Orders In Quaternion Algebras Research Articles

Ideal classes of orders in quaternion algebras

We provide an algorithm that, given any order O in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right O-ideals, including the non-invertible ones. The theory is developed for a more general kind of algebras.

On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

We give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic $3$-manifolds and we deduce the following theorem. Given an arithmetically defined cocompact subgroup $\Gamma \subset \mathrm {SL}\_2(\mathbb C)$, provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence ${\Gamma\_i}\_i$ of finite index congruence subgroups of $\Gamma$ such that the first Betti number satisfies $$ b\_1(\Gamma\_i) \gg \[\Gamma:\Gamma\_i]^{1/2} $$ as $i$ goes to infinity.

Selective orders in central simple algebras and isospectral families of arithmetic manifolds

Let k be a number field and B be a central simple algebra over k of dimension p 2 where p is prime. In the case that p = 2 we assume that B is not totally definite. In this paper we study sets of pairwise nonisomorphic maximal orders of B with the property that a $${\mathcal{O}_{k}}$$ -order of rank p embeds into either every maximal order in the set or into none at all. Such a set is called nonselective. We prove upper and lower bounds for the cardinality of a maximal nonselective set. This problem is motivated by the inverse spectral problem in differential geometry. In particular we use our results to clarify a theorem of Vigneras on the construction of isospectral nonisometric hyperbolic surfaces and 3-manifolds from orders in quaternion algebras. We conclude by giving an example of isospectral nonisometric hyperbolic surfaces which arise from a quaternion algebra exhibiting selectivity.

Monomial, Gorenstein and Bass orders

In this article we study a class of orders called monomial orders in a central simple algebra over a non-archimedean local field. Monomial orders are easily represented and they may be also viewed as a direct generalization of Eichler orders in quaternion algebras. A criterion for monomial orders to be Gorenstein or to be Bass is given. It is shown that a monomial order is Bass if and only if it is either a hereditary or an Eichler order of period two.

Rotations and units in quaternion algebras

Unit groups of orders in quaternion algebras over number fields provide important examples of non-commutative arithmetic groups. Let K = Q ( d ) be a quadratic field with d < 0 a square-free integer such that d ≡ 1 ( mod 8 ) , and let R be its ring of integers. In this note we study, through its representation in SO 3 ( R ) , the group of units of several orders in the quaternion algebra over K with basis { 1 , i , j , k } satisfying the relations i 2 = j 2 = − 1 , i j = − j i = k .

Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted sysπ 1 . P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3bound sysπ1(XPCH) ≥ 4 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

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.

Quadratic diophantine equations and orders in quaternion algebras

We first reformulate the main theorem on quadratic Diophantine equations given in our recent book in terms of the even Clifford group. In the quaternary case the group is the multiplicative group of a quaternion algebra, and the reformulated theorem connects a class of primitive solutions of a quadratic equation with a class of an order in the algebra. In the indefinite case each class is essentially a class of binary hermitian forms. In the definite case the result is an analogue of the result of Gauss on the sums of three squares.

Lattices and orders in quaternion algebras with involution

In this paper maximal orders Λ in quaternion algebras having an involution over a quadratic field extension K / F are studied. A quadratic quaternary lattice in the algebra which parametrises the optimally embedded orders in F-subalgebras is constructed. It is shown that the Clifford algebra of the dual of this lattice can naturally be embedded in the order. A theory relating quaternion orders to hermitian planes is also developed. Using these techniques, the optimally embedded suborders of Λ are classified up to genus. Finally, it is shown that the units of norm 1 in Λ maps surjectively to the spinorial kernel of the orthogonal group of the lattice.

HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS

ABSTRACT In this paper we study the possible torsion in even-dimensional higher class groups Cl 2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl 2n (Λ) can only occur for primes p dividing prime ideals ℘ of 𝒪 F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.

On embedding numbers into quaternion orders

A generalization of the chevalley-Hasse-Noether theorem from maximal orders to arbitrary Eichler orders in quaternion algebras is given. A stability property for the numbers of orbits for unit groups in quaternion orders acting on optimal embeddings of quadratic orders is proved. The results are applied to Siegel's meanvalue of integral representations by genera of integral definite ternary quadratic forms.

The genus zeta function of hereditary orders in central simple algebras over global fields

Louis Solomon introduced the notion of a zeta function ζ Θ ( s ) {\zeta _\Theta }(s) of an order Θ \Theta in a finite-dimensional central simple K-algebra A, with K a number field or its completion K P {K_P} (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ζ Θ ( s ) {\zeta _\Theta }(s) , it is sufficient to consider the zeta function of local orders Θ P {\Theta _P} . However, since these local orders Θ P {\Theta _P} are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, Z Θ P ( s ) {Z_{{\Theta _P}}}(s) , which is related to the free Θ P {\Theta _P} -ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., [ A : K ] = 4 [A:K] = 4 ). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function Z Θ P ( s ) {Z_{{\Theta _P}}}(s) of any local hereditary order Θ P {\Theta _P} in a central simple algebra over a local field. We obtain Z Θ P ( s ) {Z_{{\Theta _P}}}(s) as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.

The Genus Zeta Function of Hereditary Orders in Central Simple Algebras Over Global Fields

Louis Solomon introduced the notion of a zeta function ${\zeta _\Theta }(s)$ of an order $\Theta$ in a finite-dimensional central simple K-algebra A, with K a number field or its completion ${K_P}$ (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ${\zeta _\Theta }(s)$, it is sufficient to consider the zeta function of local orders ${\Theta _P}$. However, since these local orders ${\Theta _P}$ are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, ${Z_{{\Theta _P}}}(s)$, which is related to the free ${\Theta _P}$-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., $[A:K] = 4$). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function ${Z_{{\Theta _P}}}(s)$ of any local hereditary order ${\Theta _P}$ in a central simple algebra over a local field. We obtain ${Z_{{\Theta _P}}}(s)$ as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.

Orders in quaternion algebras over global function fields having the cancellation property

Orders in quaternion algebras over global function fields having the cancellation property

On orders in quaternion algebras

(1983). On orders in quaternion algebras. Communications in Algebra: Vol. 11, No. 5, pp. 501-522.

An explicit trace formula for Brandt matrices

An. explicit formula is found for the traces of Hecke-Brandt operators which correspond to finite-dimensional representations of the groups of units of various orders in quaternion algebras over 2e, which are ramified over ℝ; it is not assumed that the divisors of the level which are relatively prime to the discriminant of the algebra are square-free.