An alternative description of the Drinfeld p-adic half-plane

We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a non-archimedean local field F represents a moduli problem of polarized O_F-modules with an action of the ring of integers O_E in a quadratic extension E of F. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL_2(F) and SU(C)(F) for a two-dimensional split hermitian space C for E/F.Comment: 18pp, Concluding remarks section revised. Typos correcte

Towards a theory of local Shimura varieties

This is a survey article that advertizes the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also review their theory in the light of this idea. We also discuss conjectures on the $\ell$-adic cohomology of local Shimura varieties.Comment: 53 page

The supersingular locus of the Shimura variety for GU(1,n-1) over a ramified prime

We analyze the geometry of the supersingular locus of the reduction modulo p of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q, in the case that p is ramified. We define a stratification of this locus and show that its incidence complex is closely related to a certain Bruhat-Tits simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety associated to some symplectic group over F_p and some Coxeter element. The closure of each stratum is a normal projective variety with at most isolated singularities. The results are analogous to those of Vollaard/Wedhorn in the case when p is inert.Comment: A few more corrections, to appear in Math. Zeitschrif

Arithmetic diagonal cycles on unitary Shimura varieties

We define variants of PEL type of the Shimura varieties that appear in the context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.Comment: We removed all mention of an 'exotic level structure". Final version. To appear in Compositio Mathematic