Shimura Varieties Research Articles

Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series

Let V be a rational quadratic space of signature (m;2) . A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with \mathrm{SO}(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m+1 . We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel–Weil formulas at the archimedian and non-archimedian places.

On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2

We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport–Zink space for the quaternionic unitary similitude group of degree 2, with a special parahoric level. We prove that such a scheme is purely 2-dimensional, and every irreducible component is isomorphic to the Fermat surface. We also determine its connected components, irreducible components and their intersection behaviors by means of the Bruhat–Tits building of \({{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)\). In addition, we compute the intersection multiplicity of the GGP cycles associated to an embedding of the considering Rapoport–Zink space into the Rapoport–Zink space for the unramified \({{\,\mathrm{GU}\,}}_{2,2}\) with hyperspecial level for the minuscule case.

On Shimura varieties for unitary groups

This is a largely expository article based on our previous work on arithmetic diagonal cycles on unitary Shimura varieties. We define a class of Shimura varieties closely related to unitary groups which represent a moduli problem of abelian varieties with additional structure, and which admit interesting algebraic cycles. We generalize to arbitrary signature type the results of our previous work valid under special signature conditions. We compare our Shimura varieties with other unitary Shimura varieties.

Hida theory over some unitary Shimura varieties without ordinary locus

We develop Hida theory for Shimura varieties of type A without ordinary locus. In particular we show that the dimension of the space of ordinary forms is bounded independently of the weight and that there is a module of $\Lambda$-adic cuspidal ordinary forms which is of finite type over $\Lambda$, where $\Lambda$ is a twisted Iwasawa algebra.

The Plectic Weight Filtration on Cohomology of Shimura Varieties and Partial Frobenius

Abstract We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel’s work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case.

P-adic families of automorphic forms in the µ-ordinary setting

We develop a theory of $p$-adic automorphic forms on unitary groups that allows $p$-adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$-adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the $\mu$-ordinary locus, which is open and dense. By eliminating the splitting condition on $p$, our framework should allow many results employing Hida's theory to extend to infinitely many more primes. We also provide a construction of $p$-adic families of automorphic forms that uses differential operators constructed in the paper. Our approach is to adapt the methods of Hida and Katz to the more general $\mu$-ordinary setting, while also building on papers of each author. Along the way, we encounter some unexpected challenges and subtleties that do not arise in the ordinary setting.

CM liftings of surfaces over finite fields and their applications to the Tate conjecture

AbstractWe give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of$K3$surfaces over finite fields. We prove that every$K3$surface of finite height over a finite field admits a characteristic$0$lifting whose generic fibre is a$K3$surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a$K3$surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a$K3$surface of finite height and construct characteristic$0$liftings of the$K3$surface preserving the action of tori in the algebraic group. We obtain these results for$K3$surfaces over finite fields of any characteristics, including those of characteristic$2$or$3$.

Some Remarks on Ekedahl–Oort Stratifications

We study independence of symplectic embeddings of the theory of Ekedahl–Oort stratifications on Shimura varieties of Hodge type, by comparing two different embeddings with a third one. The main results are as follows. 1. The Ekedahl–Oort stratification is independent of the choices of symplectic embeddings. 2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl–Oort stratifications with respect to morphisms of Shimura varieties.

Newton Polygon Stratification of the Torelli Locus in Unitary Shimura Varieties

AbstractWe study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).

The test function conjecture for parahoric local models

We prove the test function conjecture of Kottwitz and the first-named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

Generating series of a new class of orthogonal Shimura varieties

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.

Mixed Ax–Schanuel for the universal abelian varieties and some applications

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

Dimension formula for the affine Deligne–Lusztig variety $$X(\mu , b)$$

The study of certain union $$X(\mu , b)$$ of affine Deligne–Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit dimension formula for $$X(\mu , b)$$ associated to sufficiently large dominant coweight $$\mu $$ .

CLASS NUMBERS OF CM ALGEBRAIC TORI, CM ABELIAN VARIETIES AND COMPONENTS OF UNITARY SHIMURA VARIETIES

AbstractWe give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

Smoothness of Schubert varieties in twisted affine Grassmannians

We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of ${\rm Spin}_8$ ("ramified triality") provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction.

Mod p isogeny classes on Shimura varieties with parahoric level structure

We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. We also verify most of the He-Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are non-empty for these models.

On a conjecture of Pappas and Rapoport about the standard local model for GL_ d

Abstract In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n × n {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

Unlikely intersections with Hecke translates of a special subvariety

We prove some cases of the Zilber–Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber–Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed special subvariety, conditional on arithmetic conjectures. Secondly, we prove the conjecture unconditionally for intersections between a curve and the union of Hecke correspondences on the moduli space of principally polarised abelian varieties, subject to some technical hypotheses. This generalises results of Habegger and Pila on the Zilber–Pink conjecture for products of modular curves. The conditional proof uses the Pila–Zannier method, relying on a point-counting theorem of Habegger and Pila and a functional transcendence result of Gao. The unconditional results are deduced from this using a variety of arithmetic ingredients: the Masser–Wüstholz isogeny theorem, comparison between Faltings andWeil heights, a super-approximation theorem of Salehi Golsefidy, and a result on expansion and gonality due to Ellenberg, Hall and Kowalski.