Shimura Varieties Of Abelian Type Research Articles

GALOIS REPRESENTATIONS FOR EVEN GENERAL SPECIAL ORTHOGONAL GROUPS

Abstract We prove the existence of $\mathrm {GSpin}_{2n}$ -valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$ , arising from forms of ${\mathrm {GSO}}_{2n}$ . As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$ -valued Galois representations by removing the outer automorphism ambiguity.

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

Let S be a Shimura variety. We conjecture that the heights of special points in \(S(\overline{\mathbb {Q}})\) are discriminant negligible with respect to some Weil height function \(h:S(\overline{\mathbb {Q}})\rightarrow \mathbb {R}\). Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.

Stratifications in good reductions of Shimura varieties of abelian type

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 of abelian type at a good prime. We establish several basic properties of these stratifications, including the non-emptiness, closure relation and dimension formula, generalizing those previously known in the PEL and Hodge type cases. We also study the relations between these stratifications, both in general and in some special cases, such as those of fully Hodge-Newton decomposable type. We investigate the examples of quaternionic and orthogonal Shimura varieties in details.

EKOR strata for Shimura varieties with parahoric level structure

We study the geometry of reduction modulo p of the Kisin–Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR (Ekedahl–Kottwitz–Oort–Rapoport) strata on these Shimura varieties, using the theories of G-zips and mixed characteristic local G-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.

Good and semi-stable reductions of Shimura varieties

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

On Some Generalized Rapoport–Zink Spaces

AbstractWe enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

Complex conjugation and Shimura varieties

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending these to the maximal totally real field of the reflex field. We prove the existence of such descent for many Shimura varieties whose associated adjoint group has certain factors of type A or D. This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory.

Integral canonical models for automorphic vector bundles of abelian type

We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number fields with finitely many primes inverted for Shimura varieties of abelian type with hyperspecial level at all primes we do not invert, compatible with Kisin's construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these in hand, one immediately also gets integral models for automorphic vector bundles.

Mod $p$ points on Shimura varieties of abelian type

We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.

Galois-generic points on Shimura varieties

We discuss existence and abundance of Galois-generic points for adelic representations attached to Shimura varieties. First, we show that, for Shimura varieties of abelian type, l-Galois-generic points are Galois-generic; in particular, adelic representations attached to such Shimura varieties admit (“lots of”) closed Galois-generic points. Next, we investigate further the distribution of Galois-generic points and show the Andre–Pink conjecture for them: if S is a connected Shimura variety associated to a Q-simple reductive group, then every infinite subset of the generalized Hecke orbit of a Galois-generic point is Zariski-dense in S. Our proof follows the approach of Pink for Siegel Shimura varieties. Our main contribution consists in showing that there are only finitely many Hecke operators of bounded degree on (adelic and connected) Shimura varieties. Compared with other approaches of this result, our proof, which relies on Bruhat–Tits theory, is effective and works for arbitrary Shimura varieties.

Perfectoid Shimura Varieties of Abelian Type

We prove that Shimura varieties of abelian type with infinite level at p are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at p are also perfectoid.

About the mixed André–Oort conjecture: Reduction to a lower bound for the pure case

We prove that the mixed André–Oort conjecture holds for any mixed Shimura variety if a lower bound for the size of Galois orbits of special points in pure Shimura varieties exists. This generalizes the current results for mixed Shimura varieties of Abelian type.

2-ADIC INTEGRAL CANONICAL MODELS

We use Lau’s classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points

Abstract We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of 𝒜 6 n {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

Integral models for Shimura varieties of abelian type

We construct smooth integral models of Shimura varieties of abelian type at primes where the level structure is hyperspecial.

Automorphic vector bundles on connected Shimura varieties

Introduction 0. Review of terminology concerning Shimura varieties 1. The Taniyama group, the period torsor, and conjugates of Shimura varieties 2. The compact dual symmetric Hermitian space and its conjugates 3. The principal bundle Y~ X); statement of the first main theorem 4. Automorphic vector bundles 5. Conjugates of automorphic vector bundles 6. Proof of Theorem 3.10 for the symplectic group 7. Proof of Theorem 3.10 for connected Shimura varieties of abelian type 8. First completion of the proof of Theorem 3.10 9. Second completion of the proof of Theorem 3.10 Appendix: Pairs defining connected and nonconnected Shimura varieties Bibliography 91 93 95 101 105 112 115 118 121 123 124 126 128

The Action of Complex Conjugation on a Shimura Variety

In the case that a Shimura variety has a real canonical model, complex conjugation defines an involution of the set of complex points of the variety. It is necessary to have an explicit description of this involution in order, for example, to compute the zeta function of the variety. Langlands conjectured such a description in [1, p. 417-18] and our purpose is to prove his conjecture for all Shimura varieties of abelian type. (This class of Shimura varieties is defined in S 1 of this paper; it contains all those whose canonical model is known, at the time of writing, to exist.) Recall that a Shimura variety Sh(G, X) is defined by a Q-rational reductive group G and a family X of homomorphisms COx -> G(R) satisfying certain conditions. Initially Sh (G, X) is defined as a complex variety but is expected to have a model over a certain number field E(G, X) called the reflex field. A canonical model for Sh(G, X) is a variety M(G, X) over E(G, X) satisfying certain conditions sufficient to determine it uniquely. Assume that E(G, X) is real and that the canonical model exists so that complex conjugation defines an involution 0 of Sh(G, X). When the canonical model is a moduli variety, and so has a direct description, the proof of the conjecture is straightforward. This is not usually the case, and just as the construction of the canonical model is intricate in general, so must be the proof of the conjecture. In particular, it must involve an analogous assertion for connected Shimura varieties. Such an assertion is proved in Shih [1] for connected Shimura varieties that are of primitive abelian type C, and this result is the starting point of our proof of the conjecture for all Shimura varieties of abelian type. (Since a does not preserve the connected component the analogous assertion takes on quite a different form from the original; it becomes rather a statement about the action of a negative element of G(Q).) To see how our result relates to the zeta function, consider an arbitrary