Lehner involution ω107. The quadratic twist of X23∙107 by Q( √ −23) with respect to this involution is also a counter-example to the Hasse principle over Q.