Definite Quaternion Algebra Research Articles

Presentations of the unit group of an order in a non-split quaternion algebra

We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra H(K) over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of H( Z[ 1+ −7 2 ]) . As a consequence a presentation is discovered for the orthogonal group SO 3( Z[ 1+ −7 2 ]) . These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.

Uniform distribution of Heegner points

Let E be a (modular!) elliptic curve over Q, of conductor N . Let K denote an imaginary quadratic field of discriminant D, with (N ,D) = 1. If p is a prime, then there exists a unique Zp-extension K∞/K such that Gal(K/Q) acts nontrivially on Gal(K∞/K ). The field K∞ is called the anticyclotomic Zp-extension of K . Let E(K∞ ) denote the Mordell-Weil group of E over K∞. Then a fundamental conjecture of Mazur [Maz] predicts that the size of E(K∞ ) is controlled by the prime factorization of N in K . Equivalently, Mazur’s conjecture relates the size of the Mordell-Weil group to the sign in the functional equation of certain L-series. The conjecture was verified by Greenberg, Rohrlich, and Rubin, in what Mazur calls the exceptional case, when E has complex multiplication by K . More generally, they settled the conjecture for certain abelian varieties with complex multiplication. For a discussion of this CM case, we refer the reader to [Gre], [Roh], and [Rub]. Our goal in this paper is to treat the generic case, which occurs either when E has no CM, or when the field of complexmultiplications is distinct from K . Under certain conditions on E and K , Mazur’s conjecture predicts that the group E(K∞ ) is finitely generated; our main result asserts that this is in fact the case, at least when p is an ordinary prime for E, or when the class number of K is prime to p. The main new ingredient we introduce is that of equidistribution, following ideas used by Ferrero andWashington to study the cyclotomicμ-invariant. More precisely, we show that theHeegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve X , and that the elements of a certain Galois group act independently, in a suitable sense. This, combined with a special value formula due to Gross, allows us to conclude that the special values of anticyclotomic L-functions are almost always nonzero, so that the statement about the Mordell-Weil groups follows from the machinery of Euler systems as developed by Bertolini and Darmon [BD].

Quaternionic pryms and Hodge classes

Abelian varieties of dimension 2 n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n( n−1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds.

On the indecomposability of unit groups

We consider units of orders in a simple algebra A of finite dimension over the rational field. Such A can be written as A = Mn(D) where D is a skewfield (say of index d). Let K denote the centre of A (and D), R the integral closure of K, Λ an R-order in A, Γ = Λ× the group of units of Λ and SΓ the kernel of the reduced norm map nr : A → K on Γ. Two groups G and H are said to be commensurable (denoted G ∼ H) if they have a common subgroup of finite index. Since Γ ∼ R× × SΓ, the difficulty of Γ is concentrated in SΓ. It is well known that SΓ ∼ 1 if and only if A = K or A = D is a totally definite quaternion algebra. In the search for a structure theorem it is natural to envisage group theoretical constructions which produce “large” groups out of “small” ones. The most common examples are direct products, semidirect products, free products, amalgams and HNN extensions. We show that these constructions apply to unit groups only in very few cases of small dimension, the majority of which are well-known already. This demonstrates how far away we still are from an understanding of general unit groups.

Automorphic Forms, Definite Quaternion Algebras, and Atkin–Lehner Theory on Trees

Associated to the multiplicative group of a definite quaternion algebra over Q are two notions of automorphic forms. One is a complex-valued function on the idele group. The other, a p-adic-valued function on a p-adic “upper half-plane,” arises from considering differentials on a Mumford curve associated to the quaternion algebra. Explicit connections between these forms are explored. In particular, forms of the second type are seen to arise from certain newforms of the first type. In order to define newforms, the Atkin–Lehner theory is developed for certain spaces of functions on the tree associated to PGL2(Qp). As an application, Schneider's proposed p-adic L-series associated to a differential on a Mumford curve is seen to be better interpreted as an adelic form twisted by a p-adic character.

On the Curves Associated to Certain Rings of Automorphic Forms

AbstractIn a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

Rational points on some PEL -stacks

This work studies polarized abelian varieties with endomorphism ring a maximal order in a definite quaternion algebra. The result is that over small fields of characteristic 0 the degree of the polarization can only attain certain values.

Two-dimensional complex tori with multiplication by $\sqrt {d}$

We give an elementary argument for the well known fact that the endomorphism algebra \({\rm {End}}(A)\otimes {\Bbb Q }\) of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with \({\Bbb Q }(\sqrt {d})\hookrightarrow {\rm{End_{{\Bbb Q }}}}(T)\) where \({\Bbb Q }(\sqrt {d})\) is real quadratic, is algebraic.

