Cohomology Of Arithmetic Groups Research Articles

Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups

The purpose of the workshop was to study derived generalizations of Mazur’s deformation ring of Galois representations, and the relationship of such a derived deformation ring to the homology of arithmetic groups.

Equivariant birational geometry and modular symbols

We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.

Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel–Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

A brief introduction to Quillen conjecture

Introduction: In 1971, Quillen stated a conjecture that on cohomology of arithmetic groups, a certain module structure over the Chern classes of the containing general linear group is free. Over time, many efforts has been dedicated into this conjecture. Some verified its correctness, some disproved it. So, the original Quillens conjecture is not correct. However, this conjecture still has great impacts on the field cohomology of group, especially on cohomology of arithmetic groups. This paper is meant to give a brief survey on Quillen conjecture and finally present a recent result that this conjecture has been verified by the authors.
 Method: In this work, we investigate the key reasons that makes Quillen conjecture fails. We review two of the reasons: 1) the injectivity of the restriction map; 2) the non-free of the image of the Quillen homomorphism. Those two reasons play important roles in the study of the correctness of Quillen conjecture.
 Results: In section 4, we present the cohomology of ring ​ which is isomorphic to the free module ​ over ​. This confirms the Quillen conjecture.
 Conclusion: The scope of the conjecture is not correct in Quillens original statement. It has been disproved in many examples and also been proved in many cases. Then determining the conjectures correct range of validity still in need. The result in section 4 is one of the confirmation of the validity of the conjecture.

Mini-Workshop: Computations in the Cohomology of Arithmetic Groups

Explicit calculations play an important role in the theoretical development of the cohomology of groups and its applications. It is becoming more common for such calculations to be derived with the aid of a computer. This mini-workshop assembled together experts on a diverse range of computational techniques relevant to calculations in the cohomology of arithmetic groups and applications in algebraic K -theory and number theory with a view to extending the scope of computer aided calculations in this area.

Lattices and Applications in Number Theory

This is a report on the workshop on Lattices and Applications in Number Theory held in Oberwolfach, from January 17 to January 23, 2016. The workshop brought together people working in various areas related to the field: classical geometry of numbers, packings, Diophantine approximation, Arakelov geometry, cohomology of arithmetic groups, algebraic modular forms and Hecke operators, algebraic topology. The meeting consisted of a few long talks which included an introductory part to each of the topics in the previous list, and a series of shorter talks mainly devoted to recent developments. The present report contains extended abstracts of all presentations.

Equivariant Euler–Poincaré characteristic in sheaf cohomology

Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports $${\Phi}$$ . Fix a number field F with ring of integers R. We study the class $${\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}$$ in the character group of G over F for any flat G-sheaf $${\mathcal{E}}$$ of R-modules over X. Under natural cohomological finiteness conditions we give a formula for $${\chi}$$ with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups.

A GLUING FORMULA FOR THE ANALYTIC TORSION ON HYPERBOLIC MANIFOLDS WITH CUSPS

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems, we define the Reidemeister torsion of the Borel–Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. Our main result is established via a gluing formula, and here our approach is heavily inspired by a recent paper of Lesch.

On the growth of torsion in the cohomology of arithmetic groups

In this paper we consider certain families of arithmetic subgroups of $$\mathrm{SO }^0(p,q)$$ and $$\mathrm{SL }_3(\mathbb {R})$$ , respectively. We study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules. We show that for natural sequences of such modules the torsion in the cohomology grows exponentially.

Eisenstein series, cohomology of arithmetic groups, and automorphic L-functions at half integral arguments

Abstract The automorphic cohomology of a reductive ℚ-group G, defined in terms of the automorphic spectrum of G, captures essential analytic aspects of the arithmetic subgroups of G and their cohomology. The subspace spanned by all possible residues and principal values of derivatives of Eisenstein series, attached to cuspidal automorphic forms π on the Levi factor of proper parabolic ℚ-subgroups of G, forms the Eisenstein cohomology which is a natural complement to the cuspidal cohomology. We show that non-trivial Eisenstein cohomology classes can only arise if the point of evaluation features a `half-integral' property. Consequently, only the analytic behavior of the automorphic L-functions at half-integral arguments matters whether an Eisenstein series attached to a globally generic π gives rise to a residual class or not.

