arXiv subjects
Publications and source records attributed to Ehud De Shalit.
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the μ-ordinary locus, when applied to scalar-valued forms.
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliation of height 1 and rank $m^2$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but these singularities are resolved if we pass to a natural smooth moduli problem $S^\sharp$, a successive blow-up of $S$. Over the ($μ$-)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient of $S^\sharp$ by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fibre $S_0(p)$ of a certain Shimura variety with parahoric level structure at $p$. As a result we get that this "horizontal component" of $S_0(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen-Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$, and a certain Ekedahl-Oort stratum that we denote $S_{fol}$. We conjecture that these are the only integral submanifolds.
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 $(A,ϕ,ι)$ comprised of an abelian threefold $A$ with an action $ι$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $ϕ$. 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, $\widetilde{S}$, parametrizes triples $(A,ψ,ι)$ similar to those parametrized by $S$, but where $ψ$ 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.