Finite quaternionic matrix groups

Let D \mathcal {D} be a definite quaternion algebra such that its center has degree d d over Q \mathbb {Q} . A subgroup G G of G L n ( D ) GL_n(\mathcal {D}) is absolutely irreducible if the Q \mathbb {Q} -algebra spanned by the matrices in G G is D n × n \mathcal {D}^{n\times n} . The finite absolutely irreducible subgroups of G L n ( D ) GL_n(\mathcal {D}) are classified for n d ≤ 10 nd \leq 10 by constructing representatives of the conjugacy classes of the maximal finite ones. Methods to construct the groups and to deal with the quaternion algebras are developed. The investigation of the invariant rational lattices yields quaternionic structures for many interesting lattices.

On type numbers of split orders of definite quaternion algebras

In this paper, we shall give a new relation between the arithmetic of quaternion algebras and modular forms; we shall express the type numberT q, N of a split order of type (q, N) as the sums of dimensions of some subspaces of the space of cusp forms of weight 2 with respect to Γ0(qN) which are common eigenspaces of Atkin-Lehner's involutions.

Cubic Ramanujan graphs

A fimily of cubic Ramanujan graph is explicitly constructed. They are realized as Cayley graphs of a certain free group acting on the 3-regular tree; this group is obtained from a definite quaternion algebra that splits at the prime 2 and has a maximal order of class number 1.

Group representations and lattices

This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Shl, Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irreducibility hypotheses on a rational representation V of a finite group G, then the G-stable Euclidean lattices in V are severely restricted. Unfortunately, these hypotheses are rarely satisfied when V is absolutely irreducible over Q. But there are many examples where the ring EndG(V) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the Mordell-Weil lattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves. In ? 1 we discuss lattices and Hermitian forms on Y7, and in ??2-4 the strong irreducibility hypotheses we wish to make. In ?5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean Z[G]lattices L in V with EndG(L) a maximal order in EndG(V) . We give some examples with dim V < 8 in ?6, and in ??7-9 discuss the invariants of L, such as the dual lattice and theta function. The rest of the paper is devoted to examples: in most, G is a finite group of Lie type and V is obtained as an irreducible summand of the Weil representation of G. Some of the representation theoretic problems left open by this paper are: to find all examples of pairs (G, V) satisfying the strong irreducibility hypotheses of ??2-4, and to determine the invariants (shortest nonzero vector, theta function, Thompson series, ...) of the G-lattices L so effortlessly constructed inside V.

Traces of Eichler—Brandt matrices and type numbers of quaternion orders

LetA be a totally definite quaternion algebra over a totally real algebraic number fieldF andM be the ring of algebraic integers ofF. For anyM-orderG ofA we derive formulas for the massm(G) and the type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG and of invariants ofF andJ. Applications to class numbers of quaternion orders and of ternary quadratic forms are indicated.

On maximal orders of division quaternion algebras over the rational number field with certain optimal embeddings

In this paper, we shall give explicitZ-basis of certain maximal orders of definite quaternion algebras over the rational number fieldQ(See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. LetDbe a division quaternion algebra overQ.

Siegel's modular forms and the arithmetic of quadratic forms

Recently many authors utilized indefinite quadratic forms and Weil's representations to construct automorphic forms with Euler products on metaplectic groups or on orthogonal groups. For example, Shintani [27] employed indefinite quadratic forms of signature (2, 1) and associated Weil's representations to construct automorphic forms on the metaplectic group of genus 1, answering to a question raised by Shimura [26]. His results were modulated and generalized by Niwa [17], Gelbart [-8], Oda [18] and Rallis [20]. Howe [10] has given conjectures in most general cases. In this paper, we shall be concerned with definite quadratic forms of four variables and associated Weil's representations to construct Siegel's modular forms of genus 2. More precisely, we shall construct a Siegel modular form of genus 2, which is an eigenfunction of Hecke operators, from a pair of automorphic forms on a definite quaternion algebra over Q or from an automorphic form on a totally definite quaternion algebra over a real quadratic field. To avoid excessive overlaps with the text, we shall explain our result in the simplest case. Let D be a definite quaternion algebra over Q of discriminant d 2 and R be a maximal order of D. Let D~ denote the adelization of D • For a rational prime p, we put R p = R | p and let H denote the Hamilton quaternion algebra. We set K = I I R~ x H • Let q01 and ~02 be complex valued functions on p D~ which are left D • and right K-invariant. For every rational prime p.~d, we can define the action of Hecke operator T'(p) on qh (i= 1, 2) in the same manner as in Well [33]. We assume that T'(p)~0i=2~(p)qh, 2/(p)EC(i=I, 2) if n p,t/d. Let D~ = U D • y~K be a double coset decomposition of D~, where H is i= l the class number of D. We can assume that the reduced norm of Yi is 1 (1 N i < H). For l<i<H, we put ei=lD• For l < i < H and I<j<-H, we p define a lattice L~j in D by Lij=DnYi(1-lRv)yf 1. For x~D, we put N(x)=xx* p

On Projective Modules and Automorphisms of Central Separable Algebras

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).