scholarly journals Lefschetz Properties of Subvarieties of Shimura Varieties

2002 ◽  
pp. 265-270
Author(s):  
T. N. Venkataramana
Keyword(s):  
Shimura Varieties ◽  
Lefschetz Properties

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

2021 ◽  
pp. 1-48
Author(s):  
Arvind N. Nair ◽  
Ankit Rai
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Hodge Theory ◽  
Hyperbolic Manifolds ◽  
Shimura Varieties ◽  
Restriction Maps ◽  
Locally Symmetric Spaces ◽  
Compact Congruence ◽  
Lefschetz Properties ◽  
Invent Math

Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

scholarly journals On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn 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\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.

scholarly journals Serre–Tate theory for Shimura varieties of Hodge type

2020 ◽  
Author(s):  
Ananth N. Shankar ◽  
Rong Zhou
Keyword(s):  
Shimura Varieties

scholarly journals RAPOPORT–ZINK UNIFORMIZATION OF HODGE-TYPE SHIMURA VARIETIES

2018 ◽  
Vol 6 ◽  
Author(s):  
WANSU KIM
Keyword(s):  
Shimura Varieties ◽  
Good Reduction ◽  
Canonical Models

We show that the integral canonical models of Hodge-type Shimura varieties at odd good reduction primes admits ‘$p$-adic uniformization’ by Rapoport–Zink spaces of Hodge type constructed in Kim [Forum Math. Sigma6(2018) e8, 110 MR 3812116].

Arithmetic divisors on orthogonal and unitary Shimura varieties

Astérisque ◽  
2020 ◽  
Vol 421 ◽  
Author(s):  
Jan H. Bruinier ◽  
Benjamin Howard ◽  
Stephen S. Kudla ◽  
Michael Rapoport ◽  
Tonghai Yang ◽  
...  
Keyword(s):  
Shimura Varieties
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