Indefinite Quaternion Algebra Research Articles

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.

Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms

AbstractLetπ(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebraBover a totally real fieldFwith associated Galois representationρf. LetKbe a totally complex quadratic extension ofFembedding inB. Using families of CM points on towers of Shimura curves attached toBandK, we construct an Euler system forρf. We prove that it extends top-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.

$\mathcal L$-invariants and Darmon cycles attached to modular forms

Let f be a modular eigenform of even weight k\geq 2 and new at a prime p dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module \mathbf{D}^{FM}_f and an \mathcal{L} -invariant \mathcal{L}^{FM}_f . The first goal of this paper is building a suitable p -adic integration theory that allows us to construct a new monodromy module \mathbf{D}_f and {\mathcal{L}} -invariant {\mathcal{L}}_f , in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two {\mathcal{L}} -invariants are equal.Let K be a real quadratic field and assume the sign of the functional equation of the L -series of f over K is -1 . The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Selmer group of the motive attached to f over the tower of narrow ring class fields of K . Generalizing work of Darmon for k=2 , we give a construction of local cohomology classes which we expect to arise from global classes and satisfy an explicit reciprocity law, accounting for the above prediction.

On the pullback of an arithmetic theta function

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series \({\widehat{\phi}_{\mathcal C}(\tau)}\) defined using arithmetic 0-cycles on the moduli space \({\mathcal C}\) of elliptic curves with CM by the ring of integers \({O_{\kappa}}\) of an imaginary quadratic field. The second such series \({\widehat{\phi}_{\mathcal M}(\tau)}\) has weight 3/2 and takes values in the arithmetic Chow group \({\widehat{{\rm CH}}^1(\mathcal{M})}\) of the arithmetic surface associated to an indefinite quaternion algebra \({B/\mathbb{Q}}\). For an embedding \({O_\kappa \rightarrow O_B}\), a maximal order in B, and a two sided O B -ideal Λ, there is a morphism \({j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}\) and a pullback \({j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}\). Our main result is an expression for the pullback \({j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}\) as a linear combination of products of \({\widehat{\phi}_{\mathcal C}(\tau)}\)’s and classical weight \({\frac{1}{2}}\) theta series.

Completed cohomology of Shimura curves and a p-adic Jacquet–Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet–Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur’s ‘level lowering’ principle.

POWER SERIES EXPANSIONS OF MODULAR FORMS AND THEIR INTERPOLATION PROPERTIES

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.

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.

Splitting of Abelian Varieties

We study a new local-global problem in the context of Abelian varieties: Given an absolutely simple Abelian variety over a number field K, find a necessary and sufficient condition for the existence of a place v of K (or infinitely many places, or a set of places of positive density) such that A remains absolutely simple modulo v. It is well known that for absolutely simple Abelian surfaces with multiplication by an indefinite quaternion algebra, A modulo v, denoted by A v , is always isogenous to the square of some elliptic curve. For absolutely simple Abelian varieties of complex multiplication type, we show that A v stays simple for a set of places of density 1. On the other hand, for absolutely simple Abelian varieties associated by Shimura to cusp forms of weight 2, if A v splits at a set of places of positive density, then the absolute endomorphism algebra is noncommutative. Based on these results, we then formulate a conjecture: An absolutely simple Abelian variety defined over a number field splits at a set of places of positive density if and only if its absolute endomorphism algebra is noncommutative.

On the Fourier coefficients of modular forms of half-integral weight

We prove a formula relating the Fourier coefficients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indefinite quaternion algebra over ℚ. This formula has applications to proving the nonvanishing of this lift and to an explicit version of the Rallis inner product formula.

Some explicit constructions of integral structures in quaternion algebras

Let B be an indefinite quaternion algebra over Q . Following the explicit characterization of the Eichler orders in B given by Hashimoto, we define explicit embeddings of these orders in the 2 × 2 -ring of matrices over a p -adic ring. For a natural number N and a prime q not dividing the discriminant of B , we describe the two natural inclusions of an Eichler order of level Nq in an Eichler order of level N . Moreover, we provide bases for a chain of Eichler orders in B and prove results about the rank of their intersection.

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

We prove that the theta correspondence for the dual pair $({\widetilde{\textup{SL}_2}}, \textit{PB}^\times)$ , for B an indefinite quaternion algebra over ℚ, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies between new quotients of Jacobians of Shimura curves, completely independent of Faltings’ isogeny theorem.

