scholarly journals ON THE GROWTH OF TORSION IN THE COHOMOLOGY OF ARITHMETIC GROUPS

2018 ◽  
Vol 19 (2) ◽  
pp. 537-569
Author(s):  
A. Ash ◽  
P. E. Gunnells ◽  
M. McConnell ◽  
D. Yasaki
Keyword(s):  
Symmetric Space ◽  
Arithmetic Group ◽  
Torsion Subgroup ◽  
Arithmetic Groups ◽  
Semisimple Lie Group ◽  
Congruence Subgroups ◽  
Cohomology Of Arithmetic Groups ◽  
Finite Dimensional ◽  
Rings Of Integers ◽  
Precise Expression

Let $G$ be a semisimple Lie group with associated symmetric space $D$, and let $\unicode[STIX]{x1D6E4}\subset G$ be a cocompact arithmetic group. Let $\mathscr{L}$ be a lattice inside a $\mathbb{Z}\unicode[STIX]{x1D6E4}$-module arising from a rational finite-dimensional complex representation of $G$. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup $H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$ as $\unicode[STIX]{x1D6E4}_{k}$ ranges over a tower of congruence subgroups of $\unicode[STIX]{x1D6E4}$. In particular, they conjectured that the ratio $\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$ should tend to a nonzero limit if and only if $i=(\dim (D)-1)/2$ and $G$ is a group of deficiency $1$. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including $\operatorname{GL}_{n}(\mathbb{Z})$ for $n=3,4,5$ and $\operatorname{GL}_{2}(\mathscr{O})$ for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.

scholarly journals DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS

2019 ◽  
Vol 7 ◽  
Author(s):  
AKSHAY VENKATESH
Keyword(s):  
Hecke Algebra ◽  
Galois Cohomology ◽  
Arithmetic Group ◽  
Arithmetic Groups ◽  
Cohomology Groups ◽  
Single Degree ◽  
Cohomology Of Arithmetic Groups ◽  
Favorable Conditions

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\unicode[STIX]{x1D6E4}$ . Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain $p$ -adic Galois cohomology groups on $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q}_{p})$ , and formulate the central conjecture: the motivic $\mathbf{Q}$ -lattice inside these Galois cohomology groups preserves $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$ .

Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions

2019 ◽  
Author(s):  
Anantharam Raghuram ◽  
Günter Harder
Keyword(s):  
Eisenstein Series ◽  
Symmetric Spaces ◽  
Constant Term ◽  
Local System ◽  
Arithmetic Groups ◽  
Special Values ◽  
Cohomology Of Arithmetic Groups ◽  
Finite Dimensional ◽  
Eisenstein Cohomology ◽  
Cohomological Interpretation

This book studies the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel–Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin–Selberg L-functions for GL(n) × GL(m), where n + m = N. The book carries through the entire program with an eye toward generalizations. The book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.

scholarly journals Random walks on projective spaces

2014 ◽  
Vol 150 (9) ◽  
pp. 1579-1606 ◽  
Author(s):  
Yves Benoist ◽  
Jean-François Quint
Keyword(s):  
Local Fields ◽  
Probability Measures ◽  
Stationary Probability ◽  
Semisimple Lie Group ◽  
Dimensional Representation ◽  
Dense Subgroup ◽  
Finite Dimensional Representation ◽  
Empirical Measures ◽  
Finite Dimensional ◽  
Strongly Irreducible

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a connected real semisimple Lie group, $V$ be a finite-dimensional representation of $G$ and $\mu $ be a probability measure on $G$ whose support spans a Zariski-dense subgroup. We prove that the set of ergodic $\mu $-stationary probability measures on the projective space $\mathbb{P}(V)$ is in one-to-one correspondence with the set of compact $G$-orbits in $\mathbb{P}(V)$. When $V$ is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic $\mu $-stationary measures on the flag variety of $G$.

Introduction to the Schwartz Space of T\G

1989 ◽  
Vol 41 (2) ◽  
pp. 285-320 ◽  
Author(s):  
W. Casselman
Keyword(s):  
Reductive Group ◽  
Rational Points ◽  
Schwartz Space ◽  
Parabolic Subgroups ◽  
Arithmetic Groups ◽  
Similar Question ◽  
Arithmetic Subgroup ◽  
Cohomology Of Arithmetic Groups ◽  
Image Position

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

APPROXIMATION OF -ANALYTIC TORSION FOR ARITHMETIC QUOTIENTS OF THE SYMMETRIC SPACE

2018 ◽  
Vol 19 (2) ◽  
pp. 307-350
Author(s):  
Jasmin Matz ◽  
Werner Müller
Keyword(s):  
Symmetric Space ◽  
Principal Congruence ◽  
Analytic Torsion ◽  
Congruence Subgroups ◽  
Arithmetic Subgroup ◽  
Limiting Behavior

In [31] we defined a regularized analytic torsion for quotients of the symmetric space $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behavior of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^{2}$-analytic torsion.

Mini-Workshop: Computations in the Cohomology of Arithmetic Groups

2017 ◽  
Vol 13 (4) ◽  
pp. 2941-2973
Author(s):  
Eva Bayer-Fluckiger ◽  
Philippe Elbaz-Vincent ◽  
Graham Ellis
Keyword(s):  
Arithmetic Groups ◽  
Cohomology Of Arithmetic Groups

ON FINITE GALOIS STABLE ARITHMETIC GROUPS AND THEIR APPLICATIONS

2008 ◽  
Vol 07 (06) ◽  
pp. 773-783
Author(s):  
EKATERINA S. KHREBTOVA ◽  
DMITRY MALININ
Keyword(s):  
Quantitative Result ◽  
Explicit Construction ◽  
Integral Representations ◽  
Arithmetic Groups ◽  
Group Schemes ◽  
Natural Operation ◽  
Rings Of Integers ◽  
Representations Of Finite Groups ◽  
Finiteness Theorems ◽  
Quadratic Lattices

We prove the existence and finiteness theorems for integral representations stable under Galois operation. An explicit construction of the realization fields for representations of finite groups stable under the natural operation of the Galois group is given. We also compare the representations over fields and the rings of integers, and give a quantitative result on the rarity of integral Galois stable representations. There is a series of related conjectures and applications to arithmetic algebraic geometry, finite flat group schemes, positive definite quadratic lattices and Galois cohomology.

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