Hodge Structures of Shimura Varieties Attached to the Unit Groups of Quaternion Algebras

2018 ◽  
Author(s):  
Takayuki Oda
Keyword(s):  
Shimura Varieties ◽  
Quaternion Algebras ◽  
Hodge Structures

Shimura Varieties of Weight Two Hodge Structures

1987 ◽  
pp. 1-15 ◽  
Author(s):  
James A. Carlson ◽  
Carlos Simpson
Keyword(s):  
Shimura Varieties ◽  
Hodge Structures

scholarly journals Degeneration of Hodge structures over Picard modular surfaces

2017 ◽  
Vol 13 (05) ◽  
pp. 1145-1164 ◽  
Author(s):  
Giuseppe Ancona
Keyword(s):  
Modular Forms ◽  
Shimura Varieties ◽  
Manuscripta Math ◽  
Modular Surface ◽  
Hodge Structures ◽  
Picard Modular Forms ◽  
Picard Modular Surfaces

We study variations of Hodge structures over a Picard modular surface, and compute the weights and types of their degenerations through the cusps of the Baily–Borel compactification. These computations are one of the key inputs which allow Wildeshaus [On the interior motive of certain Shimura varieties: the case of Picard surfaces, Manuscripta Math. 148(3) (2015) 351–377] to construct motives associated with Picard modular forms.

scholarly journals Ghost classes in $${\mathbb {Q}}$$-rank two orthogonal Shimura varieties

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1209-1233 ◽  
Author(s):  
Jitendra Bajpai ◽  
Matias V. Moya Giusti
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Irreducible Representations ◽  
Shimura Varieties ◽  
Hodge Structures ◽  
Highest Weight ◽  
Weight Property ◽  
Eisenstein Cohomology ◽  
Cohomology Spaces ◽  
Mixed Hodge Structures

Abstract In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature $$(1, n-1)$$ ( 1 , n - 1 ) . For the values of $$n = 4, 5$$ n = 4 , 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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 Quaternion Algebras

2021 ◽  
Author(s):  
John Voight
Keyword(s):  
Quaternion Algebras
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