Related papers: Categoricity of Shimura Varieties

We construct global cohomology classes in the middle degree cohomology of the Shimura variety of the symplectic group $GSp_6$ compatible when one varies the level at $p$. These classes are expected constituents of an Euler system for the…

We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita…

Let $(G, \Omega)$ be a Shimura datum of abelian type. It is well known that the corresponding Shimura variety $\mathrm{Sh}(G, \Omega)$ should be a moduli space of abelian motives equipped with some additional structures. In this…

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the the modularity of Kudla's generating series in the cohomology group.

In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over…

Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and $E_0$ be an imaginary quadratic field in which $p$ splits; put $E=E_0 F$. Let ${{\rm Sh}}_{1,n-1}$ be the special fiber over $\mathbb{F}_{p^2}$ of the Shimura…

We define and study a collection of special cycles on certain non-PEL Shimura varieties for $U(2,1) \times U(1,1)$ that appear naturally in the context of the recent conjectures of Gan, Gross and Prasad on restrictions of automorphic forms…

Shimura reciprocity law allows us to verify that a modular function is a class invariant. Here we present a new method based on Shimura reciprocity that allows us not only to verify but to find new class invariants from a modular function…

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective…

The authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by…

This is the English translation, done by Yihang Zhu, of the paper by Langlands and Rapoport, originally published in J. Reine Angew. Math. 378 (1987), pages 113-220. The translator also added some historical notes.

Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems…

For a Shimura variety $(G, X)$ in the superrigid regime and neat level subgroup $K_0$, we show that the canonical family of $\ell$-adic representations associated to a number field point $y \in \mathrm{Sh}_{K_0}(G, X)(F)$, \[ \left\{…

We study the characters induced by suitable level structures of abelian varieties with quaternionic multiplication following the methods of Mazur, Momose, who studied the characters induced by elliptic curves, and Arai--Momose, who studied…

We compute by a different method the generating series in Shimura's conjecture for Sp3, proved by Andrianov in 1967. We develop formulas for the Satake spherical maps for Sp_n and Gl_n.

We establish a representability criterion of $v$-sheaf theoretic modifications of formal schemes and apply this criterion to moduli spaces of parahoric level structures on local shtukas. In the proof, we introduce nice classes of…

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In…

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula involves a new class of generalized Littlewood-Richardson coefficients, all of which surprisingly seem to be…

Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory $T$ has enough structure, then the category $T\text{-}\mathbf{Mod}$ of its models carries the structure of a model…