Automorphic vector bundles with global sections on -schemes

2018 ◽  
Vol 154 (12) ◽  
pp. 2586-2605 ◽  
Author(s):  
Wushi Goldring ◽  
Jean-Stefan Koskivirta
Keyword(s):  
Vector Bundles ◽  
Shimura Varieties ◽  
Surjective Morphism ◽  
General Conjecture ◽  
Picard Modular Surfaces

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

scholarly journals HECKE OPERATORS AND THE COHERENT COHOMOLOGY OF SHIMURA VARIETIES

2021 ◽  
pp. 1-69
Author(s):  
Najmuddin Fakhruddin ◽  
Vincent Pilloni
Keyword(s):  
Hecke Operators ◽  
Shimura Varieties ◽  
Automorphic Representations ◽  
General Conjecture ◽  
Satake Parameters

Abstract We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

scholarly journals Chern classes of automorphic vector bundles, II

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Hélène Esnault ◽  
Michel Harris
Keyword(s):  
Galois Group ◽  
Vector Bundles ◽  
Shimura Varieties ◽  
Chern Classes ◽  
Toroidal Compactifications ◽  
Canonical Extensions

We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the $p$-adic field above which the variety and the bundle are defined. Comment: 28 pages

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.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

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