Orders In Central Simple Algebras Research Articles

Finiteness of elasticities of orders in central simple algebras

Let $\mathcal O$ be an order in a central simple algebra $A$ over a number field. The elasticity $\rho (\mathcal O)$ is the supremum of all fractions $k/l$ such that there exists a non-zero-divisor $a \in \mathcal O$ that has factorizations into atoms (ir

Structural properties of subadditive families with applications to factorization theory

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$ such that, for every $M\in{\bf N}$, the sequences $$ \bigl((\mathscr U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad \bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} $$ are $\mu$-periodic from some point on. The result applies, e.g., to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathscr L$ of subsets of $\bf N$ such that, for all $L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.

Metacommutation of primes in central simple algebras

In a quaternion order of class number one, an element can be factored in multiple ways depending on the order of the factorization of its reduced norm. The fact that multiplication is not commutative causes an element to induce a permutation on the set of primes of a given reduced norm. We discuss this permutation and previously known results about the cycle structure, sign, and number of fixed points for quaternion orders. We generalize these results to other orders in central simple algebras over global fields.

Arithmetical invariants of local quaternion orders

Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$. The elasticity of $R^\bullet$ is $\rho(R^\bullet) = \sup\{\, k/l : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$, $v_j$ atoms of $R^\bullet$ and $k$, $l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $\rho(R^\bullet)$ and show that the set of distances $\Delta(R^\bullet)$ and all catenary degrees $\mathsf c_\mathsf d(R^\bullet)$ are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples.

Local selectivity of orders in central simple algebras

Let [Formula: see text] be a central simple algebra of degree [Formula: see text] over a number field [Formula: see text], and [Formula: see text] be a strictly maximal subfield. We say that the ring of integers [Formula: see text] is selective if there exists an isomorphism class of maximal orders in [Formula: see text] no element of which contains [Formula: see text]. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers [Formula: see text] by leveraging the theory of affine buildings for [Formula: see text] where [Formula: see text] is a local central division algebra. Then as an application, we use the local result and a local–global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in [Formula: see text], and distinguish those which are guaranteed to contain [Formula: see text]. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra.

Factorization theory: From commutative to noncommutative settings

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n×n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.

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.

Lefschetz numbers of symplectic involutions on arithmetic groups

The reduced norm-one group G of a central simple algebra is an inner form of the special linear group, and an involution on the algebra induces an automorphism of G. We study the action of such automorphisms in the cohomology of arithmetic subgroups of G. The main result is a precise formula for Lefschetz numbers of automorphisms induced by involutions of symplectic type. Our approach is based on a careful study of the smoothness properties of group schemes associated with orders in central simple algebras. Along the way we also derive an adelic reformulation of Harder's Gauss-Bonnet Theorem.

Sets of lengths in maximal orders in central simple algebras

Let O be a holomorphy ring in a global field K, and R a classical maximal O-order in a central simple algebra over K. We study sets of lengths of factorizations of cancellative elements of R into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of O, which implies that all the structural finiteness results for sets of lengths—valid for commutative Krull monoids with finite class group—hold also true for R. If O is the ring of algebraic integers of a number field K, we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.

Hermitian Morita Theory and Unitary Groups of Hyperbolic Quadratic Modules over Maximal Orders in Central Simple Algebras

Let R = Z with quotient field say K. Let Λ and Λ1 be maximal R-orders with antistructure in central simple K-algebras A = M n (D) and A 1 = M n (D) where D and D 1 are separable over K. Let (M, h) and (N, k) be hyperbolic quadratic, hermitian Λ and Λ1 modules, respectively. Let be an arbitrary isomorphism of the corresponding projective unitary groups. Under some minimal conditions there is a category equivalence from the category of hermitian Λ-modules to that of hermitian Λ1-modules which arises fronm a hermitian Morita context and produces Φ by its action on morphisms.

An embedding theorem for orders in central simple algebras

For an n -dimensional central simple algebra defined by a generalized Hilbert symbol over a number field, we compute the number of conjugacy classes of maximal orders that contain a conjugate of the order spanned by the canonical generators of the algebra. This has applications to the study of certain finite subgroups of arithmetic orbifold groups.

