L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × ℂ, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke’s inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

A compactification of Igusa varieties

We investigate the notion of Igusa level structure for a one-dimensional Barsotti–Tate group over a scheme X of positive characteristic and compare it to Drinfeld’s notion of level structure. In particular, we show how the geometry of the Igusa covers of X is useful for studying the geometry of its Drinfeld covers (e.g. connected and smooth components, singularities). Our results apply in particular to the study of the Shimura varieties considered in Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001). In this context, they are higher dimensional analogues of the classical work of Igusa for modular curves and of the work of Carayol for Shimura curves. In the case when the Barsotti–Tate group has constant p-rank, this approach was carried-out by Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001).     

On the characterization of complex Shimura varieties

In this paper we recall basic properties of complex Shimura varieties and show that they actually characterize them. This characterization immediately implies the explicit form of Kazhdan's theorem on the conjugation of Shimura varieties. It also implies the existence of unique equivariant models over the reflex field of Shimura varieties corresponding to adjoint groups and the existence of a p-adic uniformization of certain unitary Shimura varieties. In the appendix we give a modern formulation and a proof of Weil's descent theorem.     

Modular Shimura varieties and forgetful maps

In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.

     

Integral models of certain PEL Shimura varieties with {\Gamma_1(p)}-type level structure

We study p-adic integral models of certain PEL Shimura varieties with level subgroup at p related to the \({\Gamma_1(p)}\)-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of \({\mathbb{Q}_p}\) and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are étale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort–Tate generator of a group scheme. We use these local models to find a resolution of the integral model in the case of the Siegel modular variety of genus 2. The resolution is regular with special fiber a nonreduced divisor with normal crossings.     

Moduli problem and points on some twisted Shimura varieties of PEL type

In this article we describe the moduli problem of a “twist” of some simple Shimura varieties of PEL type that appear in Kottwitz's papers [R. Kottwitz, Shimura varieties and λ-adic representations, in: Automorphic Forms, Shimura Varieties and L-Functions, part 1, in: Perspect. Math., vol. 10, Academic Press, San Diego, CA, 1990, pp. 161-209; R. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (2) (1992) 373-444] and [R. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992) 653-665] and then, using the moduli problem, we compute the cardinality of the set of points over finite fields of the twisted Shimura varieties. Using this result, we compute the zeta function of the twisted varieties. The twist of the Shimura varieties is done by a mod q representation of the absolute Galois group of the reflex field of the Shimura varieties.     

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a “twisted case”; a consequence of our results is that our construction also includes the flag varieties for Kac–Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.     

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h-hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.     

Arithmetic properties of shimura sums related to several modular forms

The paper is concerned with Shimura sums related to modular forms with multiplicative coefficients which are products of Dedekind η-functions of various arguments. Several identities involving Shimura sums are established. The type of identity obtained depends on the splitting of primes in certain imaginary quadratic number fields.     

L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × C, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke's inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

Crystalline cohomology and GL(2, ℚ)

This paper applies recent advances in crystalline cohomology to the classical case of open elliptic modular curves. In so doing control is gained over the action of inertia in the Galois representations attached to modular forms. Our aim is to study the modular Galois representations attached to automorphic forms modp of weightk≥2. We generalize to higher weightk several results which were previously accessible only in the case of weight 2 where jacobian varieties can be invoked. Additionally we reconsider Gross’s theorem on companion forms in a crystalline context. Partially supported by NSF grant DMS 90-02744. Partially supported by NSA grant MDA904-90-H-4020 and by a PSC-CUNY grant.     

Local models of Shimura varieties and a conjecture of Kottwitz

We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.     

Descent on certain Shimura curves

We give an explicit procedure for constructing Shimura curves analogous to the modular curvesX ₀(N) that are counterexamples to the Hasse principle over imaginary quadratic fields. These counterexamples are accounted for by the Manin obstruction.     

Faltings heights of CM cycles and derivatives of <Emphasis Type="Italic">L</Emphasis>-functions

We study the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.     

The planetary <Emphasis Type="Italic">N</Emphasis>-body problem: symplectic foliation,reductions and invariant tori

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.     

Integral models in unramified mixed characteristic (0, 2) of hermitian orthogonal Shimura varieties of PEL type,Part II

We construct relative PEL type embeddings in mixed characteristic (0, 2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0, 2) of hermitian orthogonal Shimura varieties of PEL type.     

On the slope stratification of certain Shimura varieties

In this paper we study the slope stratification on the good reduction of the type C family Shimura varieties. We show that there is an open dense subset U of the moduli space such that any point in U can be deformed to a point with a given lower admissible Newton polygon. For the Siegel moduli spaces, this is obtained by F. Oort which plays an important role in his proof of the strong Grothendieck conjecture concerning the slope stratification. We also investigate the p-divisible groups and their isogeny classes arising from the abelian varieties in question. Received: 10 November 2004; 13 February 2005 The research is partially supported by NSC 93-2119-M-001-018.     

Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.     

Integral models of Shimura varieties with parahoric level structure

Publications mathématiques de l'IHÉS - For a prime $p &gt; 2$ , we construct integral models over $p$ for Shimura varieties with parahoric level structure, attached to Shimura...