Shimura Surfaces Research Articles

Minimal surfaces of general type with $p_g = q = 0$ arising from Shimura surfaces

Minimal surfaces of general type with $p_g = q = 0$ arising from Shimura surfaces

Parametrizing Shimura subvarieties of $${\mathrm{A}_1}$$ Shimura varieties and related geometric problems

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, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.

Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces

Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound. Previously, this result has been shown in [BHK+13] for compact Hilbert modular surfaces using the Bogomolov-Miyaoka-Yau inequality. Our approach uses equidistribution and works uniformly for all Shimura surfaces.

The meromorphic continuation of the zeta function of a product of Hilbert and Picard modular surfaces over CM-fields

In this paper we prove in particular the meromorphic continuation to the entire complex plane of the zeta function of a product of a Hilbert modular surface and a Picard modular surface regarded over arbitrary CM-fields (see the end of the Introduction for more results regarding the meromorphic continuation of the zeta functions associated to products of Shimura curves, quaternionic Shimura surfaces and Picard modular surfaces). In order to obtain the meromorphic continuation we show the simultaneous potential modularity for an arbitrary finite number of l -adic representations associated to Hilbert modular forms and Picard modular surfaces.

Intersection theory on Shimura surfaces II

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over ℚ associated to a rational quaternion algebra into the Shimura surface associated to the base change of the quaternion algebra to a real quadratic field. After extending the associated moduli problems over ℤ we obtain an arithmetic threefold with a embedded arithmetic surface, which we view as a cycle of codimension one. We then construct a family, indexed by totally positive algebraic integers in the real quadratic field, of codimension two cycles (complex multiplication points) on the arithmetic threefold. The intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the Fourier coefficients of a (very particular) Hilbert modular form of weight 3/2. The results are higher dimensional variants of results of Kudla-Rapoport-Yang, which relate intersection multiplicities of special cycles on the integral model of a Shimura curve to Fourier coefficients of a modular form in two variables.

Intersection theory on Shimura surfaces

AbstractKudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

Plurigenera of general type surfaces in mixed characteristic

AbstractWe construct general type surfaces in mixed characteristic whose geometric genera can be made to jump by an arbitrarily prescribed positive amount under specialization. We then show that this phenomenon of jumping geometric genus presents itself in some compact Shimura surfaces. Finally, we find a set of conditions, met by the latter Shimura surfaces, that forces the higher plurigenera to remain constant in reduction modulo p.

Tate classes and L-functions on a product of a quaternionic Shimura surface and a Picard modular surface

We compute the space of Tate classes on a product of a quaternionic Shimura surface and a Picard modular surface in terms of automorphic representations including the exact determination of their field of definition and prove the equality between the dimension of the space of Tate classes and the order of the pole at s = 3 of the L-function in some special cases.

Arithmetic structure of CMSZ fake projective planes

We show that the fake projective planes that are constructed from dyadic discrete subgroups discovered by Cartwright, Mantero, Steger, and Zappa are realized as connected components of certain unitary Shimura surfaces. As a corollary we show that these fake projective planes have models defined over the number field Q ( −3 , 5 ) .

Tate classes and poles of L-functions of twisted quaternionic Shimura surfaces

In this article we generalize a result obtained by Harder, Langlands and Rapoport in the case of Hilbert modular surfaces and we prove in particular the equality between the dimension of the space of Tate classes of twisted quaternionic Shimura surfaces defined over arbitrary solvable extensions of totally real fields and the order of the pole at s = 2 of the zeta functions of these surfaces.

Automorphic Forms with Anisotropic Periods on a Unitary Group

Let G be an algebraic group over a global field F with ring of adeles, denote by Z the center of G and by C an algebraic subgroup of G over F such that the cycle CF‘\C ‘ has finite volume. Fix unitary characters ωx Z ‘/ZF‘ → 1 = unit circle in ב and ξx CF‘\C ‘ → 1, and denote by φx GF‘\G ‘ → a cusp form in the cuspidal representation π of G ‘, whose central character is ωπ = ω. By a cuspidal representation we mean an irreducible one. We say that π is C ‘-cyclic if it has a nonzero C ‘-period PC ‘φ‘ = ∫ CF‘\C ‘φc‘ξc‘dc. The overbar indicates complex conjugation. Studies of cyclic automorphic forms have applications to special values of L-functions (Waldspurger [W1, W2], Jacquet [J1, J2]), lifting problems [F], and studies of cohomology of symmetric spaces, in particular the Tate conjecture on algebraic cycles on some Shimura surfaces [FH]. The purpose of this paper—inspired by the applications to the Tate conjecture of [FH]—is to compare the notion of cyclicity by C ‘, with cyclicity by an inner form of C ‘. We let G be the quasi-split unitary group U2; 1‘ = U2; 1yE/F‘ in three variables defined by means of a quadratic separable extension E/F of global fields. The subgroup C is taken to be the