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ON THE GROWTH OF TORSION IN THE COHOMOLOGY OF ARITHMETIC GROUPS

Part of: Discontinuous groups and automorphic forms Computational number theory

Published online by Cambridge University Press:  21 March 2018

A. Ash ,
P. E. Gunnells ,
M. McConnell  and
D. Yasaki
A. Ash
Affiliation:
Boston College, Chestnut Hill, MA 02467, USA (Avner.Ash@bc.edu)
P. E. Gunnells
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305, USA (gunnells@math.umass.edu)
M. McConnell
Affiliation:
Princeton University, Princeton, NJ 08540, USA (markwm@princeton.edu)
D. Yasaki
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA (d_yasaki@uncg.edu)
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Abstract

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.

Keywords

cohomology of arithmetic groupsGalois representationstorsion in cohomology

MSC classification

Primary: 11F75: Cohomology of arithmetic groups
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F80: Galois representations 11Y99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 537 - 569
Copyright
© Cambridge University Press 2018

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Footnotes

The authors thank the Banff International Research Station and Wesleyan University, where some work was carried out on this paper. A. A. was partially supported by NSA grant H98230-09-1-0050. P. G. was partially supported by NSF grants DMS 1101640 and 1501832. D. Y. was partially supported by NSA grant H98230-15-1-0228. This manuscript is submitted for publication with the understanding that the United States government is authorized to produce and distribute reprints.

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