Let VD be the Shimura curve over Q attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of VD and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves ðover Q and over QÞ and we use it to study the set of rational points on VD over quadratic fields. Fina...

We show that the bilinear complexity of multiplication in a non-split quaternion algebra over a field of characteristic distinct from 2 is 8. This question is motivated by the problem of characterising algebras of almost minimal rank studied in [1] This paper is a translation of a report submitted by the author to the XI international seminar ”Discrete mathematics and its applications”. Definit...

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard involution. We then investigate a class of exceptional rings of degree 2 which occur in every rank n ≥ 1 and show that they essentially characterize all algeb...

We describe a Jordan-algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems. The exactness of relaxations is proved based on rank estimates. Our approach provides a unifying viewpoint on a large number of classical results related to cones of Hermitia...

Let F be a p-adic field, that is, finite extension of Qp. D dimensional division algebra over F, and let SL1(D) the group elements reduced norm 1 in D. Prasad Raghunathan proved H2(SL1(D),R/Z) is cyclic p-group whose order bounded from below by number p-power roots unity unless quaternion Q2. In this paper we give an explicit upper bound for p≥5 determine precisely when cyclotomic, p≥19, degree...

The analysis of time series in 4D commutative hypercomplex algebras is introduced. Firstly, generalized Segre’s quaternion (GSQ) random variables and signals are studied. Then, two concepts properness suggested statistical tests to check if a GSQ vector proper or not proposed. Further, method determine which specific algebra most likely achieve, possible, the properties given. Next, both linear...

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

In this paper, a quaternion firefly algorithm is utilized to solve a multi-row facility layout design problem with the objective of minimizing the total cost of transportation. A quaternion firefly algorithm takes the motion of the firefly as a quaternion one. Consequently, the solution space is explored more accurately and the answers are of higher quality. That is, the answers are considerabl...

Let D be a division algebra for a non-Archimedean local field. Given an irreducible representation π of GL2(D), we describe its restriction to the diagonal subgroup D∗ × D∗. The description is in terms of the structure of the twisted Jacquet module of the representation π. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on re...

Taking the complex nature of quantum mechanics which we observe today as a low energy effect of a broken quaternionic theory we explore the possibility that dark matter arises as a consequence of this underlying quaternionic structure to our universe. We introduce a low energy, effective, Lagrangian which incorporates the remnants of a local quaternionic algebra, investigate the stellar product...