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 $H^{\ast }(\unicode[STIX]{x1D6E4},\mathbf{Q})$ .

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.

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.

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

scholarly journals On Higher Special Elements of p-adic Representations

2020 ◽  
Author(s):  
David Burns ◽  
Takamichi Sano ◽  
Kwok-Wing Tsoi
Keyword(s):  
Galois Cohomology ◽  
Euler System ◽  
Cohomology Groups ◽  
Exterior Power ◽  
Concrete Application ◽  
Natural Generalisation ◽  
Higher Rank ◽  
Derivatives Of

Abstract As a natural generalisation of the notion of “higher rank Euler system”, we develop a theory of “higher special elements” in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet $L$-series.

scholarly journals On the Galois cohomology groups of CK/DK

1982 ◽  
Vol 8 (2) ◽  
pp. 407-415 ◽  
Author(s):  
Shin-ichi KATAYAMA
Keyword(s):  
Galois Cohomology ◽  
Cohomology Groups

Genus of abstract modular curves with level-ℓ structures

2019 ◽  
Vol 2019 (752) ◽  
pp. 25-61 ◽  
Author(s):  
Anna Cadoret ◽  
Akio Tamagawa
Keyword(s):  
Galois Cohomology ◽  
Fundamental Groups ◽  
Modular Curves ◽  
Cohomology Groups ◽  
Arbitrary Characteristic ◽  
Linear Representations ◽  
Etale Cohomology ◽  
Étale Cohomology

Abstract We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect {\mathbb{F}_{\ell}} -linear representations of étale fundamental groups of curves goes to infinity with {\ell} . This applies to the variation of the Galois image on étale cohomology groups with coefficients in {\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.

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