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 Equidistribution in supersingular Hecke orbits

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Arno Kret
Keyword(s):  
Rate Of Convergence ◽  
Hecke Operators ◽  
Shimura Varieties ◽  
Automorphic Representations ◽  
Ramanujan Conjecture ◽  
Cuspidal Automorphic Representations

AbstractWe prove that Hecke operators act with equidistribution on the basic stratum of certain Shimura varieties. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain cuspidal automorphic representations on

scholarly journals On the equality case of the Ramanujan Conjecture for Hilbert modular forms

2019 ◽  
Vol 15 (10) ◽  
pp. 2107-2114
Author(s):  
Liubomir Chiriac
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Complex Multiplication ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Automorphic Representations ◽  
Equality Case ◽  
Hilbert Modular ◽  
Ramanujan Conjecture ◽  
Satake Parameters

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations [Formula: see text] on [Formula: see text] posits that [Formula: see text]. We prove that this inequality is strict if [Formula: see text] is generated by a Hilbert modular form of weight two, with complex multiplication, and [Formula: see text] is a finite place of degree one. Equivalently, the Satake parameters of [Formula: see text] are necessarily distinct. We also give examples where the equality case does occur for places [Formula: see text] of degree two.

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 The standard sign conjecture on algebraic cycles: The case of Shimura varieties

2019 ◽  
Vol 2019 (748) ◽  
pp. 139-151 ◽  
Author(s):  
Sophie Morel ◽  
Junecue Suh
Keyword(s):  
Algebraic Cycles ◽  
Shimura Varieties ◽  
Automorphic Representations

Abstract We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur’s conjectures plus a bit more). We also indicate what is known (to us) about these statements.

scholarly journals An Euler system for GU(2, 1)

2021 ◽  
Author(s):  
David Loeffler ◽  
Christopher Skinner ◽  
Sarah Livia Zerbes
Keyword(s):  
Euler System ◽  
Quadratic Fields ◽  
Shimura Varieties ◽  
Imaginary Quadratic Fields ◽  
Automorphic Representations ◽  
Cuspidal Automorphic Representations

AbstractWe construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) .

scholarly journals Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires

2010 ◽  
Vol 146 (2) ◽  
pp. 367-403 ◽  
Author(s):  
Pascal Boyer
Keyword(s):  
Explicit Description ◽  
Shimura Varieties ◽  
Perverse Sheaves ◽  
Automorphic Representations ◽  
Local Systems ◽  
Langlands Correspondence ◽  
Princeton University ◽  
Vanishing Cycles ◽  
Local Langlands Correspondence ◽  
Invent Math

AbstractIn Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.

Arithmetic Compactifications of PEL-Type Shimura Varieties

2013 ◽  
Author(s):  
Kai-Wen Lan
Keyword(s):  
Graduate Students ◽  
Automorphic Forms ◽  
Abelian Varieties ◽  
Shimura Varieties ◽  
Modular Curves ◽  
Automorphic Representations ◽  
Langlands Program ◽  
Arithmetic Properties ◽  
Sufficient Detail ◽  
Recent Developments

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.

scholarly journals SUR LA TORSION DANS LA COHOMOLOGIE DES VARIÉTÉS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

2017 ◽  
Vol 18 (3) ◽  
pp. 499-517 ◽  
Author(s):  
Pascal Boyer
Keyword(s):  
Maximal Ideal ◽  
Cohomology Class ◽  
Hecke Algebra ◽  
Galois Representation ◽  
Local System ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Torsion Freeness ◽  
Satake Parameters ◽  
Torsion Free

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

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