Corrections to Hermitian Morita equivalences between maximal orders in central simple algebras

Corrections to Hermitian Morita equivalences between maximal orders in central simple algebras

Hermitian Morita equivalences between maximal orders in central simple algebras

Let R be a Dedekind domain with quotient field $K$. That every maximal order in a finite dimensional central simple $K$-algebra $A$, (the algebra of nxn matrices over $D$), where $D$ is separable over $K$, is Morita equivalent to every maximal order in $D$ is a well known linear result. Hahn defined the notion of Hermitian Morita equivalence (HME) for algebras with anti-structure, generalizing previous work by Frohlich and McEvett. The question this paper investigates is the hermitian analogue of the above linear result. Specifically, when are maximal orders with anti-structure in $A$, HME to maximal orders with anti-structure in $D$ in the sense of Hahn? Two sets of necessary and sufficient conditions are obtained with an application which provides the hermitian analogue under some conditions.

Local structure of Schelter-Procesi smooth orders

In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if Δ \Delta is a central simple K K -algebra of dimension n 2 n^2 , where K K is a field of trancendence degree d d , then there are only finitely many étale local classes of smooth orders in Δ \Delta . This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.

Maximal Orders in Central Simple Algebras and Bruhat–Tits Buildings

We study the affine building forSLnover a local field and give a characterization of distance involving Hecke operators. Forn=3 we give an explicitly computable distance formula. We use this local information to show that the class number of a maximal order in a central simple algebra of dimensionn2over a number fieldKis equal to the number of orbits of a group of isometries (related to the unit group of the maximal order) acting on a Bruhat-Tits building forSLn(K). This generalizes results of Serre and Vignéras who considered the quaternion case in which the Bruhat-Tits building is a tree.

Homology of maximal orders in central simple algebras

and moreover that HH2(O/R) = 0. These two facts greatly facilitate the computation of the cyclic homology HC∗(O/R) ([3]). This paper presents non-commutative analogues of the main results of [3]. In particular, L is to be replaced by a central simple algebra D over K, while O is taken to be a maximal R-order in D. Of course, O is not, in general, uniquely defined by R and D, but it turns out that the homology of O/R is independent of the choice of maximal order. We prove that (0.1) remains valid in the non-commutative context but that the odd Hochschild homology groups vanish. The first section is devoted to generalities on Hochschild homology. We do not claim to prove any new result but only to establish notation and give self-contained proofs of various facts for which we could not find convenient references. The main results, the periodicity theorem and the vanishing theorem, are proved in §2 when R is a complete discrete valuation ring with perfect residue field. We construct an element in Hochschild cohomology such that Yoneda product gives the desired periodicity. The calculations needed to compute the low-dimensional Hochschild homology groups are laborious, but the resulting formulae are quite simple. The globalization is carried out in §3 and is completely standard. The vanishing of odd Hochschild homology guarantees that the Loday-Quillen spectral sequence degenerates at E, so it is easy to compute cyclic homology, in both the local and the global case, up to extension. This is explained in §4, where the extension problem is partially solved by an application of the universal coefficient theorem.

Solomon's second conjecture: A proof for local hereditary orders in central simple algebras

Solomon’s second conjecture concerns partial zeta functions of local orders. The exact definitions of orders and ideals can be found in [R], and we use the partial zera functions, as defined in [BR]. We briefly recall the definitions. Let R, be a discrete valuation ring in a local field K with valuation v. A local order 0 is an R,-order in a central simple K-algebra A = M,(D). If IZ = 1 the valuation u has a unique extension vD in the skew field D. We denote with d the unique maximal R,-order in D and with fi the maximal ideal in A. Then # (A/b) = q is finite. We fix a uniformizing element n of A, i.e., uD(rc) = 1. If n # 1 the local orders are not so easy to describe. The maximal R,-orders /i in A are isomorphic with M,(A). The partial zeta functions of a local order 0 correspond to a choice of two left @-ideals 2, -4 in A (cf. [BR, p. 1371):

On two classical theorems in the theory of orders

The paper contains generalizations of the Latimer-MacDuffee theorem and the Chevalley-Hasse-Noether theorem. It shows that the two theorems are closely related to each other by means of a duality, which depends on simultaneous actions of the idèle groups on maximal orders in central simple algebras and on embeddings of maximal commutative subrings into such orders.

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.

A duality theorem for orders in central simple algebras over function fields

A duality theorem for orders in central simple algebras over function fields