Indefinite Quaternion Algebra Research Articles

Small diameters and generators for arithmetic lattices in textrm{SL}_2(mathbb {R}) and certain Ramanujan graphs

We show that arithmetic lattices in textrm{SL}_{2}(mathbb {R}), stemming from the proper units of an Eichler order in an indefinite quaternion algebra over mathbb {Q}, admit a ‘small’ covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.

Square integrals of the logarithmic derivatives of Selberg’s zeta functions in the critical strip

In our previous work (Y. Hashimoto, Selberg’s zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268 ), we proposed an upper bound of the logarithmic derivative of Selberg’s zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.

Power series expansions of modular forms and p-adic interpolation of the square roots of Rankin–Selberg special values

Let [Formula: see text] be a newform of even weight [Formula: see text] for [Formula: see text], where [Formula: see text] is a possibly split indefinite quaternion algebra over [Formula: see text]. Let [Formula: see text] be a quadratic imaginary field splitting [Formula: see text] and [Formula: see text] an odd prime split in [Formula: see text]. We extend our theory of [Formula: see text]-adic measures attached to the power series expansions of [Formula: see text] around the Galois orbit of the CM point corresponding to an embedding [Formula: see text] to forms with any nebentypus and to [Formula: see text] dividing the level of [Formula: see text]. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic [Formula: see text]-level structure. Also, we restrict these [Formula: see text]-adic measures to [Formula: see text] and compute the corresponding Euler factor in the formula for the [Formula: see text]-adic interpolation of the “square roots” of the Rankin–Selberg special values [Formula: see text], where [Formula: see text] is the base change to [Formula: see text] of the automorphic representation of [Formula: see text] associated, up to Jacquet-Langland correspondence, to [Formula: see text] and [Formula: see text] is a compatible family of grössencharacters of [Formula: see text] with infinite type [Formula: see text].

Wiles defect for Hecke algebras that are not complete intersections

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$. If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

A quaternionic construction of p-adic singular moduli

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 construction of cohomology classes inspired by that of Darmon–Vonk, 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. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture.

Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.

Galois extensions and a conjecture of Ogg

Let N = p q N=pq be a product of two distinct primes. There is an isogeny J 0 ( N ) n e w → J N J_0(N)^\mathrm {new}\to J^N defined over Q \mathbf {Q} between the new quotient J 0 ( N ) J_0(N) and the Jacobian of the Shimura curve attached to indefinite quaternion algebra of discriminant N N . In the case when p = 2 , 3 , 5 , 7 , 13 p=2,3,5,7,13 , Ogg made predictions about the kernels of these isogenies. We show that Ogg’s conjecture is false in general. Afterwards, we propose a strategy for proving results toward Ogg’s conjecture in certain situations. Finally, we discuss this strategy in detail for N = 5 ⋅ 13 N=5\cdot 13 .

Virtual Abelian varieties of $\mathrm{GL}_2$-type

This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of actions of Atkin-Lehner operators and the class group of $K$. Thus we deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties. These new families of moduli spaces are further analysed when they are of dimension $2$. We provide explicit numerical bounds for when they are surfaces of general type. In addition, for two particular examples, we show that they are both rational surfaces by computing the coordinates of inequivalent elliptic points and studying the intersections of Hirzebruch cycles with exceptional divisors.

CM-points and lattice counting on arithmetic compact Riemann surfaces

Let X(D,1)=Γ(D,1)\\H denote the Shimura curve of level N=1 arising from an indefinite quaternion algebra of fixed discriminant D. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant d<0 on X(D,1) as d→−∞. We prove that if |d| is sufficiently large compared to the radius r≈log⁡X of the circle, we can improve on the classical O(X2/3)-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.

On optimal embeddings and trees

We apply the theory of Bruhat–Tits trees to the study of optimal embeddings from orders of rank two and three to quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in an indefinite quaternion algebra yield optimal representations of such orders. This completes previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as suborders. The same technique is used in the second part of this work to compute local embedding numbers, which extends previous results by J. Brzezinski.

On Ribet’s isogeny for $J_0(65)$

Let $J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $65$. We study the isogenies $J_0(65)\rightarrow J^{65}$ defined over $\mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7$, as is predicted by a conjecture of Ogg.

