Related papers: On The Hecke Orbit Conjecture for PEL Type Shimura…

We prove a $p$-adic analogue of the Andr\'{e}-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$.…

We generalize some of the results of Andreatta, Iovita, and Pilloni and the author to Hodge type Shimura varieties having non-empty ordinary locus. For any $p$-adic weight $\kappa$, we give a geometric definition of the space of…

We review the Newton stratification and Ekedahl-Oort stratification on the special fiber of a smooth integral model for a Shimura variety of Hodge type at a prime of good reduction. We show that the \mu-ordinary locus coincides with the…

Our main theorem describes the degree 0 cohomology of Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well known analogue…

We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of…

For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke…

In this paper we study the geometry of good reductions of Shimura varieties of abelian type. More precisely, we construct the Newton stratification, Ekedahl-Oort stratification, and central leaves on the special fiber of a Shimura variety…

We construct canonical non-vanishing global sections of powers of the Hodge bundle on each Ekedahl-Oort stratum of a Hodge type Shimura variety. In particular we recover the quasi-affineness of the Ekedahl-Oort strata. In the projective…

We establish quantitative strengthenings of Mazur's conjecture regarding the non-torsion property of higher Heegner points on modular and Shimura curves, confirming both a vertical version for sufficiently large powers $n$ and a horizontal…

We give a new definition -- and in some cases, a new construction -- of integral canonical models of Shimura varieties that uses the notion of an aperture appearing in work of Gardner--Madapusi on some conjectures of Drinfeld. This applies…

We study the Ekedahl-Oort stratification for good reductions of Shimura varieties of PEL type. These generalize the Ekedahl-Oort strata defined and studied by Oort for the moduli space of principally polarized abelian varieties (the "Siegel…

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of…

For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a…

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…

Systematic choice of the Hecke eigenforms of half-integral weight is an interesting problem in the theory of modular forms. In this paper, we find all Dedekind-eta products of half-integral weight which are Hecke eigenforms up to weight…

We define variants of PEL type of the Shimura varieties that appear in the context of the Arithmetic Gan-Gross-Prasad conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral…

We prove, assuming the Generalized Riemann Hypothesis for imaginary quadratic fields, that irreducible curves in the product of two modular curves that contain infinitely many complex multiplication points are either a Hecke correspondence…

We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of…

In \cite{Shimura}, Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm's bounds give…

Let $F$ be a totally real field in which a prime number $p>2$ is inert. We continue the study of the (generalized) Goren--Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb F_p$. We prove that, when the…