Related papers: Constructing non-maximal orders in quaternion alge…

We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.

A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may…

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…

Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show…

In this note, we give a concrete realization of the Jacquet-Langlands correspondence for non-Eichler orders of indefinite quaternion algebras defined over $\mathbb Q$. To be more precise, we consider a special type of index-two suborder of…

The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a…

We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra.

We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related…

The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper we consider the related problems of computing all minimal overorders…

A major current goal of noncommutative geometry is the classification of noncommutative projective surfaces. The generic case is to understand algebras birational to the Sklyanin algebra. In this work we complete a considerable component of…

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…

We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra A on X. Then we study the moduli scheme of torsion free A-modules of rank one. Finally we prove that this moduli…

We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.

A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…

Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements.…

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a…

We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees…

Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field,…

We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…