Quaternion Division Research Articles

The density hypothesis for horizontal families of lattices

abstract: We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations are uniform in the volume and spectral aspects.

On Motohashi’s formula

We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of L L -functions for G L 1 GL_1 with the third moment of L L -functions for G L 2 GL_2 over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on M 2 ( A ) M_2(\mathbb {A}) instead of G L 2 ( A ) GL_2(\mathbb {A}) , on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi’s other proof of Motohashi’s formula. We give a variation of our method in the case of division quaternion algebras instead of M 2 M_2 , yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.

Remarks Regarding Computational Aspects in Algebras Obtained by Cayley–Dickson Process and Some of Their Applications

Due to the computational aspects which appear in the study of algebras obtained by the Cayley–Dickson process, it is difficult to obtain nice properties for these algebras. For this reason, finding some identities in such algebras plays an important role in obtaining new properties of these algebras and facilitates computations. In this regard, in the first part of this paper, we present some new identities and properties in algebras obtained by the Cayley–Dickson process. As another remark regarding the computational aspects in these algebras, in the last part of this paper, we solve some quadratic equations in the real division quaternion algebra when their coefficients are some special elements. These special coefficients allowed us to solve interesting quadratic equations, providing solutions directly, without using specialized softs.

On quaternion algebras over some extensions of quadratic number fields

Let p and q be two positive primes, let \(\ell\) be an odd positive prime and let F be a quadratic number field. Let K be an extension of F of degree \(\ell\) such that K is a dihedral extension of \({\mathbb {Q}}\), or else let K be an abelian \(\ell\)-extension of F unramified over F whenever \(\ell\) divides the class number of F. In this paper, we provide a complete characterization of division quaternion algebras \(H_{K}(p, q)\) over K.

On quaternion algebras over the composite of quadratic number fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

Some properties of the norm in a division quaternion algebra

In this paper we provide some applications of the norm form in some quaternion division algebras over rational field, and we give some properties of Fibonacci sequence and Fibonacci sequence in connection to quaternion elements. We provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four‐square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quaternions defined on some division quaternion algebras, and we define and give properties of some special quaternions by using Fibonacci sequences.

Yang–Mills-like field theories built on division quaternion and octonion algebras

Yang–Mills-like field theories are developed using both division quaternion and octonion algebras. Quaternionic field theory is the same as the Yang–Mills gauge theory. The octonionic field theory is an expansion from a similar field theory built with split octonions algebra that uses a complex-form basis as proposed by Gunaydin and Gursey (J Math Phys 14:1651, 1973), now with division octonions. This proposed octonionic field theory is the last possible field theory built using the division algebra of octonions, as permitted by the Hurwitz theorem. The interpretation of this “octonionic field” is not straightforward, except that it is similar to the quaternionic Yang–Mills field. A possible interpretation for this octonionic field theory for spin-1/2 particles is proposed in terms of quantum mechanics.

Counting and equidistribution for quaternion algebras

We aim at studying automorphic forms of bounded analytic conductor in the division quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato-Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term in the totally definite case.

Maass space for lifts to GL(2) over a division quaternion algebra

Muto, Narita and Pitale construct counterexamples to the Generalized Ramanujan Conjecture for GL2(B) over the division quaternion algebra B with discriminant two via a lift from SL2. In this paper, we try to exactly characterize the image of this lift. The previous methods of Maass, Kohnen or Kojima do not apply here, hence we approach this problem via a combination of classical and representation theory techniques to identify the image. Crucially, we use the Jacquet Langlands correspondence described by Badulescu and Renard to characterize the representations.

Dirac operators in tensor categories and the motive of quaternionic modular forms

We define a motive whose realizations afford modular forms (of arbitrary weight) on an indefinite division quaternion algebra. This generalizes work of Iovita–Spiess to odd weights in the spirit of Jordan–Livné. It also generalizes a construction of Scholl to indefinite division quaternion algebras, and provides the first motivic construction of new-subspaces of modular forms.

LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATIONS

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

About division quaternion algebras and division symbol algebras

In this paper, we find a class of division quaternion algebras over the field Q (i) and a class of division symbol algebras over a cyclotomic field.

A Clifford algebra associated to generalized Fibonacci quaternions

In this paper, using the construction of Clifford algebras, we associate to the set of generalized Fibonacci quaternions a quaternion algebra A (i.e., a Clifford algebra of dimension four). Indeed, for the generalized quaternion algebra , denoting , if , therefore the algebra A is split. If , then the algebra A is a division algebra. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice versa. MSC:11E88, 11B39.

Heegner points and Jochnowitz congruences on Shimura curves

Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1)) is -1 (respectively, +1), we prove a ``Jochnowitz congruence'' between the algebraic part of L'_K(E,1) (expressed in terms of Heegner points on Shimura curves) and the algebraic part of L_K(g,1). This establishes a relation between Zhang's formula of Gross-Zagier type for central derivatives of L-series and his formula of Gross type for special values. Our results extend to the context of Shimura curves attached to division quaternion algebras previous results of Bertolini and Darmon for Heegner points on classical modular curves.

Special values of L-functions and the arithmetic of Darmon points

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over without complex multiplication and is a ring class character such that , we prove a Gross–Zagier type formula relating Darmon points to a suitably defined algebraic part of ; this generalizes results of Bertolini, Darmon and Dasgupta to the case of division quaternion algebras and arbitrary characters. Finally, as an application of this formula, assuming the rationality conjectures for Darmon points we obtain vanishing results for Selmer groups of E over extensions of K contained in narrow ring class fields when the analytic rank of E is zero, as predicted by the Birch and Swinnerton-Dyer conjecture.

On computing quaternion quotient graphs for function fields

Let Λ be a Fq[T ]-maximal order in a division quaternion algebra over Fq(T ) which is split at the place∞. The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ∗ on the Bruhat-Tits tree T associated to PGL2(Fq((1/T ))). This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group Λ∗ in terms of generators and relations. Moreover we determine an upper bound for its running time using that Λ∗\T is almost Ramanujan.

ON GENERATORS OF ARITHMETIC GROUPS OVER FUNCTION FIELDS

Let F = 𝔽q(T) be the field of rational functions with 𝔽q-coefficients, and A = 𝔽q[T] be the subring of polynomials. Let D be a division quaternion algebra over F which is split at 1/T. For certain A-orders in D we find explicit finite sets generating their groups of units.

A new quaternion: Unit vector division quaternion

It is well known that, quaternion algebra allows the defining division of 3-D vectors. Division of vectors has many applications in applied mathematics and mechanics. However, indicial notation is widely used in the branches mentioned above. It is shown in this paper that division of orthonormal vectors can be conveniently handled by a simple operator. This operator can be used in the division of two arbitrary vectors and allows the expression of vector division using indicial notation. This new operator, which is a quaternion itself, can be also used in expressing quaternion product of two vectors, and the derivative of a vector with respect to a vector.

On abelian surfaces with potential quaternionic multiplication

An abelian surface A over a field K has potential quaternionic mul- tiplication if the ring End ¯ K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

On the set of discriminants of quadratic pairs

Let F be any field of characteristic 2. In this paper, we prove that, for any central simple F-algebra ( A, σ) with a symplectic involution, and for any element α∈F/℘(F), there exists a quadratic pair of the form ( σ, f) of discriminant equal to α, as soon as A is not a division quaternion algebra.