Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $ξ_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{3,4,6,7,8,9,11,12\right\}$ and also we obtain a characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{5,10\right\}$. In the 4th section we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{n}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $n=l^{k},$ with $l$ a prime integer, $l\equiv 3$ (mod $4$) and $k$ a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{l}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $l$ is a Fermat prime number.
{\small In this paper, we find a class of division quaternion algebras over the field }$\mathbb{Q}\left( i\right) ${\small \ and a class of division symbol algebras over a cyclotomic field.}
Muto, Narita and Pitale construct counterexamples to the Generalized Ramanujan Conjecture for GL(2,B) over the division quaternion algebra B with discriminant two via a lift from SL(2). 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.
The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such group explicitly by lifting from Maass cusp forms for the congruence subgroup of level 2. We show that this lifting is non-zero and Hecke-equivariant. This allows us to determine each local component of such a cuspidal representation. We then know that our cuspidal representations provide examples of CAP representations, and in fact, counterexamples of the Generalized Ramanujan conjecture.
In this paper we determine sufficient conditions for a quaternion algebra to split over a quadratic field. In the last section of the paper, we find a class of division symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq 3$) over a $p-$ adic field or over a cyclotomic field.
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.
Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of quaternion division algebras H_K(p,q) over some dihedral fields K.
Let F be a non-Archimedean local field of residue characteristic p. In this paper, we first compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands and Jacquet-Langlands correspondence are not compatible with the reduction, except in the cases considered by Vigneras.
To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely determined by its maximal subfield spectrum. However in this paper, we show that there are fields for which the maximal subfield spectrum says little to nothing about a quaternion division algebra's isomorphism class. We give an explicit construction of a division algebra with infinite genus. Along the way we introduce the notion of a "linking field extension,…
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.
In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve E which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in subfields of such a division field. Such fields include a large class of non-solvable quintic extensions and our application provides an arithmetic counterpart to Klein's "solution" of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the elliptic curve reduced modulo p.
A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, the maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chindburg and Friedman in the quaternionic case.
Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an abelian $\ell$-extension unramified over $F$ assuming $\ell$ divides the class number of $F$. In this paper, we obtain a complete characterization of division quaternion algebras $H_{K}(p, q)$ over $K$.
We give a new proof of Howard's $Λ$-adic Gross-Zagier formula, which we extend to the context of indefinite Shimura curves over $\mathbf{Q}$ attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two-variable $p$-adic $L$-function restricted to the anticyclotomic line to the cyclotomic $p$-adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the original approaches of Gross-Zagier and Perrin-Riou, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and their expli…
Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $ H_K(α, m)$ a quaternion algebra over $K$, where $α\in \mathcal{O}_K$. In this paper we give necessary and sufficient conditions for $ H_K(α, m)$ to split over $K$ for some values of $α$, and we obtain a complete characterization of division quaternion algebras $ H_K(α, m)$ over $K$ whenever $α$ and $m$ are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers.
We study origami $f: C \rightarrow E$ with $G$-Galois cover $Q_8$. For a point $P \in E(\mathbb{Q}) \backslash \left\{ \mathcal{O} \right\}$, we study the field obtained by adjoining to $\mathbb{Q}$ the coordinates of all of the preimages of $P$ under $f$. We find a defining polynomial, $f_{E, Q_8,P}$, for this field and study its Galois group. We give an isomorphism depending on $P$ between a certain subfield of this field and a certain subfield of the 4-division field of the elliptic curve.
Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(ζ_n+\barζ_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ and $\operatorname{PSU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $Δ=Δ_{\mathfrak{p}}$ for $\operatorname{SL_{2}}(F_\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8…
In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensi…
A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}(ζ_{p}) = \mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue for the division fields of an elliptic curve. More precisely, for a prime number $p \geq 5$ and a positive integer $n$, we study the $p$-divisibility of the class number of the $p^{n}$-division field $\mathbb{Q}(E[p^{n}])$ of an elliptic curve $E$ of the form $y^{2} = x^{3}-(s^{4}+t^{2})x$. In particular, we construct a certain infinite subfamily consisting of curves with novel properties that t…
An abelian surface A over a field K has potential quaternionic multiplication if the ring End_\bar 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.
Let $Λ$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. 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 $\mathcal{T}$ associated to $PGL_2(\mathbb{F}_q((1/T)))$. 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 run…
Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(μ_\ell)$ be the cyclotomic field generated by the group of $\ell^{th}$-roots of unity. We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is "small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent. We show that if $\ell=2$ or 3 and $H$ is nilpotent then the reduction of $A$ at $p$ is totally toroidal, so its conductor is $p^{\dim A}$. T…
Let D be a quaternion division algebra over a totally real number field F which splits exactly at one infinite place. We assume that there is a p-adic place where D doesn't split. Then the associated Shimura curve has a Cherednik unifomization by the p-adic upper half plane. We define an integral model of the Shimura curve an over the integers O of this p-adic place. We prove an uniformization theorem for the completion of the model over the formal spectrum Spf O.
We give examples of quaternion and octonion division algebras over a field $F$ of characteristic $2$ that split over a purely inseparable extension $E$ of $F$ of degree $\geq 4$ but that do not split over any subextension of $F$ inside $E$ of lower exponent, or, in the case of octonions, over any simple subextension of $F$ inside $E$. Thus, we get a negative answer to a question posed by Bernhard Mühlherr and Richard Weiss. We study this question in terms of the isotropy behaviour of the associated norm forms.
Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.
Let $F=\mathbb{F}_q(T)$ be the field of rational functions with $\mathbb{F}_q$-coefficients, and $A=\mathbb{F}_q[T]$ be the subring of polynomials. Let $D$ be a division quaternion algebra over $F$ which is split at $1/T$. Given an $A$-order $Λ$ in $D$, we find an explicit finite set generating $Λ^\times$.
In this paper, we deal with the conjecture of 'Quantum Unique Ergodicity'. Z. Rudnick and P. Sarnak showed that there is no 'strong scarring' on closed geodesics for arithmetic congruence surfaces derived from a quaternion division algebra. We extend this result to a class of three-dimensional Riemannian manifolds X=Gamma\H^3 that are again derived from quaternion division algebras. We show that there is no 'strong scarring' on closed geodesics or on Gamma-closed imbedded totally geodesics surfaces of X.
We propose polynomial-time algorithms for finding nontrivial zeros of quadratic forms with four variables over rational function fields of characteristic 2. We apply these results to find prescribed quadratic subfields of quaternion division division algebras and zero divisors in $M_2(D)$, the full matrix algebra over a division algebra, given by structure constants. We also provide an implementation of our results in MAGMA which shows that the algorithms are truly practical.
Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan proved that $H^2(SL_1(D),\mathbb R/\mathbb Z)$ is a cyclic $p$-group whose order is bounded from below by the number of $p$-power roots of unity in $F$, unless $D$ is a quaternion algebra over $\mathbb Q_2$. In this paper we give an explicit upper bound for the order of $H^2(SL_1(D),\mathbb R/\mathbb Z)$ for $p\geq 5$ and determine $H^2(SL_1(D),\mathbb R/\mathbb …
Spread-spectrum systems are presented, which are based on Finite Field Fourier Transforms. Orthogonal spreading sequences defined over a finite field are derived. New digital multiplex schemes based on such spread-spectrum systems are also introduced, which are multilevel Coding Division Multiplex. These schemes termed Galois-field Division Multiplex (GDM) offer compact bandwidth requirements because only leaders of cyclotomic cosets are needed to be transmitted.
There are 3 examples in these notes. The first one is the standard example of the cubic resolvent of a quartic. The second example is exactly from Adelmann \cite{Adelmann} and gives a defining polynomial corresponding to the unique $S_4$-quotient of $\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})$. The splitting field of the Adelmann polynomial over $\mathbb{Q}$ is a subfield of the 4-division field of an elliptic curve, that contains the 2-division field of the elliptic curve. The third example is new and needed in the study of the field theory of quaternion origami. Associated to an elliptic curve de…
Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves E/Q such that Q(E[n]) is as small as possible, that is, when Q(E[n])=Q(zeta_n), and we prove that this is only possible for n=2,3,4, or 5. More generally, we classify all curves such that Q(E[n]) is contained in a cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem), when Q(E[n])/Q is an abelian extension. In particular, we prove that…
The first aim of this note is to fill a gap in the literature by proving that, given a global field $K$ and a finite set $\mathcal{S}$ of primes of $K$, every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/K)$ over $K$ with nilpotent kernel has a solution ${\rm{Gal}}(F/K) \rightarrow G$ such that all primes in $\mathcal{S}$ are totally split in $F/L$. We then apply this to inverse Galois theory over division rings. Firstly, given a number field $K$ of level at least $4$, we show that every finite solvable group occurs as a Galois group over the division ring $H_K$ of quaternions wi…
Let $F$ be the maximal totally real subfield of $\mathbf{Q}(ζ_{32})$, the cyclotomic field of $32$nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptio…
Cohn and Kumar showed the quadratic character of $q$ modulo $p$ gives the sign of the permutation of Hurwitz primes of norm $p$ induced by the Hurwitz primes of norm $q$ under metacommutation. We demonstrate that these permutations are equivalent to those induced by the right standard action of $\operatorname{PGL}_2 (\mathbb{F}_p)$ on $\mathbb{P}^1 (\mathbb{F}_p)$. This equivalence provides simpler proofs of the results of Cohn and Kumar and characterizes the cycle structure of the aforementioned permutations. Our methods are general enough to extend to all orders over the quaternions with a d…
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 χis a ring class character such that L(E/K,χ,1) is not 0 we prove a Gross-Zagier type formula relating Darmon points to a suitably defined algebraic part of L(E/K,χ,1); this generalizes results of Bertolini, Darmon and Dasgupta to the case of division quaternion algebr…
