Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper we prove a variety of number theoretic results about Brauer equivalent number fields (e.g., they must have the same signature). These results are then applied to the geometry of certain arithmetic locally symmetric spaces. As an example, we construct incommensurable arithmetic locally symmetric spaces containing exactly the same set of proper immersed totally geodesic surfaces.
We study "quadratic periods" on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products of automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is that this integral refinement is implied by another conjecture (Conjecture D below) regarding integrality of theta lifts between certain quaternionic unitary groups.
We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara's Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond-Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara's Lemma.
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.
In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.
Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(\mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).
In this article, we use deformation theory of Galois representations valued in the symplectic group of degree four to prove a freeness result for the cohomology of certain quaternionic unitary Shimura variety over the universal deformation ring for certain type of residual representation satisfying a property called rigidity. This result plays an important role in the proof of the arithmetic level raising theorem for the symplectic similitude group of degree four over the field of rational numbers by the author.
In this note we study the special fiber of the Rapoport-Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat-Tits stratification and is intimately related to the Bruhat-Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.
Let D be a quaternion division algebra over a totally real number field F which splits exactly at one infinite place. We assume that there is a p-adic place where D doesn't split. Then the associated Shimura curve has a Cherednik unifomization by the p-adic upper half plane. We define an integral model of the Shimura curve an over the integers O of this p-adic place. We prove an uniformization theorem for the completion of the model over the formal spectrum Spf O.
Let F be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables p-adic families of finite slope quaternionic automorphic forms over F. It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves. Secondly, we attach p-adic L-functions to triples of ordinary p-adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu-Zhang-Zhang to our fa…
We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree $2$ over a ramified odd prime $p$ if the level at $p$ is given by a special maximal compact open subgroup. More precisely, we show that such a locus has purely $2$-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport--Zink space for the q…
Locally analytic vectors in the completed cohomology of quaternionic Shimura curves
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
We study the Hodge-Tate period domain associated to a quaternionic Shimura curve at a prime of bad reduction, and give an explicit description of its Ekedahl-Oort stratification.
This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1$, examples of manifolds/Shimura varieties with precisely $n$ commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic $3$-manifolds and quaternionic Shimura surfaces, wher…
We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to obtain (conjectural) equations for the canonical embedding of Shimura curves.
Using the Rost invariant for torsors under Spin groups one may define an analogue of the Arason invariant for certain hermitian forms and orthogonal involutions. We calculate this invariant explicitly in various cases, and use it to associate to every orthogonal involution with trivial discriminant and trivial Clifford invariant over a central simple algebra of even co-index a cohomology class $f_3$ of degree 3 with $μ_2$ coefficients. This invariant $f_3$ is the double of any representative of the Arason invariant; it vanishes when the algebra has degree at most 10, and also when there is a q…
We determine the Galois representations inside the $l$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.
In this paper we generalize Cherednik's method and prove that certain Shimura varieties corresponding to groups of unitary similitudes and automorphic vector bundles over them have p-adic uniformization. This is proved for Shimura varieties, uniformized by the complex unit ball, when the central simple algebra over a CM-field defining the group of unitary similitudes has Brauer invariant 1/d at p. In this case, Shimura varieties can be uniformized by Drinfeld's covers of p-adic upper half-spaces.
We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties.
Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Z…
Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and CM points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin-Selberg convolution associated with a cusp form of half-integral weight $n/2 +1 $, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shi…
Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus Q_O of quaternionic multiplication by O in the moduli space A_g of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe Q_O as a union of Atkin-Lehner quotients of Shimura varieties and we compute the number of irreducible components of Q_O in terms of class numbers of CM-fields.
We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field $K$ and a CM number field $E$, returns a finite lists of groups which contains $\mathrm{Br}(\overline{X})^{G_K}$ f…
We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic twisted Hilbert modular surface. This quantity is then related to the arithmetic volume of a Shimura curve, via the arithmetic Grothendieck-Riemann-Roch theorem and the Jacquet-Langlands correspondence.
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…
