Definite Quaternion Algebra Research Articles

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form f with coefficients in a maximal order of a definite quaternion algebra over Q, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f , as s tends to +∞. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad’s general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Liftings of reduction maps for quaternion algebras

We construct liftings of reduction maps from CM points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras.

Ribet bimodules and the specialization of Heegner points

For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X 0(D, N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P ∈ CM(R) specializes to a singular point (resp. a irreducible component) of the special fiber $\tilde X$ of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X 0(D, N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence Φ between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebra that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in $\tilde X$ . We also illustrate our results with an explicit example.

On the representation of integers by indefinite binary Hermitian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts

Given an elliptic cusp form f and an automorphic form f′ on a definite quaternion algebra over ℚ, there is a theta lifting from (f, f′) to an automorphic form ℒ(f, f′) on the quaternion unitary group GSp(1, 1) generating quaternionic discrete series at the Archimedean place. The aim of this paper is to provide an explicit formula for Fourier coefficients of ℒ(f, f′) in terms of periods of f and f′ with respect to a unitary character χ of an imaginary quadratic field. As an application, we show the existence of (f, f′) with ℒ(f, f′) ≢ 0.

Serre weights for quaternion algebras

AbstractWe study the possible weights of an irreducible two-dimensional mod p representation of ${\rm Gal}(\overline {F}/F)$ which is modular in the sense that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.

Quaternion algebras, Heegner points and the arithmetic of Hida families

Given a newform f, we extend Howard’s results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovář’s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures à la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.

Level Raising and Completed Cohomology

We describe an application of Poincaré duality for completed homology spaces (as defined by Emerton) to level raising for p-adic modular forms. This allows us to give a new description of the image of Chenevier’s p-adic Jacquet–Langlands map between an eigencurve for a definite quaternion algebra and an eigencurve for GL2. The points on the eigencurve at which we “raise the level” are (nonsmooth) points of intersection between an “old” and a “new” component.

Geometric level raising for p-adic automorphic forms

AbstractWe present a level-raising result for families of p-adic automorphic forms for a definite quaternion algebra D over ℚ. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor’s theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such a pairing on the infinite dimensional space of p-adic automorphic forms, so instead we introduce a space defined with respect to a dual coefficient system and work with a pairing between the usual forms and the dual space. A key ingredient is an analogue of Ihara’s lemma which shows an interesting asymmetry between the usual and the dual spaces.

On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups

Let H be a definite quaternion algebra over Q with discriminant DH and R a maximal order of H. We denote by Gn a quaternionic unitary group and put Γn=Gn(Q)∩GL2n(R). Let Sκ(Γn) be the space of cusp forms of weight κ with respect to Γn on the quaternion half-space of degree n. We construct a lifting from primitive forms in Sk(SL2(Z)) to Sk+2n−2(Γn) and a lifting from primitive forms in Sk(Γ0(d)) to Sk+2(Γ2), where d is a factor of DH. These liftings are generalizations of the Maass lifting investigated by Krieg.

Norm Euclidean Quaternionic Orders

Abstract.We determine the norm Euclidean orders in a positive definite quaternion algebra over

Small zeros of hermitian forms over a quaternion algebra

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 (19), and presents an extension of the classical theorem of Cassels (1) on small zeros of rational quadratic forms to the context of quaternion algebras.

Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E / F . The Birch and Swinnerton–Dyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F.

Newforms of squarefree level and theta series

We show that the cuspidal part of the span of the theta series associated to maximal integral lattices on a definite quaternion algebra \({\mathfrak {A}}\) over \({\mathbb {Q}}\) is precisely the space of newforms. Furthermore, using the Eichler commutation relation and an idea coming out of the Yoshida lifting, we show how to express each newform as an explicit linear combination of such theta series. The coefficients of these linear combinations come from cuspidal eigenvectors of Brandt matrices.

On Level-Raising Congruences

In this paper we rewrite a work of Sorensen to include nontrivial types at the infinite places. This work extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D×, where D is a definite quaternion algebra over a totally real field F . We do this for any automorphic representations π of an arbitrary reductive group G over F which is compact at infinity. We do not assume π∞ is trivial. If λ is a finite place of Q, and w is a place where πw is unramified and πw ≡ 1 (mod λ), then under some mild additional assumptions (we relax requirements on the relation between w and ` which appear in previous works) we prove the existence of a π ≡ π (mod λ) such that πw has more parahoric fixed vectors than πw. In the case where Gw has semisimple rank one, we sharpen results of Clozel, Bellaiche and Graftieaux according to which πw is Steinberg. To provide applications of the main theorem we consider two examples over F of rank greater than one. In the first example we take G to be a unitary group in three variables and a split place w. In the second we take G to be an inner form of GSp(2). In both cases, we obtain precise satisfiable conditions on a split prime w guaranteeing the existence of a π ≡ π (mod λ) such that the component πw is generic and Iwahori spherical. For symplectic G, to conclude that πw is generic, we use computations of R. Schmidt. In particular, if π is of Saito-Kurokawa type, it is congruent to a π which is not of Saito-Kurokawa type.

Shimura correspondence for level [formula omitted] and the central values of L-series

Given a weight 2 and level p 2 modular form f, we construct two weight 3/2 modular forms (possibly zero) of level 4 p 2 and non-trivial character mapping to f via the Shimura correspondence. Then we relate the coefficients of the constructed forms to the central value of the L-series of certain imaginary quadratic twists of f. Furthermore, we give a general framework for our construction that applies to any order in definite quaternion algebras, with which one can, in principle, construct weight 3/2 modular forms of any level, provided one knows how to compute ideal classes representatives.

Module supersingulier et homologie des courbes modulaires

We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic. We compare this Z -module to the homology of the modular curve X 0 ( p ) . We give an interpretation of Gross formula for special values of L-functions of modular forms in this context. As an application, we obtain a formula for the sum of the squares of the optimal embeddings numbers of quadratic orders in a definite quaternion algebra.

Cancellation in totally definite quaternion algebras

In [M.-F. Vignéras, Simplification pour les ordres des corps de quaternions totalement définis, J. reine angew. Math. 286/287 (1976), 257–277.], M.-F. Vignéras proves that only a finite number of Eichler orders of a totally definite quaternion algebra over a number field can have the cancellation property. In this article, we draw up the complete list of the genus of these orders.

Heights and Determinants over Quaternion Algebras#

In Liebendörfer (2004) we defined a height function for matrices over a positive definite rational quaternion algebra. In this article, we prove that this height can be expressed, like the well-known height over number fields, as a product of local factors involving the maximal minors of the matrix. Part of our proof uses a quaternionic determinant invented by Moore (1922) and yields a non-commutative analogue of the Cauchy–Binet formula. The new formula of the height enables us to improve a result of Liebendörfer (2004).

Duality of Heights Over Quaternion Algebras

Given a number field K, it is well-known that the height of a subspace in KN and of its orthogonal complement coincide. We prove the analogous fact when K is replaced by a positive definite rational quaternion algebra with respect to the heights recently introduced by the first author. Since quaternion algebras are non-commutative, we cannot just follow the classical proof but have to work with localizations and certain finite rings.