Georgios Pappas

Local models in the ramified case. III unitary groups

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p -adic integral models of these Shimura varieties and study their etale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

Rapoport-Zink spaces for spinor groups

We develop a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of the canonical integral models of Shimura varieties of Hodge type at primes of good reduction. We then apply the general theory to the special case of Shimura varieties associated with groups of spinor similitudes, and, in the basic case, determine explicitly the reduced scheme underlying the Rapoport-Zink formal scheme.

On the supersingular locus of the GU(2,2) Shimura variety

We describe the supersingular locus of a GU(2,2) Shimura variety at a prime inert in the corresponding quadratic imaginary field.

On arithmetic class invariants

Let F be a number field with ring of integers OF , and let S be a finite set of places of F . Assume that S contains the set S∞ of archimedean places of F , and write Sf for the set of finite places contained in S. Let OS (or O, when there is no danger of confusion) denote the ring of Sf -integers of F . Write F for an algebraic closure of F . Let Y be any scheme over Spec(O). Suppose thatG is a finite, flat commutative group scheme over Y of exponent N , and let G denote the Cartier dual of G. Let π : X → Y be aG-torsor, and write π0 : G → Y for the trivialG-torsor. ThenOX is anOG-comodule, and so it is also anOGD -module (see [12]). As an OGD -module,OX is locally free of rank one, and it therefore gives a line bundle Mπ over G. Set Lπ := Mπ ⊗M−1 π0 . Then the map ψ : H (Y,G) → Pic(G) ; [π ] → [Lπ ]

Integral Grothendieck–Riemann–Roch theorem

We show that, in characteristic zero, the obvious integral version of the Grothendieck–Riemann–Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck’s strategy: we use resolution of singularities and the weak factorization theorem for birational maps.

Duality and Hermitian Galois Module Structure

Suppose

Galois Module Structure and the γ-Filtration

\mathcal{O}

Nearly Perfect Complexes and Galois Module Structure

is either the ring of integers of a number field, the ring of integers of a

The Group Logarithm Past and Present

p

ε-constants and equivariant Arakelov–Euler characteristics

-adic local field, or a field of characteristic