On maps between modular Jacobians and Jacobians of Shimura curves

Fix a squarefree integer N, divisible by an even number of primes, and let Γ′ be a congruence subgroup of level M, where M is prime to N. For each D dividing N and divisible by an even number of primes, the Shimura curve X D (Γ0(N/D) ∩ Γ′) associated to the indefinite quaternion algebra of discriminant D and Γ0(N/D) ∩ Γ′-level structure is well defined, and we can consider its Jacobian J D (Γ0(N/D) ∩ Γ′). Let J D denote the N/D-new subvariety of this Jacobian. By the Jacquet-Langlands correspondence [J-L] and Faltings’ isogeny theorem [Fa], there are Hecke-equivariant isogenies among the various varieties J D defined above. However, since the isomorphism of Jacquet-Langlands is noncanonical, this perspective gives no information about the isogenies so obtained beyond their existence. In this paper, we study maps between the varieties J D in terms of the maps they induce on the character groups of the tori corresponding to the mod p reductions of these varieties for p dividing N. Our characterization of such maps in these terms allows us to classify the possible kernels of maps from J D to J D′, for D dividing D′, up to support on a small finite set of maximal ideals of the Hecke algebra. This allows us to compute the Tate modules $$T_\mathfrak{m} $$ J D of J D at all non-Eisenstein $$\mathfrak{m}$$ of residue characteristic l > 3. These computations have implications for the multiplicities of irreducible Galois representations in the torsion of Jacobians of Shimura curves; one such consequence is a “multiplicity one” result for Jacobians of Shimura curves.

Non-elliptic Shimura curves of genus one

We present explicit models for non-elliptic genus one Shimura curves X0(D, N) with Γ0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan’s work [10, Ch. III] on points with complex multiplication on Shimura curves.

The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II

We prove a general subconvex bound in the level aspect for Rankin–Selberg L-functions associated with two primitive holomorphic or Maass cusp forms over Q. We use this bound to establish the equidistribution of incomplete Galois orbits of Heegner points on Shimura curves associated with indefinite quaternion algebras over Q.

On QM-abelian surfaces with model of [formula omitted]-type over [formula omitted

The purpose of this paper is to characterize QM-abelian surfaces which has a model of GL 2 -type over Q . The “special” involutions on a corresponding indefinite quaternion algebra (which is defined in Section 2) play an essential role in this paper.

Quaternions, polarizations and class numbers

We study abelian varieties $A$ with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number $\pi_0(A)$ of isomorphism classes of principal polarizations on $A$ in terms of relative class numbers of CM fields by means of Eichler's theory of optimal embeddings. As a consequence, we exhibit simple abelian varieties of any even dimension admitting arbitrarily many non-isomorphic principal polarizations. On the other hand, we prove that $\pi_0(A)$ is uniformly bounded for simple abelian varieties of odd square-free dimension.

Special points on products of two Shimura curves

Let S-1 and S-2 be two Shimura curves over Q attached to rational indefinite quaternion algebras B-1Q and B-2Q with maximal orders B-1 and B-2 respectively. We consider an irreducible closed algebraic curve C in the product (S-1 x S-2)(C) such that C(C) boolean AND (S-1 x S-2)(C) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields. that C is of Hedge type.

Some remarks on a spectral correspondence for maass waveforms

We prove a detailed result in classical language on a special case of the Jacquet-Langlands correspondence: Let O1 be the (cocompact) Fuchsian group consisting of the units of norm one in a maximal order O in an indefinite quaternion algebra over ℚ. Let d=d(O) be the discriminant of O⁠. Our result involves a certain integral transform Θ constructed by Dennis Hejhal (1985). We prove that when chosen appropriately, the map Θ gives a linear bijection between each discrete eigenspace of the Laplacian on O1\ℋ and the space of newforms of the same eigenvalue on the congruence group Γ0(d). The proof uses trace formulas for the Hecke operators on O1 and Γ0(d)

On the Sato-Tate conjecture for QM-curves of genus two

An abelian surface A A is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve C C of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms.

Parametrizations of elliptic curves by Shimura curves and by classical modular curves.

Fix an isogeny class of semistable elliptic curves over Q. The elements A of have a common conductor N, which is a square-free positive integer. Let D be a divisor of N which is the product of an even number of primes--i.e., the discriminant of an indefinite quaternion algebra over Q. To D we associate a certain Shimura curve X(0)D(N/D), whose Jacobian is isogenous to an abelian subvariety of J0(N). There is a unique A [symbol; see text] A in for which one has a nonconstant map piD : X(0)D(N/D) --> A whose pullback A --> Pic0(X(0)D(N/D)) is injective. The degree of piD is an integer deltaD which depends only on D (and the fixed isogeny class A). We investigate the behavior of deltaD as D varies.