This paper describes new issues of the Mathematica standard package Quaternions for implementing Hamilton’s Quaternion Algebra. This work attempts to endow the original package with the ability to perform operations on symbolic expressions involving quaternion-valued functions. A collection of new functions is introduced in order to provide basic mathematical tools necessary for dealing with re...

In an earlier paper [14] I have adumbrated amethod for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it wi...

We generalize the classical Cayley-Dickson doubling process starting with a quaternion algebra over a field F by allowing the scalar in the doubling to be an invertible element in the algebra. We investigate the resulting eight-dimensional algebras over F and show that they are division algebras for all scalars chosen in D outside of the base field F , if D is a division algebra.

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras amo...

A free bimodule resolution is constructed for the integral group ring of the dihedral group of order 4m. This resolution is applied for a description, in terms of generators and defining relations, of the Hochschild cohomology algebra of this group ring. Introduction Let K be a commutative ring with unity, let R be an associative K-algebra that is a finitely generated projective K-module, let Λ...

We prove that the abelian K-surfaces whose endomorphism algebra is an indefinite 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

algebra. First, we recall some definitions in the theory of associative algebras. LetA 6= 0 be an algebra over the fieldK. If the equations ax = b, ya = b, ∀a, b ∈ A, a 6= 0, have unique solutions, then the algebra A is called a division algebra. If A is a finite-dimensional algebra, then A is a division algebra if and only if A is without zero divisors (x 6= 0, y 6= 0 ⇒ xy 6= 0).(see [9]) Let ...