scholarly journals ENDOSCOPY AND COHOMOLOGY IN A TOWER OF CONGRUENCE MANIFOLDS FOR

2019 ◽  
Vol 7 ◽  
Author(s):  
SIMON MARSHALL ◽  
SUG WOO SHIN
Keyword(s):  
Symmetric Spaces ◽  
Unitary Groups ◽  
Automorphic Representations ◽  
Locally Symmetric Spaces ◽  
Hermitian Forms ◽  
Work In Progress ◽  
Endoscopic Classification

By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to$U(n,1)$. In the case of lattices arising from Hermitian forms, we expect that the growth exponents we obtain are sharp in all degrees.

scholarly journals RAISING THE LEVEL OF AUTOMORPHIC REPRESENTATIONS OF OF UNITARY TYPE

2021 ◽  
pp. 1-24
Author(s):  
Christos Anastassiades ◽  
Jack A. Thorne
Keyword(s):  
Unitary Groups ◽  
Automorphic Representations ◽  
Endoscopic Classification

Abstract We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.

scholarly journals On the image of complex conjugation in certain Galois representations

2016 ◽  
Vol 152 (7) ◽  
pp. 1476-1488 ◽  
Author(s):  
Ana Caraiani ◽  
Bao V. Le Hung
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Real Field ◽  
Complex Conjugation ◽  
Automorphic Representations ◽  
Locally Symmetric Spaces ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Totally Real Field

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

On the cohomology of uniform arithmetically defined subgroups in SU*(2n)

2011 ◽  
Vol 151 (3) ◽  
pp. 421-440 ◽  
Author(s):  
JOACHIM SCHWERMER ◽  
CHRISTOPH WALDNER
Keyword(s):  
Lie Group ◽  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Close Relation ◽  
Unitary Representations ◽  
Geometric Constructions ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Irreducible Unitary Representations

AbstractWe study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.

scholarly journals Classification of invariant AHS-structures on semisimple locally symmetric spaces

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Jan Gregorovič
Keyword(s):  
Symmetric Spaces ◽  
Equivalence Classes ◽  
Locally Symmetric Spaces ◽  
Local Symmetries

AbstractWe discuss which semisimple locally symmetric spaces admit an AHS-structure invariant under local symmetries. We classify them for all types of AHS-structures and determine possible equivalence classes of such AHS-structures.

scholarly journals On some recent progress in the classification of $$(P\hbox { and }Q)$$-polynomial association schemes

2019 ◽  
Author(s):  
Alexander L. Gavrilyuk ◽  
Jack H. Koolen
Keyword(s):  
Symmetric Spaces ◽  
Recent Progress ◽  
Association Schemes ◽  
Compact Symmetric Spaces ◽  
Expository Paper

AbstractThe problem of classification of $$(P\hbox { and }Q)$$(PandQ)-polynomial association schemes, as a finite analogue of E. Cartan’s classification of compact symmetric spaces, was posed in the monograph “Association schemes” by E. Bannai and T. Ito in the early 1980s. In this expository paper, we report on some recent results towards its solution.

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