arXiv subjects
Publications and source records attributed to Yousheng Shi.
We use the idea of splitting models to define and study a semi-stable model for unitary Shimura varieties of signature $(n-1,1)$ with maximal parahoric level structure at ramified primes. In this case, the ``naive'' splitting model defined by Pappas and Rapoport fails to be flat in a crucial way. We prove that the genuine splitting model in this case is flat with semi-stable reduction.
We construct the Bruhat--Tits stratification of the reduced locus of the ramified unitary Rapoport--Zink space of signature $(n-1,1)$, with the level being the stabilizer of a vertex lattice. We develop the local model theory for Bruhat--Tits strata, proving their normality and Cohen--Macaulayness, and provide precise dimension formulas. Additionally, we establish an explicit scheme-theoretical isomorphism between Bruhat--Tits strata and Deligne--Lusztig varieties.
This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modulartiy result on $U(1, 1)$ Shimura curves.
We prove the Kudla--Rapoport conjecture for Krämer models of unitary Rapoport--Zink spaces at ramified places. It is a precise identity between arithmetic intersection numbers of special cycles on Krämer models and modified derived local densities of hermitian forms. As an application, we relax the local assumptions at ramified places in the arithmetic Siegel--Weil formula for unitary Shimura varieties, which is in particular applicable to unitary Shimura vartieties associated to unimodular hermitian lattices over imaginary quadratic fields.
In this paper, we proved a local arithmetic Siegel-Weil formula for a $U(1, 1)$-Shimura variety at a ramified prime, a.k.a. a Kudla-Rapoport conjecture at a ramified case. The formula needs to be modified from the original Kudla-Rapoport conjecture. In the process, we also gives an explicit decomposition of the special divisors of the Rapoport-Zink space of unitary type $(1, 1)$ (Krämer model). A key ingredient is to relate the Rapoport-Zink space to the Drinfeld upper plane.
In this paper, we propose a modified Kudla-Rapoport conjecture for the Krämer model of unitary Rapoport-Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla-Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies arithmetic Siegel-Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice.
In this paper, we study special cycles on the basic locus of certain unitary Shimura varieties over the ramified primes and their local analogues on the corresponding Rapoport-Zink spaces. We study the support and compute the dimension of these cycles.
We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.
We study generalized special cycles on Hermitian locally symmetric spaces $Γ\backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R}) $ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator representation and a construction which essentially comes from the thesis of Greg Anderson, we show that Poincaré duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermtian locally symmetric manifolds associated to $G$ to vector valued automorphic forms associated to the groups $G'=\mathrm{U}(m,m)$, $\mathrm{O}(m,m)$ or $\mathrm{Sp}(m,m)$ which forms a reductive dual pair with $G$ in the sense of Howe.
In this paper, we study special cycles on the Krämer model of $\mathrm{U}(1,1)(F/F_0)$-Rapoport-Zink spaces where $F/F_0$ is a ramified quadratic extension of $p$-adic number fields with the assumption that the $2$-dimensional hermitian space of special quasi-homomorphisms is anisotropic. We write down the decomposition of these special cycles and prove a version of Kudla-Rapoport conjecture in this case. We then apply the local results to compute the intersection numbers of special cycles on unitary Shimura curves and relate these intersection numbers to Fourier coefficients of central derivatives of certain Eisenstein series.