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 details of a formerly known relation between ternary quadratic forms and quaternion orders through the even Clifford algebra. Based on this and classifications of ternary quadratic forms we give a completely explicit classification of quaternion orders in the padic case.
Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was proved universal by Hurwitz using a special ring of quaternions. Here seven other quaternary quadratic forms are shown universal by investigation of appropriate rings of quaternions.
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.
Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of representations of a positive integer by sums of four squares using quaternions. The result is the formula for the number of representations of a positive integer by one of the above mentioned quaternary quadratic forms.
We develop an elementary divisor theory for the unimodular and the modular group over quadratic field extensions and quaternion algebras. In particular, we investigate which sets of elementary divisors can occur. Under an additional hypothesis we establish a correspondence of unimodular and modular double cosets.
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.
We study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use geometry of numbers to prove that certain ideals of the ring of quaternions contain elements of a small norm. We prove that $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal over $\mathbb{Q}(ζ_7+ζ_7^{-1})$ and that $x^2+xy+y^2+z^2+zw+w^2$ represents all totally positive multiples of certain special elements.
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 2x2-matrix ring M_2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.
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 a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined …
Quaternion extensions are often the smallest extensions to exhibit special properties. In the setting of the Hasse-Arf Theorem, for instance, quaternion extensions are used to illustrate the fact that upper ramification numbers need not be integers. These extensions play a similar role in Galois module structure. To better understand these examples, we catalog the ramification filtrations that are possible in totally ramified extensions of dyadic number fields. Interestingly, we find that the catalog depends, for sharp lower bounds, upon the refined ramification filtration, which is associated…
We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers.
Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple $L$-algebra.
Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\mathrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\mathrm{SL}_2(\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. The…
A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with a type number 3 or larger. The proof extends to a few other closely related orders.
Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants of these orders including the class number, the index $w(B)$ and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras.
Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner poi…
We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study maximal arithmetic subgroups of such groups, and show that examples can be produced by studying orders of the quaternion algebra. In both cases, we relate the algebraic properties of the underlying rings to sufficient and necessary conditions for the groups to be isomorphic and/or conjugate to one another.
We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus our attention on the characterization of elements having a finite continued fraction expansion. By means of a suitable notion of quaternionic height, we prove a criterion for finiteness. Furthermore, we draw some consequences about the solutions of a family of quadratic polynomial equations with coefficients in $B$.
We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral $D_{2n}$ or (generalized) quaternion $\mathbb H_{2n}$ of two-power order. In the spirit of recent work of Fiorilli and Jouve arXiv:2001.05428, we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical aspect, where the degree goes to infinity. Our main contribution uncovers in families of extensions a phenomenon, for which Ng gave numerical evidence : real zeros…
We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the $E_8$ lattice (resp. $E_8^2$ and $Λ_{16}$) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for co…
An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the algorithms for ideal manipulation. However, this is not the only option, as an ideal can also be represented by its generating set. Moreover, in certain quadratic orders of the ring of integral quaternions, every ideal can be represented by a single quaternion, called a pseudo generator. In this paper, we develop algorithms which allow us to manipulate ideals using s…
Waldspurger's formula gives an identity between the norm of a torus period and an L-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding L-functions agree, (the norms of) these periods---which occur on different quaternion algebras---are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.
Let $D$ be a positive definite quaternion algebra over a totally real number field $K$, $F(X,Y)$ a hermitian form in 2N variables over $D$, and $Z$ a right $D$-vector space which is isotropic with respect to $F$. We prove the existence of a small-height basis for $Z$ over $D$, such that $F(X,X)$ vanishes at each of the basis vectors. This constitutes a non-commutative analogue of a theorem of Vaaler, and presents an extension of the classical theorem of Cassels on small zeros of rational quadratic forms to the context of quaternion algebras.
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.
We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can consider the signed intersection number of pairs of embeddings. Using the Hecke operators and the intersection pairing, we construct a generating series that is demonstrated to be a classical modular form of weight two.
We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q).
Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q…
Let $F$ be a totally real field in which a prime number $p>2$ is inert. We continue the study of the (generalized) Goren--Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb F_p$. We prove that, when the dimension of the quaternionic Shimura variety is even, the Tate conjecture for the special fiber of the quaternionic Shimura variety holds for the cuspidal $π$-isotypical component, as long as the two unramified Satake parameters at $p$ are not differed by a root of unity.
Let $A$ be a quaternion algebra over a number field $F$, and $\mathcal{O}$ be an $O_F$-order of full rank in $A$. Let $K$ be a quadratic field extension of $F$ that embeds into $A$, and $B$ be an $O_F$-order in $K$. Suppose that $\mathcal{O}$ is a Bass order that is well-behaved at all the dyadic primes of $F$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus of $\mathcal{O}$. This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan [Optimal embeddings in quaternion algebras. J. Number Theory, 128(10):285…
Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ form a cocompact discrete subgroup $Γ_{p,a}\subset SL(2,\mathbb{R})$ and is commensurable with the group of units of an order in a quaternion algebra over $\mathbb{Q}$. The problem addressed in this paper is an estimate for the traces of a set of generators for $Γ_{p,a}$. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a g…
Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.
Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ form a cocompact discrete subgroup $Γ_{p,a}\subset SL(2,\mathbb{R})$ and is commensurable with the group of units of an order in a quaternion algebra over $\mathbb{Q}$. The problem addressed in this paper is an estimate for the traces of a set of generators for $Γ_{p,a}$. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a g…
Let $k$ be a number field. In this paper, we give a formula for the mean value of the square of class numbers times regulators for certain families of quadratic extensions of $k$ characterized by finitely many local conditions. We approach this by using the theory of the zeta function associated with the space of pair of quaternion algebras.
Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gaussian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R. An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the quest…
Let k be an imaginary quadratic number field, let F be a rational quaternion algebra and M an extension of F as a quaternion k-algebra. We are going to classify the F-orders which arise as an intersection of F with a maximal M-order; and we are going to prove that the discriminant of such an intersection determines uniquely the isomorphism type of the corresponding maximal M-order. Building on this, we are going to relate this intersection to the intersection of a second rational quaternion algebra F' in M with a second maximal M-order. This allows us to determine whether the Bianchi group ove…
For every finite $p$-group $G_p$ that is non-cyclic and non-quaternion and every positive integer $\ell\neq p$ that is greater than $2$, we prove the first non-trivial bound on $\ell$-torsion in class group of every $G_p$-extension. More generally, for every nilpotent group $G$ where every Sylow-$p$ subgroup $G_p\subset G$ is non-cyclic and non-quaternion, we prove a non-trivial bound on $\ell$-torsion in class group of every $G$-extension for every integer $\ell>1$.
Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representa…
We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^\prime$, when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p, \infty}$, the definite quaternion a…
An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a co…
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/Q be a totally real field extension of degree g and let D be a definite quaternion algebra with center F. Fix an odd prime p which is unramified in F and D. We produce weight shiftings between (mod p) automorphic forms on the multiplicative group of D having fixed level. When the starting weight does not contain any (2,...,2)-block, we obtain these shiftings via maps induced in cohomology by intertwining operators acting on F_{p}-representations of GL(2,O_{F}/pO_{F}). Shiftings by (p-1,...,p-1) for weights containing (2,...,2)-blocks are obtained following the methods of Edixhoven-Khare,…
