Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K... more
Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and \chi is a ring class character such that L(E/K,\chi,1) is
Aragona-Fernadez-Juriaans showed that a generalized holomorphic function has a power series. This is one of the ingredients use to prove the identity theorem.
Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an... more
Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence... more
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.
We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing of commutators and anticommutators of Clifford algebra... more
We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing of commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.
Is "Gravity" a deformation of "Electromagnetism"? Deformation theory suggests quantizing Special Relativity: formulate Quantum Information Dynamics SL(2,C)_h-gauge theory of dynamical lattices, with unifying gauge... more
Is "Gravity" a deformation of "Electromagnetism"? Deformation theory suggests quantizing Special Relativity: formulate Quantum Information Dynamics SL(2,C)_h-gauge theory of dynamical lattices, with unifying gauge "group" the quantum bundle obtained from the Hopf monopole bundle underlying the quaternionic algebra and Dirac-Weyl spinors. The deformation parameter is the inverse of light speed 1/c, in duality with Planck's constant h. Then mass and electric charge form a complex coupling constant (m,q), for which the quantum determinant of the quantum group SL(2,C)_h expresses the interaction strength as a linking number 2-form. There is room for both Coulomb constant k_C and Newton's gravitational constant G_N, exponentially weaker then the reciprocal of the fine structure constant α. Thus "Gravity" emerges already "quantum", in the discrete framework of QID, based on the quantized complex harmonic oscillator: the quantized q...
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence... more
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.
Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to... more
Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of ...
In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to reveal... more
In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to reveal and prove a number of new properties of Clifford algebra. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.
I consider differential of mapping f of continuous division ring as linear mapping the most close to mapping f. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative... more
I consider differential of mapping f of continuous division ring as linear mapping the most close to mapping f. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent. I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function of original number a̅=a+iai+jaj+kak . In quaternion algebra this difference leads to the absence of analogue of the Cauchy Riemann equations that are well known in the theory of complex function.
We propose an algebraic framework for studying coherent space-time codes, based on arithmetic lattices on central simple algebras. For two transmit antennas, this algebra is called a quaternion algebra. For this reason, we call these... more
We propose an algebraic framework for studying coherent space-time codes, based on arithmetic lattices on central simple algebras. For two transmit antennas, this algebra is called a quaternion algebra. For this reason, we call these lattices quaternionic lattices. The design criterion is the one described by V. Tarokh et al. (see IEEE Trans. Inf. Theory, vol.44, p.744-65, 1998).
Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $ E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are... more
Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $ E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which the generators are send to conjugate elements, by analyzing the points of an algebraic variety, that we call the \emph{variety of affine c-representations of}$G$. Each point in this variety correspond to a representation in the unit group of a quaternion algebra and their affine deformations.
The basics on the arithmetic on quaternion algebras is introduced: (maximal) orders, (principal) ideals, (reduced) norm/discriminant, ideal classes, etc.
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence... more
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.
We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on... more
We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locally compact field of characteristic 0 and each $\Gamma_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.
We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup... more
We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locally compact field of characteristic 0 and each $\Gamma_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.
In this paper, we analyze the performance of an Algebraic Space Time Codes (ASTC), called the Golden code. Due to its Algebraic construction based on Quaternionic algebra, the code has a full rate, full diversity, non-vanishing constant... more
In this paper, we analyze the performance of an Algebraic Space Time Codes (ASTC), called the Golden code. Due to its Algebraic construction based on Quaternionic algebra, the code has a full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. In this paper, we analyze the performances of the Golden codes in correlated Rayleigh channel. We consider a coherent demodulator and we analyze the channel estimation error impact.