Special values of hypergeometric functions and periods of CM elliptic curves

Let X 0 6 ( 1 ) / W 6 X_0^6(1)/W_6 be the Atkin–Lehner quotient of the Shimura curve X 0 6 ( 1 ) X_0^6(1) associated to a maximal order in an indefinite quaternion algebra of discriminant 6 6 over Q \mathbb {Q} . By realizing modular forms on X 0 6 ( 1 ) / W 6 X_0^6(1)/W_6 in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer’s formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over Q ¯ \overline {\mathbb {Q}} with complex multiplication.

Surface abéliennes à multiplication quaternionique munie d'une structure de niveau Γ0(N)

A theorem of Mazur gives the set of possible prime degrees for rational isogenies between elliptic curves. In this paper, we are working on a similar problem in the case of abelian surfaces of [Formula: see text]-type over [Formula: see text] with quaternionic multiplication (over [Formula: see text]) endowed with a [Formula: see text] level structure. We prove the following result: for a fixed indefinite quaternion algebra [Formula: see text] of discriminant [Formula: see text] and a fixed quadratic imaginary field [Formula: see text], there exists an effective bound [Formula: see text] such that for a prime number [Formula: see text], not dividing the conductor of the order [Formula: see text], there do not exist abelian surfaces [Formula: see text] such that [Formula: see text] is a maximal order of [Formula: see text] and [Formula: see text] is endowed with a [Formula: see text] level structure.

Dimension formulas of paramodular forms of squarefree level and comparison with inner twist

In this paper, we give an explicit dimension formula for the spaces of Siegel paramodular cusp forms of degree two of squarefree level. As an application, we propose a conjecture on symplectic group version of Eichler–Jacquet–Langlands type correspondence. It is a generalization of the previous conjecture of the first named author for prime levels published in 1985, where inner twists corresponding to binary quaternion hermitian forms over definite quaternion algebras were treated. Our present study contains also the case of indefinite quaternion algebras. Additionally, we give numerical examples of $L$ functions which support the conjecture. These comparisons of dimensions and examples give also evidence for conjecture on a certain precise lifting theory. This is related to the lifting theory from pairs of elliptic cusp forms initiated by Y. Ihara in 1964 in the case of compact twist, but no such construction is known in the case of non-split symplectic groups corresponding to quaternion hermitian groups over indefinite quaternion algebras and this is new in that sense.

Rational torsion points on Jacobians of Shimura curves

Let p and q be distinct primes. Consider the Shimura curve X associated to the indefinite quaternion algebra of discriminant pq over Q. Let J be the Jacobian variety of X , which is an abelian variety over Q. For an odd prime `, we provide sufficient conditions for the non-existence of rational points of order ` on J. As an application, we find some non-trivial subgroups of the kernel of an isogeny from the new quotient of J0(pq) to J.

Local Points on Shimura Coverings of Shimura Curves at Bad Reduction Primes

Let XD be the Shimura curve associated with an indefinite rational quaternion algebra of reduced discriminant D>1. For each prime l|D, there is a natural cyclic Galois covering of Shimura curves XD,l → XD constructed by adding certain level structure at l. The main goal of this note is to study the existence of local points at primes p≠l of bad reduction on the intermediate curves of these coverings and their Atkin–Lehner quotients.

Roots of unity in definite quaternion orders

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.

QUATERNIONIC MODULAR FORMS OF ANY WEIGHT

In this work we give a geometric definition, as sections of line bundles, of p-adic analytic families of overconvergent modular forms attached to an indefinite quaternion algebra over ℚ. As a consequence of this, we obtain the existence of an eigencurve in this context. Our theory includes the interpretation of a modular form as a rule on test objects. We introduce the Hecke operators U and T l, both in families and for a single weight. We show that the U -operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of eigenvalues associated to overconvergent eigenforms of finite slope with respect to the U -operator.

An algorithm for the principal ideal problem in indefinite quaternion algebras

AbstractDeciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.

Serre weights and Shimura curves

We compare the cohomology of Shimura curves constructed using quaternion algebras over a totally real field with the cohomology of related unitary Shimura curves. This allows us to pass information about generalized Serre weight conjectures from unitary groups to indefinite quaternion algebras.