On the bad reduction of certain U(2, 1) Shimura varieties

Abstract

Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,\varphi ,\iota )$ comprised of an abelian threefold $A$ with an action $\iota$ of the ring of integers ${\mathcal{O}}_E$, and a principal polarization $\varphi$. The second surface, $S_0(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^2$. The third Picard surface, $\tilde{S}$, parametrizes triples $(A,\psi ,\iota )$ similar to those parametrized by $S$, but where $\psi$ is a polarization of degree $p^2$. We study the components, singularities and naturally defined stratifications of these surfaces, and their behaviour under the morphisms. A particular role is played by a foliation we define on the blow-up of $S$ at its superspecial points.

59 pages, 3 figures