On Eisenstein series and the cohomology of arithmetic groups

The automorphic cohomology of a reductive Q -group G captures essential analytic aspects of the arithmetic subgroups of G. The subspace spanned by all possible residues and principal values of derivatives of Eisenstein series, attached to cuspidal automorphic forms π on the Levi factor of proper parabolic Q -subgroups of G, forms the Eisenstein cohomology. We show that non-trivial classes can only arise if the point of evaluation features a “half-integral” property. Consequently, only the analytic behavior of the automorphic L-functions at half-integral arguments matters whether an Eisenstein series attached to a globally generic π gives rise to a residual class or not.

Erratum to: Cohomology of arithmetic groups with infinite dimensional coefficient spaces

A correction of an error in the proof of Lemma 5.3 is given.

Cohomology of arithmetic groups with infinite dimensional coefficient spaces

The cuspidal cohomology groups of arithmetic groups in certain infinite dimensional modules are computed. As a result we get a simultaneous generalization of the Patterson-Conjecture and the Lewis-Correspondence.

Smith theory and Hecke operators

Of the connections between the cohomology of arithmetic groups and representations of Galois groups, some are known and more are conjectured. This paper proves one of these conjectures for certain monomial Galois representations. Let p be a prime. We show that the representation of the absolute Galois group of Q induced from an F p × -valued ray class character of the cyclotomic field Q (ζ p ) is attached to a Hecke eigenclass in the mod p cohomology of a torsion-free congruence subgroup Γ ( M ) of GL(p−1, Z ) . The method involves constructing an action of the Hecke algebra on the fundamental exact sequence of Smith theory arising from the action by conjugation of an element of order p in GL(p−1, Z ) on Γ ( M ).

On the reductive Borel-Serre compactification: L p -cohomology of arithmetic groups (for large p )

The L 2 -cohomology of an arithmetic quotient of an Hermitian symmetric space (i.e., of a locally symmetric variety) is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the L p -cohomology of an arithmetic quotient, for p finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratified spaces.

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p 2, p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices.

Eisenstein series and cohomology of arithmetic groups: The generic case

Eisenstein series and cohomology of arithmetic groups: The generic case

Doubly cuspidal cohomology for principal congruence subgroups of 𝐺𝐿(3,𝑍)

The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the f-cuspidal cohomology—spanned by the classes that generate representations of the associated finite Lie group which are cuspidal in the sense of finite Lie group theory. Few concrete examples of f-cuspidal cohomology have been computed geometrically, outside the cases of rational rank 1, or where the symmetric space has a Hermitian structure. This paper presents new computations of the f-cuspidal cohomology of principal congruence subgroups Γ ( p ) \Gamma (p) of GL ( 3 , Z ) {\text {GL}}(3,\mathbb {Z}) of prime level p. We show that the f-cuspidal cohomology of Γ ( p ) \Gamma (p) vanishes for all p ⩽ 19 p \leqslant 19 with p ≠ 11 p \ne 11 , but that it is nonzero for p = 11 p = 11 . We give a precise description of the f-cuspidal cohomology for Γ ( 11 ) \Gamma (11) in terms of the f-cuspidal representations of the finite Lie group GL ( 3 , Z / 11 ) {\text {GL}}(3,\mathbb {Z}/11) . We obtained the result, ultimately, by proving that a certain large complex matrix M is rank-deficient. Computation with the SVD algorithm gave strong evidence that M was rank-deficient; but to prove it, we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields.

The Steinberg module and the Cohomology of arithmetic groups

Let G be a connected algebraic Q-group, S, the Steinberg representation of G= G(Q). Recall that SG may be realized on the reduced integral homology of the Tits building of parabolic Q-subgroups of G (see Section 1). In this article, we combine facts about SG with the Borel-Serre Duality Theorem (see (3.1)) and basic Lie theory to derive new results on the cohomology of arithmetic subgroups r of G. These are summarized below. Our sharpest results apply to H’(T, -) (v is the virtual cohomological dimension of r) where G is split over E. Some of our work generalizes that of Ash, Ash and Rudolph, and Lee and Szczarba on &5,(Z). See [Al, A-R, L-S, L-S1 1. In earlier papers on this topic, an important ingredient has been a simplicial complex Y of dimension equal to vcd S&(E) on which &5,(Z) acts cocompactly with finite cell stabilizers. The existence of such a Y for a general Chevalley group has not been verified. (There has been recent success with SP,(Z) in [M-M].) Roughly speaking, we avoid this issue by using SG as a substitute for the top chain group of Y. Assume for now the G is semisimple, split over H, and has no factor of We 4.

Non-square-integrable cohomology of arithmetic groups

Non-square-integrable cohomology of arithmetic groups