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 Upper bounds for geodesic periods over rank one locally symmetric spaces

2018 ◽  
Vol 30 (5) ◽  
pp. 1065-1077 ◽  
Author(s):  
Jan Frahm ◽  
Feng Su
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Locally Symmetric Spaces ◽  
Ambient Manifold ◽  
Rank One ◽  
General Rank ◽  
Laplace Eigenvalues ◽  
Periods Of Automorphic Forms

AbstractWe prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

scholarly journals Brauer equivalent number fields and the geometry of quaternionic Shimura varieties

2018 ◽  
Vol 70 (2) ◽  
pp. 675-687
Author(s):  
Benjamin Linowitz
Keyword(s):  
Symmetric Spaces ◽  
Number Fields ◽  
Shimura Varieties ◽  
Brauer Groups ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Equivalent Number ◽  
Geodesic Surfaces

Abstract Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper, we prove a variety of number theoretic results about Brauer equivalent number fields (for example, they must have the same signature). These results are then applied to the geometry of certain arithmetic locally symmetric spaces. As an example, we construct incommensurable arithmetic locally symmetric spaces containing exactly the same set of proper immersed totally geodesic surfaces.

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 Shimura varieties at level and Galois representations

2020 ◽  
Vol 156 (6) ◽  
pp. 1152-1230 ◽  
Author(s):  
Ana Caraiani ◽  
Daniel R. Gulotta ◽  
Chi-Yun Hsu ◽  
Christian Johansson ◽  
Lucia Mocz ◽  
...  
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Hecke Algebras ◽  
Derived Category ◽  
Shimura Varieties ◽  
Nilpotent Ideal ◽  
Locally Symmetric Spaces ◽  
Compactly Supported ◽  
Symmetric Varieties

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

scholarly journals Lp spectral theory and heat dynamics of locally symmetric spaces

2010 ◽  
Vol 258 (4) ◽  
pp. 1121-1139 ◽  
Author(s):  
Lizhen Ji ◽  
Andreas Weber
Keyword(s):  
Spectral Theory ◽  
Symmetric Spaces ◽  
Locally Symmetric Spaces
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