Shimura Varieties Of Type Research Articles

Differential operators mod p : analytic continuation and consequences

This paper concerns certain $\mod p$ differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the $\mod p$ reduction of the $p$-adic theta operators previously studied by some of the authors. In the characteristic $0$, $p$-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole Shimura variety. On the other hand, our $\mod p$ operators extend ("analytically continue", in the language of de Shalit and Goren) to the whole Shimura variety. As a consequence, motivated by their use by Edixhoven and Jochnowitz in the case of modular forms for proving the weight part of Serre's conjecture, we discuss some effects of these operators on Galois representations. Our focus and techniques differ from those in the literature. Our intrinsic, coordinate-free approach removes difficulties that arise from working with $q$-expansions and works in settings where earlier techniques, which rely on explicit calculations, are not applicable. In contrast with previous constructions and analytic continuation results, these techniques work for any totally real base field, any weight, and all signatures and ranks of groups at once, recovering prior results on analytic continuation as special cases.

Complex multiplication cycles and Kudla-Rapoport divisors, II

This paper is about the arithmetic of {\it Kudla-Rapoport divisors} on Shimura varieties of type ${\rm GU}(n-1,1)$. In the first part of the paper we construct a toroidal compactification of N.~Kr\"amer's integral model of the Shimura variety. This extends work of K.-W.~Lan, who constructed a compactification at unramified primes. In the second, and main, part of the paper we use ideas of Kudla to construct Green functions for the Kudla-Rapoport divisors on the open Shimura variety, and analyze the behavior of these functions near the boundary of the compactification. The Green functions turn out to have logarithmic singularities along certain components of the boundary, up to log-log error terms. Thus, by adding a prescribed linear combination of boundary components to a Kudla-Rapoport divisor one obtains a class in the arithmetic Chow group of Burgos-Kramer-K\"uhn. In the third and final part of the paper we compute the arithmetic intersection of each of these divisors with a cycle of complex multiplication points. The computation is quickly reduced to the calculations of the author's earlier work {\it Complex multiplication cycles and Kudla-Rapoport divisors}. The arithmetic intersection multiplicities are shown to appear as Fourier coefficients of the diagonal restriction of the central derivative of a Hilbert modular Eisenstein series.

Topological automorphic forms

The authors apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type $U(1,n-1)$. These cohomology theories of topological automorphic forms ($\mathit{TAF}$) are related to Shimura varieties in the same way that $\mathit{TMF}$ is related to the moduli space of elliptic curves. Table of Contents: $p$-divisible groups; The Honda-Tate classification; Tate modules and level structures; Polarizations; Forms and involutions; Shimura varieties of type $U(1,n-1)$; Deformation theory; Topological automorphic forms; Relationship to automorphic forms; Smooth $G$-spectra; Operation on $\mathit{TAF}$; Buildings; Hypercohomology of adele groups; $K(n)$-local theory; Example: chromatic level $1$; Bibliography; Index. (MEMO/204/958)

Spectre et homologie des variétés hyperboliques complexes de congruence

In the first part of this Note, we show that the first non-zero eigenvalue of the Laplace operator on 1-forms of a standard congruence arithmetic complex hyperbolic n-manifold is always ⩾ 10n−11 25 . The following parts of this Note concern homological applications of this result. We prove, in particular, that if Sh 0 H⊂Sh 0 G are two Shimura varieties of type U( n−1,1) and U( n,1), the natural map H 2 n−3 (Sh 0 H)→ H 2 n−3 (Sh 0 G) is injective, first step of a “Lefschetz theorem” for Shimura varieties. To cite this article: N. Bergeron, L. Clozel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 995–998.