Related papers: Special cycles on unitary Shimura curves at ramifi…
In this paper, we compute the cohomology sheaves of the $\ell$-adic nearby cycles on the local model of the PEL $\mathrm{GU}(n-1,1)$ Shimura variety over a ramified prime, with level given by the stabilizer of a self-dual lattice. This…
We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally…
We consider Shimura varieties associated to a unitary group of signature $(n-s,s)$ where $n$ is even. For these varieties, by using the spin splitting models from Zachos-Zhao, we construct flat, Cohen-Macaulay, and normal $p$-adic integral…
In this work, we study the supersingular locus of the Shimura variety associated to the unitary group $\mathrm{GU}(2,4)$ over a ramified prime. We show that the associated Rapoport-Zink space is flat, and we give an explicit description of…
We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the…
In this article, for nilpotent orbits of ramified quasi-split unitary groups with two Jordan blocks, we give closed formulas for their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. These…
We study some automorphic cohomology classes of degree one on the Griffiths-Schmid varieties attached to some unitary groups in 3 variables. Using partial compactifications of those varieties, constructed by K. Kato and S. Usui, we define…
We compute the arithmetic volumes of integral models of unitary Shimura curves. This establishes the base case of an inductive argument to compute the arithmetic volumes of unitary Shimura varieties of higher dimension, to appear in…
We give a description of the $\mathrm{GL}_4$ Rapoport-Zink space, including the connected components, irreducible components, intersection behavior of the irreducible components, and Ekedahl-Oort stratification. As an application of this,…
Around 2000 Kudla presented conjectures about deep relations between arithmetic intersection theory, Eisenstein series and their derivatives, and special values of Rankin $L-$series. The aim of this text is to work out the details of an old…
We construct universal $G$-zips on good reductions of the Pappas-Rapoport splitting models for PEL-type Shimura varieties. We study the induced Ekedahl-Oort stratification, which sheds new light on the mod $p$ geometry of splitting models.…
On an orthogonal Shimura variety, one has a collection of special cycles in the Gillet-Soule arithmetic Chow group. We describe how these cycles behave under pullback to an embedded orthogonal Shimura variety of lower dimension. The bulk of…
Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a…
Let K be a complete field of unequal characteristics $(0,p)$. The aim of this paper is to describe the the semi-stable models for covers $\bold P^1_K@>>>\bold P^1_K$ of degree p, unramified outside $r\leq p$ points and totally ramified…
Special birational transformations $\Phi:\p^r\da Z$ defined by quadric hypersurfaces are studied by means of the variety of lines $\mathcal L_z\subset\p^{r-1}$ passing through a general point $z\in Z$. Classification results are obtained…
We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We…
We study Shimura curves of PEL type in the space of polarised abelian varieties $A^{\delta}_p$ generically contained in the ramified Prym locus. We generalise to ramified double covers, the construction done in [10] in the unramified case…
In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke…
In various contexts, the zeta function of an object splits into a product of $L$-functions. We categorify this product formula for quadratic covers of objects in the following contexts: quadratic extensions of number fields, ramified double…
We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy…